Improved Tomographic Refractive Index Profile Evaluation of Asymmetric Glass Fiber Preforms and the Fibers Themselves

Abstract

To provide a method for determining a refractive index profile (RIP) of a preform when the RIP is substantially asymmetrical.SOLUTION: In the method: (i) the preform is scanned, starting with a first projection angle, and raw data are created representing an object through measured data; (ii) optionally, the object is rotated, and the step (i) is 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 a step (v); (iv) the object is rotated, and the 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; and (viii) finally, refractive index steps / gradients and dimensions are determined.SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] The present disclosure generally relates to refractive index measurement, and more particularly, to a method for measuring the refractive index profile of a transparent cylindrical object such as a fiber preform.

Background Art

[0002] Transparent cylindrical objects such as fiber preforms, optical fibers, light pipes, and optical tubes are used in various optical applications. In many cases, it is desirable to know the refractive index profile (RIP) of such an object. For example, an optical fiber is formed by heating a fiber preform and drawing the molten end into a thin glass thread. The RIP of the preform defines the RIP of the resulting optical fiber, and accordingly, the waveguide characteristics of the optical fiber are determined. Therefore, it is important to be able to accurately measure the RIP of the fiber preform.

[0003] There are various methods for determining the radial RIP of a cylindrical optical object, particularly a preform for an optical fiber. A cylindrical optical object often has a cylindrical longitudinal axis, and at least one layer k having a layer radius r k and a layer refractive index n k extends symmetrically and uniformly in the radial direction around the axis. Of course, there are also many objects having a non-radial RIP, an asymmetric RIP, or a gradient RIP, and the present disclosure focuses on objects having an asymmetric RIP. A deflection angle distribution ψ(ρ,θ) is measured, and the RIP is reconstructed from the deflection angle distribution (ρ is the scanning position and θ is the measured projection angle). The RIP is also known as the refractive index distribution and is represented by the symbol n(x,y), where x and y are Cartesian coordinates.

[0004] The Cartesian coordinate system uniquely specifies each point within a three-dimensional space by three Cartesian numerical coordinates (x, y, z), where the three Cartesian numerical coordinates are signed distances from a point to three fixed, mutually perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or simply an axis of the system, and the point where they intersect is its origin, which is typically the ordered triplet (0, 0, 0). Coordinates can also be defined as the positions of the perpendicular projections of the point onto the three axes, expressed as signed distances from the origin.

[0005] Unfortunately, the refractive index distribution cannot be measured directly. The refractive index distribution is usually determined indirectly as the deflection or interference of a light beam passing through the volume region of an optical element, and the stepwise transmission of the light beam through the optical element is called "scanning" and is performed in the direction of ρ. The spatial refractive index distribution within the optical element can be inferred from the interference or deflection of the output light beam (output beam) based on the beam direction at the beam incidence point (incident beam). The set of deflection angles ψ measured during the scanning of a light beam for a single fixed projection angle θ in the direction (ρ direction) across the longitudinal axis forms a deflection angle distribution ψ(ρ, θ) (where θ is fixed).

[0006] For better visualization and illustration, the geometric relationship is schematically shown in FIG. 1. An object with a uniform refractive index n1 surrounded by a refractive index adjusting fluid having a refractive index n0 such that n0 < n1 is shown. The radius of the object is r1. The deflection angle ψ is defined as the angle between the output beam and the incident beam, and y is defined as the distance between the cylindrical longitudinal axis and the incidence point of the incident beam. When scanning the object, the beam refracts as soon as it contacts the object and is directed towards the center of the object. The shortest distance of the beam to the center of the object is given by the radius r * which appears in the formulation of subsequent numerical integration (often called the inverse Abel transform in the literature).

[0007] For an axially symmetric object having a refractive index distribution that is completely step-shaped, the deflection angle distribution ψ(y) can be mathematically described with reference to the following equation (1).

[0008] [Mathematics]

[0009] In Equation (1), m is the number of layers of the object, n0 is the refractive index of the surrounding medium, n k is the refractive index of the k-th layer, and r k is the radius of the k-th layer. A known mathematical method for calculating the refractive index profile from the deflection angle distribution based on measurement data according to Equation (1) is based on the well-known numerical integration (referred to as the "inverse Abel transform" in the literature) calculated from the following Equation (2).

[0010] [Mathematics] Here, r is the shortest distance from the longitudinal axis of the cylinder of the object to the beam path, that is, defined by Equation (3),

[0011] [Mathematics] R is the reference point of the refractive index distribution, that is, the radial position of the reference refractive index (atmosphere, refractive index adjustment fluid, or reference glass plate surrounding the object). By applying the partial derivative of ψ from the position Δt, the formulation takes the form of the well-known inverse Abel transform in mathematics. In principle, the inverse Abel transform can be applied not only to the ideal step refractive index profile as reflected in Equation (1), but also to any type of deflection angle distribution of axisymmetric RIP.

[0012] U.S. Patent No. 4,227,806 describes a method for non-destructively determining the parameters of an optical fiber preform. The preform is scanned by a laser beam incident laterally on a core-clad structure, the deflection angle of the output beam is measured, and then the refractive index distribution is compared with the theoretical or empirical deflection angle distribution of a preform with a known refractive index distribution. During the measurement, the preform is placed in a tank containing a refractive index adjustment fluid to prevent the deflection angle from becoming too large.

[0013] U.S. Patent No. 4,441,811 describes a method and apparatus for determining the refractive index distribution of a cylindrical transparent optical preform. Also in this case, the preform inserted into the refractive index adjusting fluid is scanned by a laterally incident light beam extending perpendicular to the optical axis. The light beam is deflected by the glass of the preform and imaged by an optical device onto a positionable detector. The refractive index profile is calculated from the deflection angle distribution using numerical integration. Other preform parameters such as the preform diameter, core diameter, eccentricity, and CCDR value (clad-to-core diameter ratio) can also be determined from the reconstructed RIP.

[0014] Tomography is imaging by sectioning or slicing using any kind of transmission wave (such as light waves). It is well known to use computer tomography (CT) to determine the RIP of an optical fiber preform. See, for example, Y. Zhao et al., "Nondestructive Measurement for Arbitrary RIP Distribution of Optical Fiber Preforms", Journal of Lightwave Technology, Vol. 22, No. 2, pp. 478-486 (2004), Y. Zhao et al., "Nondestructive Measurement of Refractive Index Profile for Holey Fiber Preforms", Optics Express, Vol. 11, No. 20, pp. 2474-2479 (2003), Y. Zhao et al., "Tomographic Reconstruction for Arbitrary Refractive Index Distribution of Optical Fiber Preforms", Symposium on Optical Fiber Measurements, at 51-54 (2004), and A. Novozamsky, "Tomography Reconstruction of Geometry and Refractive Index Profile of Highly Asymmetric Optical Fiber Preforms", Proc. SPIE, Vol. 7746, 77461O-1 - 77461O-6 (2010).

[0015] A method of reconstructing RIP from a laterally measured deflection angle distribution by using an inverse Abel transform can also be found in U.S. Patent Nos. 4,744,654, 5,078,488, and 4,515,475. The following technical papers describe methods using an inverse Radon transform: Michael R. Hutsel and Thomas K. Gaylord, "Concurrent three-dimensional characterization of the refractive-index and residual-stress distributions in optical fibers", Applied Optics, Vol. 51, No. 22, pp. 5442-52 (2012), and S. Fleming et al., "Nondestructive Measurement for Arbitrary RIP Distribution of Optical Fiber Preforms", Journal of Lightwave Technology, Vol. 22, No. 2, pp. 478-86 (2004).

[0016] P. Chu et al., "Nondestructive Determination of Refractive Index Profile of an Optical Fiber: Fast Fourier Transform Method", Applied Optics, Vol. 182, No. 7, pp. 1117-22 (1979) (incorporated herein by reference) presents another method that utilizes a fringe pattern. This method uses a fast Fourier transform (FFT) to solve the inverse Abel transform and convert the path length to a refractive index profile. The advantage of the FFT method is that it can process calculations quickly.

[0017] However, the difference from the actual RIP due to the simple reconstruction of the RIP or the refractive index profile n(r) (in the symmetric case) or n(x,y) (in the asymmetric case) from the laterally measured deflection angle distribution using either an inverse Abel transform or an inverse Radon transform is not negligible. The reason for this error is a known measurement artifact that occurs at the refractive index discontinuity at the boundary between a transparent object and the environment or at the boundary between radial refractive index steps. Measurements made at a boundary where the refractive index jumps from a low refractive index to a high refractive index in the volume region near the boundary of an optical object (when viewed from the outside to the inside) result in a region that cannot be illuminated or characterized in principle. The occurrence of a non-measurable region caused by the measurement method in the case of an upward refractive index jump is typical for an optical fiber where the refractive index of the core is relatively higher than that of the inner cladding layer.

[0018] The error source is illustrated in the schematic of Figure 2 by referring to a simple case, i.e., a rod with a uniform refractive index distribution n1, which is inserted into a refractive index adjusting fluid (also called a refractive index matching fluid or immersion fluid) having a refractive index n0, where n0 is less than n1. During the scanning of the rod, a beam that collides tangentially at the point of incidence is refracted towards the center of the rod and exits the rod as an outgoing beam with different propagation directions, resulting in a beam path as shown in Figure 2. Thus, there exists an area or region within the object where the light beam cannot pass tangentially (i.e., is never irradiated by the beam during the measurement). This area is shown hatched in Figure 2 and is indicated by the radius r * and the angle β (beta), where the angle β is 90 degrees - ψ / 2. As a result, it is impossible to measure the deflection angle in the region r * < r < r1, and due to this measurement error, it becomes clear that the reconstructed refractive index value is lower than the actual refractive index.

[0019] For example, typical differences and errors in the reconstruction RIP of a stepped refractive index profile are round and too small step heights of the profile. The publication "Index profile reconstruction of fiber preforms form data containing a surface refraction component" by Werner J. Glantschnig, Applied Optics, Volume 29, Number 19, pages 2899 - 2907 (1990) (the "Glantschnig publication" incorporated herein by reference) is one of the publications that addresses the problems caused by unmeasurable regions. Glantschnig suggests that the actually missing deflection angles in unmeasurable regions can be inferred by interpolation based on three measurement points inside the deflection angle distribution immediately before the discontinuity.

[0020] Interpolation based on three measurement points does not always yield good results. Therefore, some of the methods for measuring RIP cannot provide accurate measurements of the RIP of a simple homogeneous rod. One reason for this drawback is the presence of refractive index discontinuities at the boundaries or edges of the rod. The Glantschnig publication explains why refractive index discontinuities are not accurately reconstructed from the deflection function data. The Glantschnig publication proposes a method for measuring RIP, but requires precise measurement of the deflection angle at the edge of the object, which is so difficult as to be infeasible.

[0021] To solve these problems, U.S. Patent No. 8,013,985 issued to Corning Incorporated (incorporated herein by reference) proposes a modification to this reconstruction method. To measure the RIP of a transparent cylindrical object such as a fiber preform, a beam deflection angle function is measured. An estimated RIP representing the actual RIP is fitted to the measured deflection angle distribution by a numerical model. In the measurement, the fiber preform to be measured is placed between a laser and a conversion lens. The preform has a central axis and a cylindrical surface defining a preform radius R. An incident beam impinging on the cylindrical surface at height x is deflected within the preform and exits at another angle as an exit beam, which is detected by a photodetector and processed by a controller. The deflection angle is defined as the angle between the exit beam and the incident beam and is changed by varying the height x of the laser beam, and the deflection angle distribution is measured.

[0022] For this purpose, a symmetric correlation is completed for the measured deflection function to define the central coordinates. The measured deflection function is bisected about the central coordinates, and a refractive index half-profile is obtained for each of the bisected portions, resulting in an estimated refractive index profile for each half. The parameters relevant to the RIP calculation are the radius r of the preform k and the refractive index n of the layer k which is. The target angle distribution ψ t is iteratively fitted to the measured deflection function, and measurement points close to the boundary (refractive index discontinuity) are omitted within or on the edge of the preform. This arithmetic iterative fitting method of a mathematical function can be called "fitting".

[0023] According to U.S. Patent No. 8,013,985, fitting is performed by inserting into the equation the above equation (1) (without considering the arccosine part shown in the second line of the equation) the unknown parameters of the RIP, i.e., the value of the radius R of the preform (or the value of the radius of the refractive index discontinuity), and the unknown refractive index values, and the unknown parameters are such that the resulting target angle distribution ψ t is the measured deflection angle distribution ψm It is changed to best match the m . Therefore, the target angular distribution is fitted to the measured deflection angle distribution using unknown parameters.

[0024] Based on the thus-fitted and simulated target angular distribution, a reconstructed refractive index profile is derived. This profile extends to a reconstructed preform radius R that is larger than the radius of the inner object region. * In the case of a cylindrical object where the RIP has at least one discontinuity, the method is applied to the various object regions defined by the discontinuities respectively.

[0025] In this method, the simulated target angular distribution ψ t is fitted with unknown parameters to the measured deflection angle distribution ψ m and a radial refractive index distribution that can extend to the boundaries of the discontinuities located further outside the refractive index profile is derived from the simulated target angular distribution.

[0026] Therefore, the detection of the complete RIP of an optical object having several layers radially separated by refractive index discontinuities requires the continuous measurement, calculation, and estimation of the layers defined by each discontinuity from the outside to the inside. Systematic errors and numerical errors can result in the fitting of the simulated target angular distribution. Furthermore, the comparison of the deflection angle distributions, i.e., the comparison between the simulated and the measured ones, is not very illustrative and requires advanced expertise to determine whether the fitting is optimal and, optionally, how it is optimal, or whether post-correction or further modification is required, and, optionally, which values require post-correction or further modification.

[0027] Similar to the '985 patent, U.S. Patent No. 10,508,973, issued to the assignee of the present application (Heraeus Quarzglas GmbH & Co. KG of Hanau, Germany) and incorporated herein by reference, discloses a fitting procedure for an axially symmetric preform. The '973 patent teaches a method for determining the refractive index profile of a cylindrical optical object, particularly a preform for an optical fiber. The method includes: (a) adjusting a measured deflection angle distribution, including determining the extrema of the deflection angle distribution, to obtain an adjusted deflection angle distribution; (b) converting the adjusted deflection angle distribution to an adjusted refractive index profile; (c) evaluating the adjusted refractive index profile for the orientation values of the layer radius and layer refractive index of a virtual refractive index profile; (d) generating a simulated deflection angle distribution based on the virtual refractive index profile having the orientation values and converting the deflection angle distribution to a simulated refractive index profile; (e) fitting the simulated refractive index profile to the adjusted refractive index profile by iterative fitting of parameters to obtain a fitted simulated refractive index profile defined by the fitted parameters; and (f) obtaining the refractive index profile as a virtual refractive index profile having the fitted parameters. The method represents the state of the art with respect to determining the RIP when the RIP is substantially or completely step-index.

[0028] The latest methods result in at least one of two errors. The first type of error is a RIP measurement artifact. Since all reconstructed profiles (including tomographic reconstruction two-dimensional or 2D RIP) have typical profiling measurement artifacts (underevaluated refractive index steps, distorted profiles, rounded edges, etc.), accurate evaluation is not possible. The second type of error is a systematic error due to asymmetry. Ignoring the asymmetry of a nearly symmetric preform by simply fitting a symmetric distribution (which can usually handle measurement artifacts in the case of symmetry) results in a large systematic error.

[0029] Therefore, there is still a need for a method of determining the RIP of a cylindrical transparent object that does not have an axially symmetric or nearly axially symmetric refractive index distribution. The RIP may or may not be substantially or completely stepped. There is also a need for a method of minimizing measurement artifact errors. Of course, this method must also be improved with respect to validity, accuracy, reliability, and reproducibility. The related need is for a method of improving the assembly of preforms and meeting the requirements of preform customers, especially for preforms of increasingly complex designs. SUMMARY OF THE INVENTION

[0030] To meet these and other needs, and in view of that object, the present disclosure provides a method for determining the refractive index profile of an object, such as a preform, especially when the RIP is not substantially symmetric. The method includes the following steps. (a) Providing an object that includes an optical object having a longitudinal axis, with at least one layer extending outwardly about the longitudinal axis and at least one layer lacking complete rotational symmetry in cross-section; (b) Scanning the object with a light beam starting from a first projection angle and creating raw data representing the object through the measurement data; (c) Optionally, rotating the object and repeatedly iterating step (b) until all projection angles are scanned and all measurement data are created; (d) Processing the measurement data to form a sinogram and proceeding to step (f) if optional step (c) is completed; (e) Rotating the object and repeatedly iterating steps (b) and (d) until all projection angles are scanned; (f) Calculating a 2D RIP; (g) Selecting a target line section within the 2D RIP; (h) Applying a fitting procedure to the target line section; (i) Determining the refractive index step / gradient and dimensions.

[0031] It should be understood that both the foregoing general description and the following detailed description are exemplary of the present disclosure and not restrictive. **Brief Description of the Drawings**

[0032] The present disclosure is best understood from the following detailed description when read in conjunction with the accompanying drawings. It is emphasized that, in accordance with common practice, the various features of the drawings are not to scale. On the contrary, the dimensions of the various features are arbitrarily enlarged or reduced for clarity. The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawings will be provided by the Patent Office upon request and payment of the necessary fee. The drawings include the following figures.

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DETAILED DESCRIPTION OF THE INVENTION

[0033] In this specification and the following claims, reference is made to several terms, which are defined as having the following meanings attributed to them. Terms such as "include", "includes", "including", "have", "has", "having", "comprise", "comprises", "comprising", etc. mean including but not limited to, that is, they are inclusive and not exclusive. The term "substantially" as used in this specification is a descriptive term indicating approximation, meaning "to a considerable degree" or "not the whole but most of the specified thing", and is intended to avoid a strict numerical boundary for the specified parameter.

[0034] The term "about" (or "approximately") means that the quantity, size, formulation, parameter, and other quantities and characteristics are not exact and need not be exact, but may be approximate and / or larger or smaller as desired, reflecting tolerances, conversion factors, rounding, measurement errors, etc., and other factors known to those skilled in the art. If a value is described as being approximately a particular number or approximately equal to a particular number, the value is within ±10% of that number. For example, a value of about 10 refers to a value from 9 to 11 (including 9 and 11). When the term "about" is used to describe the endpoint of a value or range, the present disclosure should be understood to include the particular value or endpoint. Whether or not the numerical or range endpoints in this specification are described as "about", the numerical or range endpoints are intended to include two embodiments: those modified by "about" and those not modified by "about". It will be further understood that each endpoint of the range is significant both in relation to and independent of the other endpoint.

[0035] The term "about" or "substantially" refers further to all terms within the scope, unless specifically stated otherwise. For example, about 1, 2, or 3 is equivalent to about 1, about 2, or about 3, and further includes about 1 - 3, about 1 - 2, and about 2 - 3. The compositions, components, ingredients, additives, and similar aspects, as well as the specific values and preferred values disclosed for their ranges, are for illustrative purposes only and do not exclude other defined values or other values within the defined range. The compositions and methods of the present disclosure include any value or any combination of values, those having specific values, more specific values, and the described preferred values.

[0036] The indefinite articles "a" or "an" and their corresponding definite article "the" used in the present disclosure mean at least one, or one or more, unless otherwise specified. The terms indicating directions used in the present disclosure (e.g., up, down, right, left, front, back, top, bottom) are used only with respect to the drawn figures and the coordinate axes given to those figures and are not intended to imply an absolute direction.

[0037] It is well known to use computed tomography (CT) to determine the refractive index profile (RIP) of an optical fiber preform. The variables applicable to determining the RIP using CT are defined by the figures provided in Figure 3. An object with a refractive index distribution n(x, y) surrounded by a refractive index adjusting fluid having a refractive index n0 such that n0 < n(x, y) is shown. The Cartesian coordinates x and y belong to the physical object (e.g., a glass body). When scanning the object, the beam refracts as soon as it contacts the object and is directed towards the center of the object along the path indicated by the arrow. The deflection angle ψ is defined as the angle between the outgoing beam and the incoming beam.

[0038] During CT, the object is rotated. The mechanism for holding and rotating the sample (typically a rotating stage) is not shown in Figure 4 for simplicity. However, tomography requires an accurate method of rotating the sample. Manual rotation is possible but not practical. An electric rotating stage with an encoder is more suitable.

[0039] Rotation of the object by an angle θ, called the projection angle, gives the Cartesian axes x’ and y’. The scanning procedure is performed in the direction of y’ along the scanning position ρ. At a given fixed projection angle, such a scan gives a measured deflection angle ψ(ρ, θ = fixed). It is necessary to measure the deflection angle distribution ψ(ρ, θ) for a large set of projection angles θ in order to determine the exact RIP. Thus, the deflection angle distribution ψ(ρ, θ) is measured and the RIP is reconstructed from the deflection angle distribution (ρ is the scanning position and θ is the measured projection angle).

[0040] The method will be described below with reference to a transparent cylindrical object in the form of a fiber preform. However, those skilled in the art will understand that the described method can be generally applied to any cylindrical object having a refractive index in the radiation of a given wavelength, the corresponding deflection angle distribution can be measured through lateral transmission of radiation of a certain wavelength, and there exists a target deflection angle distribution function that can be represented as a function that can fit the measurement data. Referring now to the drawings, like reference numerals refer to like elements throughout the various figures constituting the drawings, and FIG. 4 is a schematic diagram showing an exemplary embodiment of a basic deflection angle distribution measurement system 100 that can be used to establish a measured deflection angle distribution function. The deflection angle is calculated by the formula: ψ = arctan(projection on the camera / distance to the camera).

[0041] 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 "optical beam") and provide one or more laser beams to one or more beam combiner fibers 2, and the beam combiner fiber 2 delivers one or more laser beams to a beam adjuster 3. (In optical fiber communication, a single-mode optical fiber is an optical fiber designed to carry only single-mode light, that is, light in only the transverse mode. Standard G.652 defines the most widely used form of single-mode optical fiber). One or more beam combiner fibers 2 are either a Y-shaped fiber with two inputs and one output or an X-shaped fiber with crosstalk in the splice region. The beam adjuster 3 is aligned along the first optical axis 15 and changes the characteristics of the laser beam. The beam emitted by the beam adjuster 3 travels along the first optical axis 15 and enters an off-axis parabolic reflector having a pair of mirrors 4 and an aperture (or "pinhole") 5. The first mirror 4 generates a parallel beam that passes through the aperture 5. The second mirror 4 focuses these beams along the second optical axis 15.

[0042] The focused beam reflected from the second mirror 4 of the off-axis parabolic reflector passes through the optical shutter 6 and enters the measurement cell 9. As long as the control voltage to the shutter 6 remains high, the shutter 6 remains open. However, as soon as the voltage drops, the shutter 6 closes, providing a built-in "fail-safe" operation, that is, security.

[0043] The measurement cell 9 is planar and has opposing side surfaces that are perpendicular to the second optical axis 15. Inside the measurement cell 9, typically, in the form of an optical fiber preform having a core with a higher refractive index surrounded by at least one cladding layer with a lower refractive index, a transparent cylindrical object 11 having a uniform or step-type RIP to be measured is disposed. A refractive index adjusting fluid 10 surrounds the object 11 within the measurement cell 9. In an exemplary embodiment, the refractive index adjusting fluid 10 is an oil having a refractive index close to but not the same as that of the object 11. However, for simplicity, the refractive index of the refractive index adjusting fluid 10 matches that of the measurement cell 9, and as a result, no additional deflection occurs at the entry and exit points in the illustrated embodiment.

[0044] As an alternative to the widely used paraffin RI matching oil, a glycerol-water mixture can also be used. The mixing ratio allows for a relatively easy and accurate adjustment or regulation of the RI to a desired RI level within the range of 1.333 (water RI) to 1.48 (glycerol RI).

[0045] The laser beam enters the measurement cell 9 and first impinges on the object 11 at the first edge of the object 11, undergoing a first refraction. The first refracted laser beam then proceeds through the object 11 and exits the object 11 from the opposite edge where the laser beam undergoes a second refraction and exits the object 11. The deflection angle, identified by the Greek letter Ψ (psi), is defined by the angle of the exiting laser beam with respect to the incident laser beam. The exiting 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 prevent ambient light from adversely affecting the measurement. A suitable photodetector unit includes a line-scanning camera 13 having an optically active sensor 14. (It is also possible to use alternatives to the line-scanning camera. Among other suitable cameras are 2D cameras and time-delay integration or "TDI" line-scanning cameras. The TDI line-scanning camera is somewhat hybrid and, for example, a 2D sensor having 16,384×256 pixels, but its architecture and sensor are extended one-dimensionally like a line-scanning camera.) The photodetector unit then transmits the corresponding detector signal to the controller 16 via a data connection 17 for processing.

[0046] The measurement cell 9 is attached to a linear stage 7 configured to support the measurement cell 9 and move the measurement cell 9 in a movement direction 8 (e.g., vertically as shown in FIG. 4). By performing measurements of the deflection angle over a range of heights of the laser beam with respect to the central horizontal axis of the measurement cell 9 and the object 11 within the measurement cell 9, the corresponding detector signals received and processed by the controller 16 generate the measured deflection angle, as will be described in more detail below. In other words, the movement of the linear stage 7 enables the height of the laser beam to be varied with respect to the measurement cell 9 and the object 11 such that the measured deflection function encompasses the radius range of the object 11. Of course, it is also possible to move the camera with respect to the laser beam so that the camera remains aligned with the laser beam, and vice versa.

[0047] In the embodiment shown in FIG. 4, the object 11 within the measurement cell 9 is moved (e.g., scanned), while in another embodiment, all other components (such as laser diodes 1a, 1b, 1c and the photodetector unit) can be simultaneously moved (e.g., scanned) relative to the object 11 (which is held in a stationary state) so that the height of the laser beam can be changed to send the laser beam through different parts of the object 11.

[0048] The controller 16 is, for example, a computer including a processor unit (e.g., a CPU), a memory unit, and support circuits, all of which are operably interconnected. The processor may be any form of general-purpose computer processor that can be used in an industrial environment, or may include such a general-purpose computer processor. The memory unit includes a computer-readable medium capable of storing instructions (e.g., software) that direct the processor to execute the methods described in detail below. The memory unit may be, for example, random access memory, read-only memory, a floppy or hard disk drive, or other forms of digital storage devices. In an exemplary embodiment, the instructions stored in the memory unit, when executed by the processor, are in the form of software that converts the processor into a special-purpose processor that controls (i.e., directs or causes to execute) the system 100 to execute one or more of the methods described below. The support circuits are operably (e.g., electrically) coupled to the processor and may include a cache, a clock circuit, an input / output subsystem, a power supply, control circuits, and the like.

[0049] The laser diodes 1a, 1b, 1c, the shutter 6, the linear stage 7, and the line scan camera 13 are each configured to transmit 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, and any conventional data connection 17 known to those skilled in the art is suitable.

[0050] To obtain the deflection function, many other deflection function measurement systems similar to those described above can be used. Regardless of the system used, several methods are possible for determining the RIP of the cylindrical optical object 11. The deflection angle is calculated by the formula: ψ = arctan(projection onto the camera / distance to the camera).

[0051] As described above, CT is based on the well-known inverse Radon transform. This transform can be written as in Equation (4) when applied to the RIP field.

[0052]

Equation

[0053] In contrast to interference RI profiling, the phase shift diagram η(ρ,θ) is typically not directly measurable. Instead, the deflection function ψ(ρ,θ) is accessible and can be transformed to the phase shift diagram η(ρ,θ) by the approximation given in Equation (5).

[0054]

Equation

[0055] In tomography, typically, the measurement data is preprocessed to improve the quality of reconstruction. Such preprocessing also applies in the field of RI profiling where the importance of an accurately defined deflection function origin is well known. Regardless of how precisely all the mechanical parts and components of the measurement system 100 are manufactured and assembled, it is always beneficial to numerically fine-tune the origin. For the evaluation of a radially symmetric profile, there are different approaches to achieve a so-called prepared deflection function, which is a measured deflection function ψ prepared (y)=ψ measured (y - y shift ) - ψ Offset as follows. The two parameters ψ Offset and y shift are very small but typically non-zero for any particular measurement system 100. Some methods for defining these two parameters for a radially symmetric RIP are described in U.S. Patent No. 10,508,973.

[0056] In an asymmetric RIP, the deflection function for a fixed projection angle θ can be complex. Such complexity reduces the possibility of achieving a prepared deflection function ψ prepared / (ρ,θ)=ψ measured / (ρ - ρ shift ) - ψ Offset . However, generally in tomography, there are many techniques for improving the quality of the reconstructed image. Most of the research and algorithms for improving the quality of the reconstructed image focus on X-ray and NMR-CT, but some of the methods are applicable in the field of tomographic RIP. For simplicity, only a few examples of improving the quality of the reconstructed image are given in the following disclosure, and those skilled in the art can choose and select additional or alternative methods.

[0057] For each measured deflection angle distribution ψ(ρ,θ), ρ shiftProperly defining this can be understood as a well-known radius runout correction in tomography. Radius runout occurs when there is an orbital movement instead of just a rotation along the Z-axis that changes the projection angle θ. Mathematically, this adds a shift for each projection angle θ. Tomographic reconstruction images affected by radius runout typically exhibit blurred edges, unclear features, or transitions. As a result, radius runout correction can be achieved by minimizing the entropy of the reconstructed image. In this case, it is necessary to vary ρ shift for each single projection and calculate the reconstruction. This iterative process requires a lot of time-consuming reconstruction calculations and is not very efficient. Another preferred approach is described below.

[0058] In most cases, one or more features within the measured sample can be used for radius runout correction. Those skilled in the art can identify appropriate features according to the following criteria. The feature should have a sharp RI step (preferably with an RI difference of at least 10 -3 over several micrometers), so that any edge detection algorithm functions well. Also, it is preferable that the feature defines or 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 the outer RI variation. In practice, the measured value for each projection angle θ is shifted only by the individual ρ shift , and as a result, the features are aligned concentrically around the origin regardless of how chaotic the interior may appear. As shown in the sinogram included in the drawing, this means that each column is shifted vertically so that the feature pairs are aligned. Figure 17 shows, for example, that a circular feature appears as a horizontal line of a straight line in the sinogram

Number

[0059] ρ shiftThe measured deflection function ψ measured In addition to the proper alignment of measured , the quality of the reconstructed 2D RIP can also be improved by properly defining ψ Offset for each measured projection. As shown in FIG. 4, the measurement system 100 uses two beam perpendicular planes and a surrounding reference material with a constant refractive index, so the phase shifts η resulting from passing through the references on both sides (ρ = 0 and ρ = ρ ref ) must be equal. This constraint can be used to fix ψ Offset for each measured projection. In particular, the deflection function offset ψ Offset is changed by integrating over the reference materials on both sides (by setting the upper integration limit ρ in Equation (5) to the reference position ρ ref ) so that equal phase shifts of η(ρ = 0) = η(ρ = ρ ref ) are obtained. By properly defining ρ shift and ψ Offset , the prepared deflection function ψ prepared (ρ, θ), and thus the calculation of the prepared phase shift diagram η prepared (ρ, θ) becomes possible.

[0060] In addition to RIP-specific mathematical tools for improving the quality of reconstruction, more general CT tools also exist. The number of tools is not limited to the following given example regarding sinogram interpolation. The inverse Radon transform has an integration over the scanning position ρ and the projection angle θ. Ideally, both the step sizes dρ and dθ are very small. However, the finer mesh size defined by dρ and dθ increases the overall measurement time required, so there is a trade-off.

[0061] In addition to reducing the mesh size by using smaller step sizes dρ, dθ, or both dρ and dθ, it is possible to interpolate within the sinogram. Interpolation of additional projection angles θ yields particularly excellent results with respect to improving the reconstruction quality. The reason is that the sinogram typically does not vary much in the direction of dθ on a small scale. In contrast, discontinuities typically exist in the direction of dρ. Starting from one or several additional interpolated projection angles θ between the measured projections, the number of projections in the prepared sinogram can reach 10 times, 20 times, or even more than the number of measured projections. Many different methods can be applied to two-dimensional interpolation, including, but not limited to, nearest neighbor interpolation, bilinear interpolation, cubic interpolation, and spline interpolation.

[0062] Frequency domain filters need to be used to apply the inverse Radon transform to the prepared phase shift diagram η(ρ,θ). Different types of filters are suitable for image reconstruction. Among such filters are the Ram-Lak filter, the Shepp-Logan filter, the Cosine filter, the Hamming window filter, and the Hann window filter. The Ram-Lak filter is also known as the Ramp filter. Since this filter is sensitive to noise, it is multiplied by an appropriate window to improve the results. The 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 filter and the Hann filter are obtained by multiplying the Ram-Lak filter by the Hamming window and the Hann window, respectively. The Ram-Lak filter and the Shepp-Logan filter are high-pass filters that keep the edge information intact. The Cosine filter, the Hamming filter, and the Hann filter are band-pass filters. They are used to smooth the image and remove extra edges from the image.

[0063] In modern CT, there are many methods for improving the quality of reconstruction, which can also be applied here by those skilled in the art. However, such improvements always depend on the specific object under investigation, and a particular method may be beneficial for a particular object, simply result in additional computation time, or even worsen the results. In the case of large deflections due to large refractive index steps within the object, it may be beneficial to apply backpropagation instead of direct backprojection. However, this iterative procedure requires many resources and results in longer computation time.

[0064] The measurement of the RIP of an optical fiber preform is important for the inspection of the quality of an optical fiber waveguide and also for the specification of additional processing (e.g., overcoating or etching) of the preform before fiber drawing. In the case of preforms, non-destructive techniques are particularly useful. The method disclosed in this document is based on the measurement of the deflection of a laterally directed light beam scanned through the preform. The method further is based on the numerical processing of a set of measured deflection functions at many angular projections. The method disclosed above calculates the RIP further based on the assumption that the preform is circular. However, a method for tomographic reconstruction of the RIP for preforms with highly asymmetric cross-sections is desired.

[0065] Attempts have been made to address the RI characteristics of asymmetric 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, pp. 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, pp. 316-27 (2005). These two references are incorporated herein by reference.

[0066] The following disclosure describes steps of an improved method for providing an accurate RIP of an object that is not symmetric or at least not substantially symmetric. Such a preform may be a preform having an unintentional asymmetry. Further, this method can be used for multi-core preforms, Panda preforms (used to form a common style of polarization-maintaining fiber having circularly symmetric stress rods on both sides of the core to induce polarization), or preforms for fiber lasers having non-circular interfaces.

[0067] An improved method for evaluating an asymmetric glass fiber preform has two main stages. The first stage involves the determination of the tomographic refractive index. The second stage involves the correction of measurement artifacts.

[0068] In the first stage of the method, the deflection function is measured for different orientations (a single scan provides unhelpful information about an asymmetric preform). The number of measurements to be made must balance the data accuracy (more measurements are desired to give a higher resolution) against the time required to make the measurements (fewer measurements are quicker). For example, system 100 might take about 18 minutes to scan a preform every 2° using one light source, or about 3.6 hours to scan a preform every 0.5° using three light sources (a scan at 0.5° intervals means 720 measurements are made over the entire 360° around the preform). From the measurements, a deflection angle distribution ψ(ρ, θ) is obtained. FIG. 5A is a graph of an exemplary, simplified deflection angle distribution ψ(ρ, θ) (in degrees) plotted against position (in millimeters) along the ρ-axis for a given projection angle (θ = 8°). FIG. 5B is a graph highlighting the central portion of the graph of FIG. 5A using a reduced scale along both the horizontal and vertical axes.

[0069] The measurements are then subjected to a tomographic evaluation once they are made. Such an evaluation includes steps of calculating a phase shift and stacking to form a sinogram. A sinogram provides a convenient way to represent the complete set of data acquired during the scan and is useful for checking for systematic errors during data acquisition or for verifying that a preprocessing step (e.g., radius runout correction) is functioning correctly.

[0070] As shown in FIG. 6, the Radon transform is an integral transform that transforms a function f defined on the x-y plane into a function Rf (i.e., the Radon transform) defined on the two-dimensional (θ, s) space of lines in the plane, and its value at a particular line is equal to the line integral of the function over that line. The parameter ρ is the length of a portion of the straight line A - A', s is the distance of the line from the origin, and θ is the angle that the normal or perpendicular to the line makes with the x-axis. This transform was introduced by Johann Radon in 1917 and he also provided the formula for the inverse transform. Radon further included formulas for the transform in three dimensions where the integration is performed over the plane (the integration over lines is known as the X-ray transform). This has since been generalized to higher-dimensional Euclidean spaces and more broadly in the context of integral geometry. The inverse Radon transform is widely applicable to tomography for creating images from projection data associated with cross-sectional scans of an object.

[0071] When the function f represents an unknown, the inverse Radon transform represents the projection data obtained as the output of a tomographic scan. Thus, the inverse of the Radon transform can be used to reconstruct the original quantity from the projection data and thus forms the mathematical basis for tomographic reconstruction, also known as iterative reconstruction. Since the inverse Radon transform of a point source off-center is a sine wave, the depiction of inverse Radon transform data is often called a sinogram. As a result, the inverse Radon transform of a number of scans appears graphically as a number of sine waves with different amplitudes and phases called a sinogram.

[0072] FIG. 7 is a sinogram obtained as a result of calculating a phase shift and stacking the measurement data shown in FIG. 5A. The phase shift is expressed in arbitrary units (abbreviated as a.u.). An arbitrary unit is a relative unit of measurement for indicating the ratio of an amount of a quantity to a predetermined reference measurement value. "Predetermined" means determined in advance, and thus a predetermined characteristic must be determined prior to some event, i.e., either selected or at least known. FIG. 8 shows a plot of the relative refractive index in the x-y plane (i.e., for the cross-section of the preform) after applying the inverse Radon transform. The refractive index is shown as a relative value based on the refractive index of the surrounding reference glass plate (n0 = 1.4587).

[0073] Accordingly, the refractive index distribution is determined using known techniques (inverse Radon transform is preferred). These steps complete the first "tomographic" stage of an improved method for evaluating an asymmetric glass fiber preform. Unfortunately, the results show typical measurement artifacts such as underestimated refractive index steps, distorted profiles, and rounded edges.

[0074] Measurement artifacts are corrected in the second stage of the method. Artifact correction is achieved by applying a method such as that of the '973 reference to selected line sections of the 2D RIP obtained from the tomographic data (i.e., the "starting RIP"). The "modeled RIP" is created after assuming a symmetric preform based on several parameters (which will be changed). The "distorted RIP" is calculated from the modeled RIP by (i) simulating the deflection angle distribution of the modeled RIP and (ii) calculating the distorted RIP from the simulated distribution. The parameters are changed with respect to the modeled RIP until the distorted RIP approaches the starting RIP. This approach is an approximation. The correction calculation is performed as if the starting RIP were measured on a preform with rotational symmetry, which is not the case. The important underlying assumption is that the measurement artifacts for the actual sample are similar to those for a rotationally symmetric preform.

[0075] The starting point of the improved evaluation method is 2D-RIP based on tomographic measurements and reconstructions (including all CT techniques available today for improved imaging). One or more line sections of the area of interest (core, side pit, edge, etc.) are selected. For each of the selected line sections, an abstract, fictitious but symmetric preform is "fitted" to the selected line section. For the sake of a simplified explanation, it is assumed that an abstract, fictitious symmetric profile can be directly fitted. However, since there is no analytical representation for such RIPs (including typical measurement artifacts), the profile cannot be directly fitted. Therefore, the fitting procedure includes the following steps for each iteration: (i) assuming a symmetric preform based on some parameters (to be changed), (ii) simulating the angular deviation distribution, (iii) reconstructing the symmetric RIP, and (iv) calculating the deviation of the selected cross-section. By this measurement and evaluation procedure, the tomographic 2D RIP reconstruction first addresses all asymmetries (so that no systematic errors occur), and the fitting procedure for the line sections takes care of the rest, in particular, dealing with typical measurement artifacts in refractive index profiling.

[0076] FIG. 9 is a flowchart showing the steps of a particular embodiment of an improved method 500 that provides an accurate refractive index profile of an object that does not have a symmetric cross-section or at least does not have a substantially symmetric cross-section. Method 500 determines the radial RIP of a cylindrical optical object 11 having a cylindrical longitudinal axis, and at least one layer extends axially asymmetrically about the cylindrical longitudinal axis. In step 501 of method 500, the deflection function is measured starting from a projection angle (theta) 0. In step 502, object 11 is scanned. After completion of step 502, it is possible to proceed directly to step 504 of method 500.

[0077] Optionally, but not necessarily, the original image can be created from a scan. In optional step 503, one or more of various techniques can be used to improve the quality of the original image. For example, a scanning camera has an "image ROI" function that enables the operator to specify the portion of the sensor array used for image acquisition. ROI is the acronym for "region of interest". When an image ROI is specified, the camera transmits only pixel data from within that region. In most cameras, this specification significantly increases the maximum frame rate of the camera. The ROI can be brightened. Another technique can be applied to collect several images with different exposures.

[0078] As another example of a technique for improving the quality of the original image, a scanning camera with high dynamic range (or HDR) can be used. In short, HDR helps to produce "better" pixels. The range of luminance levels has been expanded by HDR, thereby enabling enhanced discrimination between white and black. The range of colors has also been expanded, so colors can also be made more detailed.

[0079] As yet another example of a technique for improving the quality of the original image, filtering algorithms can be used to remove OVD diffraction. Outside vapor deposition (OVD) is a process in which glass is deposited in layers to produce a porous deposit (glass soot). The porous glass deposit can be cleaned, for example, by flowing chlorine gas (dehydrogenation) to reduce the OH percentage. At the same time, the porous deposit becomes a glass preform (vitrification). Despite further processing steps such as vitrification, the micro-layer structure associated with the process may remain and may also affect RI variations at the microscale. These very frequent RI variations result in diffraction when scanned with a laser and even a non-coherent light beam.

[0080] Higher diffraction orders degrade the accuracy of the determination of the deflection angle up to a very severe case where the 0th order beam cannot be distinguished from higher orders. The high-frequency RI micro-layer structure caused by the OVD manufacturing process is typically aperiodic and curved, so the diffraction pattern can be very irregular up to the point where higher diffraction orders can become brighter than the diffraction order from the 0th order beam. It is well known to those skilled in the art that only the evaluation of the 0th order beam yields a useful deflection angle distribution and a proper RIP reconstruction. Depending on the level of severity of the OVD diffraction (i.e., the intensity of the light within the higher order diffraction orders), additional preprocessing by applying an OVD filter algorithm may be required.

[0081] After step 502, which includes scanning the object 11 with a light beam (starting from a first projection angle) and creating raw data representing the object through the measurement data, method 500 can proceed along one of two alternative paths. The first alternative path includes rotating the object 11 by a small angle (theta), such as π / 180, and repeatedly iterating step 502 until all projection angles within the full range (0 to 2π) are scanned and all measurement data are acquired. Next, the method processes and stacks the measurement data to create a phase-shifted sinogram and proceeds to step 508.

[0082] The second alternative path after step 502 is shown in FIG. 9. In step 504 of method 500, the deflection function and the phase shift diagram are calculated (as described above). It is also possible to directly calculate the phase shift diagram without an intermediate step of calculating the deflection function (i.e., the intermediate step is optional). The phase shift diagram is exported and stacked as described above to form a sinogram in step 505. Next, method 500 asks in step 506 whether all projections within the full range (0 to 2π) have been measured. If the full range has not yet been measured, method 500 performs step 507. In step 507, object 11 is rotated by a small angle (theta), such as π / 180, and method 500 returns to step 502 to scan object 11. Steps 502, 503 (optional), 504, 505, 506, and 507 are repeatedly iterated until all projections within the full range (0 to 2π) have been measured, and method 500 can proceed to step 508.

[0083] In optional step 508, the quality of the sinogram can be improved using one or more of various techniques. As an example, radius runout can be corrected. As another example, the sinogram can be interpolated to a fine mesh.

[0084] In step 509, the 2D RIP is calculated from the sinogram by applying the inverse Radon transform. This calculation has been described above. Step 509 completes the first stage of method 500.

[0085] Step 510 begins the second stage of method 500 and asks whether there are any significant RIP measurement artifacts present in the 2D RIP. Measurement artifacts occur when the refractive index step transitions from a low value to a high value (in the direction from the outside to the inside). This transition can be relatively easily determined for each selected line segment. The term "significant" can refer to, for example, the sharpness of the RI step (i.e., the sharper the step, the more significant the artifact). If the step is a gradient and not steep, as in the case of a germanium-doped core with a parabolic-shaped RI, then there are no artifacts or they can be ignored.

[0086] If the RIP measurement artifacts are not significant, method 500 performs step 520. In step 520, before method 500 ends at step 580, a specific geometry and refractive index step are determined. If significant RIP measurement artifacts are present in the 2D RIP, method 500 proceeds to step 530. In step 530, a target line segment (also called the target region or ROI) is selected within the 2D RIP. Next, in step 540, a fitting procedure is applied to the selected line segment. Further details about the fitting procedure are provided in connection with FIG. 10 below.

[0087] When the fitting procedure applied in step 540 meets one or more stopping criteria, method 500 proceeds to step 560. The stopping criteria can be determined by comparing the result of the fitting procedure to a predetermined accuracy level for the selected line segment of the object 11 being evaluated. The predetermined accuracy level can depend on the specific 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 the most preferred predetermined accuracy is about 99% or more.

[0088] In step 560, fitting parameters such as RI step, geometric shape, gradient, and dimensions are exported. Next, method 500 asks, in step 570, whether all target regions within the preform have been characterized. If fewer than all target regions have been characterized, method 500 returns to step 530, and a new target line section is selected within the 2D RIP. Steps 530, 540, and 560 are repeatedly iterated until all target regions are characterized, and method 500 can proceed to step 580 and complete.

[0089] Figure 10 is a flowchart summarizing the steps of a particular embodiment of the fitting procedure applied in step 540 of the improved method 500. The target line section (or ROI) selected within the 2D RIP in step 530 is used as the start of the fitting procedure in step 541. Next, in step 542, start parameters for the fitting procedure are extracted from the line section, and in step 543, a first fitting region is defined. Next, method 500 asks, in step 544, whether the actual layer is stepped. If the answer to the question presented in step 544 is negative, method 500 asks, in step 545, whether there is another inner layer. If the answer to the question presented in step 545 is positive, in step 546, the fit region is moved to the next inner layer, and method 500 returns to step 544. Thus, a loop including steps 544, 545, and 546 is made until the question presented in step 544 is answered affirmatively or the question presented in step 545 is answered negatively.

[0090] If the answer to the question presented in step 545 is negative, method 500 asks, in step 547, whether the line section is stepped. If the answer to the question presented in step 547 is positive, in step 548, the measurement artifact-compensated parameters and the line section are achieved, and fitting procedure 540 ends. If the answer to the question presented in step 547 is negative, in step 549, the measurement artifact-compensated RIP is determined, and in step 550, the measurement artifact-compensated line section is achieved.

[0091] If the answer to the question presented in step 544 is positive, in step 551, an analytical bias curve of the axisymmetric RIP intended to mimic the measurement artifacts as an equivalent axisymmetric RIP is generated. In step 552, the bias is converted to a bias RIP. Next, in step 553, the accuracy of the bias RIP is calculated compared to the initial line section. The accuracy can be measured by the R-squared statistic that quantifies the prediction accuracy of the statistical model. The statistic indicates the proportion of the variance in the result variable explained by the prediction. This is the coefficient of determination R 2 , r 2 , and also known as r-squared. R 2 typically has a value in the range from 0 to 1 (or 100%). A value of 1 indicates that the prediction is identical to the observed value. It is not possible to have an R 2 value greater 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 there may still be a non-linear relationship between the observed and predicted values. Finally, a value of 0.5 means that half of the variance in the result variable is explained by the model. Sometimes, R 2 is expressed as a percentage (e.g., 50%).

[0092] Next, method 500 asks, in step 554, whether certain stopping criteria have been met (e.g., a certain R-squared value has been calculated in step 553). With each iteration of the fitting procedure, the variation in the initial parameters (n k and r k ) becomes progressively smaller. When the variation becomes small enough to fall below a certain threshold, further continuation of the fitting iteration is unnecessary. In practice, there are other inherent errors that limit precision, so for example, it makes no sense to make variations in n -5 smaller than 10 k . The same view applies to the layer radius r k .

[0093] If the answer to the question presented in step 554 is negative, method 500 proceeds to step 555, where the actual fitting parameters n m and r m are changed before method 500 returns to step 551. Thus, a loop including steps 551, 552, 553, 554, and 555 is made until the question presented in step 554 is answered affirmatively. If the answer to the question presented in step 554 is affirmative, the method returns to step 545.

[0094] The figure in FIG. 11 shows, for a preform having a simple step profile, the adjusted refractive index profile n'(r) and the virtual refractive index profile n * (r) modeled by its evaluation. The refractive index is shown as a relative value based on the refractive index of the refractive index adjusting fluid (n0 = 1.4587). In the figure, the relative refractive index n(r) - 1.4587 is plotted against the radius r in millimeters.

[0095] The virtual refractive index profile n *(r) already shows or is close to the refractive index profile of the preform actually expected. The virtual refractive index profile is based on the adjusted refractive index profile n’(r) and the orientation value derived from the profile, and the orientation value includes the estimated values of the refractive index and radius from the unmeasurable region.

[0096] Using Equation (1), the simulated deflection angle distribution Ψ”(y) is generated from the virtual refractive index profile n * (r) in the following method steps. Therefore, the simulated deflection angle distribution ψ * (y) is based on the assumption of the refractive index profile of the preform (i.e., the virtual refractive index profile n * (r)), which is then derived from the adjusted refractive index profile n’(r) after correction and evaluation of the original measurement values.

[0097] The simulated refractive index profile n”(r) plotted in this designation in FIG. 11 is obtained again by converting the simulated deflection angle distribution ψ”(y) by numerical integration of Equation (2). The profile has a rounded region 53 between the cladding region 52 and the core region 51. (As described above, such a core rod is often surrounded in the preform by an inner cladding layer of fluorine-doped or germanium-doped quartz glass and an outer cladding layer of undoped quartz glass). Except for this rounded region 53, the simulated refractive index n”(r) approximately coincides with the adjusted refractive index profile n’(r). Considering that the assumed refractive index distribution n * (r) is very different, this is remarkable. This similarity suggests that the assumption underlying the virtual refractive index profile n * (r) is already very close to the actual refractive index profile n(r) of the preform. That is, the virtual refractive index profile n * (r) in FIG. 11 accurately or at least sufficiently accurately reflects the actual refractive index profile n(r).

[0098] In practice, an exact match between the simulated refractive index profile n”(r) and the adjusted refractive index profile n’(r) cannot be achieved. However, by iteratively fitting the simulated refractive index profile n”(r) to the adjusted refractive index profile n’(r), it is possible to achieve a suitable and arbitrarily accurate fit. An alternative generalized fitting procedure can be described with reference to FIG. 12.

[0099] The alternative fitting procedure allows for a more accurate assessment of the RIP that has current measurement artifacts due to steep RI steps from low to high (in the direction from outside to inside), which are not completely step-like. In FIG. 12, a specific relative refractive index n(r) - 1.4587 is plotted against the radius r (in millimeters). The measurement data 301 are indicated by “X”. Typical measurement artifacts of the RIP appear at r = ±22 mm, and as a result, the reconstructed measurement values are different from the actual RIP in that region. The goal of the fitting procedure is to determine the RI step and its associated artifact error, which can be used later to correct the measurement data 301. For this purpose, a simulation helper 302 and an associated simulation fit or fitting curve 303 are provided, similar to the fitting procedure described above. The simulation helper 302 is a hypothesized RIP that is generated with the goal of estimating and covering the RI step at r = ±22 mm. The hypothesized RIP of the simulation helper 302 enables the generation of a deflection function by using Equation (1). By transforming this deflection function using Equation (2), the fitting curve 303 is obtained.

[0100] For a fixed range (e.g., 21 mm < |r|), the parameters of the simulation helper 302 are changed so that the fitting curve 303 matches the measurement data 301 around the RI step. The region must be small enough so that any inner RI gradient does not affect the fit, but must be wide enough so that this rounded curve shape can be fitted robustly enough. Depending on the visual feedback, one skilled in the art can adjust the selected region to improve the fitting procedure.

[0101] Adding the difference between the simulation helper 302 and the fitting curve 303 to the measurement data 301 results in 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 for each occurring RI step. In this case, the RIP of the simulation helper 302, the fitting curve 303, and the fit result 304 are joined together according to each of the different regions.

[0102] The characterized preform itself does not have such described RI steps, but even in certain cases having an RI of undoped fused silica near the outer surface, this method is also very useful for the following reasons. A setup such as apparatus 100 usually has an undoped fused silica measurement cell 9 because this material is homogeneously available and its absolute RI is well known including its dispersion, temperature dependence, and other properties. It is also state of the art to use a refractive index adjusting fluid 10 to keep the generated deflection angles small enough for apparatus 100 to be able to capture them. When the RI of the refractive index adjusting fluid 10 is above the measurement cell 9 and the RI is in the outer region of the glass object 11, measurement artifacts appear at the boundary between the measurement cell 9 and the refractive index adjusting fluid 10 and also affect the entire inner region. The only way to completely avoid this artifact is for the refractive index adjusting fluid 10 to exactly match the measurement cell 9 and the outer region of the glass object 11. In addition to a general practicality problem (due to the strong temperature dependence of the liquid suitable as the refractive index adjusting fluid 10), this eliminates the boundary completely, resulting in the distinction between the glass object 9, the refractive index adjusting fluid 10, and the measurement cell 11 becoming impossible.

Examples

[0103] The following examples are included to more clearly show the overall nature of the present disclosure. These examples are illustrative of the present disclosure and not limiting.

[0104] Figures 13 - 16 show Example 1 of a preform having a single elliptical core. Figure 13 is a sinogram created by applying method 500, i.e., by rotating the preform, acquiring measurement values, and stacking the measurement values from left to right. A single sine wave shape reflecting a single core is shown before application of the Radon transform. Note that the vertical lines reflect the first line section 310 and the second line section 312. The sinogram shows the magnitude of the deflection angle distribution ψ(ρ,θ) in the ρ - θ plane in arbitrary units.

[0105] FIG. 14 is a plot of the relative refractive index n(x,y) - 1.446 in the x-y plane showing the reconstructed 2D RIP of a preform having a single elliptical core. The first line section 310 is taken along the major axis (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.

[0106] Data taken along the second line section 312 drawn through the core of the FIG. 14 figure is shown in the graph of FIG. 15 showing the line section fitting procedure. The graph plots the relative refractive index n(r) - 1.446 against the position r in millimeters. FIG. 15 shows the main concept of method 500. The graph of FIG. 15 includes three curves, namely, curve 320 through the selected tomographic reconstruction line section data, simulation fit curve 321, and fit result (or hypothesis) curve 322. The assumed RI steps are 0.002 and 0.01, which can be very well evaluated in method 500.

[0107] FIG. 16 emphasizes the need for tomographic measurements 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 a preform having a single elliptical core and compares two different tomographic line sections with a single relevant measurement value. Curve 330 shows a single measurement value along the first line section 310, and curve 332 shows a single measurement value along the second line section 312. Curve 334 shows the tomographic measurement value along the first line section 310, and curve 336 shows the tomographic measurement value along the second line section 312. Thus, method 500 can be compared with a simple evaluation (basic inverse Abel transformation). Depending on the orientation during a single measurement, the results can vary greatly. In contrast, the level of RIP is independent of the selected line section. The fit function of method 500 and the resulting profile are omitted to avoid overloading the graph.

[0108] Figures 17 to 20 show Example 2 of a preform having a single core with an octagonal cladding that can be considered to be approximately rotationally symmetric. The core is made of fluorine-doped quartz glass sold under the registered trademark Fluosil (registered trademark) by Heraeus Quarzglas GmbH & Co. KG of Germany. Figure 17 is a sinogram created by applying method 500, that is, by rotating the preform, acquiring measurement values, and stacking the measurement values from left to right. Note that the vertical lines reflect the first line section 340 and the second line section 342. The sinogram shows the magnitude of the deflection angle distribution ψ(ρ,θ) in the ρ-θ plane in arbitrary units.

[0109] Figure 18 is a diagram of the relative refractive index n(x,y) - 1.446 in the x-y plane showing the reconstructed 2D RIP of the preform having an octagonal cladding. The first line section 340 is taken along the major axis passing through the opposing vertices of the octagonal cladding. The second line section 342 is taken along the minor axis passing through the opposing flat sections between the vertices of the octagonal cladding.

[0110] The data taken along the line section 342 drawn through the core of the figure in Figure 18 is shown in the graph of Figure 19 showing 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 Figure 19 includes four curves, namely, curve 350 passing through the measured line section data, simulation fit curve 351, fit result (or hypothesis) curve 352, and single measurement reconstruction curve 353. The graph shows the advantages of method 500 and emphasizes the need for tomographic evaluation compared to a single measurement.

[0111] Data taken along line section 340 drawn through the core of the figure of FIG. 18 is shown in the graph of FIG. 20 showing 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, namely, curve 360 passing through the measured line section data, simulation fit curve 361, fit result (or hypothesis) curve 362, and single measurement reconstruction curve 363. The graph shows the advantages of method 500 and emphasizes the need for tomographic evaluation as compared to a single measurement.

[0112] FIGS. 21 - 24 show Example 3 of a more complex preform having seven cores. FIG. 21 is a sinogram created by applying method 500, i.e., by rotating the preform, acquiring measurements, and stacking the measurements from left to right. Six sine wave shapes are shown, one for each of the non - central cores. Note that the vertical lines reflect the first line section 370 and the second line section 372. The sinogram shows the magnitude of the deflection angle distribution ψ(ρ,θ) in the ρ - θ plane in arbitrary units.

[0113] FIG. 22 is a diagram of the relative refractive index n(x,y) - 1.4587 in the x - y plane showing the reconstructed 2D RIP of the preform having seven cores. The first line section 370 is a shorter line section taken through the central core. The second line section 372 is a longer line section taken through the central core.

[0114] Data taken along the longer line section 372 drawn through the central core of the figure of FIG. 22 is shown in the graph of FIG. 23 showing 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, namely, a curve 380 passing through the measured line section data, a simulation fit curve 381, and a fit result (or hypothesis) curve 382. This graph shows the advantages of method 500. The Radon transform and its inverse transform can be applied to the transition between the sinogram of FIG. 21 and the graph of FIG. 23.

[0115] Data taken along the shorter line section 370 drawn through the central core of the figure of FIG. 22 is shown in the graph of FIG. 24 showing 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, namely, a curve 384 passing through the measured line section data, a simulation fit curve 385, and a fit result (or hypothesis) curve 386. This graph shows the advantages of method 500.

[0116] FIGS. 25 to 28 show Example 4 of a preform having six cores and no central core. FIG. 25 is a sinogram created by applying method 500, that is, by rotating the preform, acquiring measurement values, and stacking the measurement values from left to right. Six sine wave shapes are shown, one for each core. Note that the curves reflect the first line section 390 and the second line section 392. The line sections 390, 392 shown in the sinogram of FIG. 25 appear curved because the line sections 390, 392 in the 2D RIP (see FIG. 26) do not pass through the center. The sinogram shows the magnitude of the deflection angle distribution ψ(ρ,θ) in the ρ-θ plane in arbitrary units.

[0117] FIG. 26 is a plot of the relative refractive index n(x,y)-1.4587 in the x-y plane showing the reconstructed 2D RIP of a preform having six cores. The first line section 390 is a shorter line section passing through one of the cores. The second line section 392 is a longer line section passing through another one of the cores and including the preform boundary of about ±32 mm.

[0118] The data taken along the longer line section 392 are shown in the graph of FIG. 27 which shows 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, namely, a curve 394 passing through the measured line section data, a simulation fit curve 395, and a fit result (or hypothesis) curve 396. This graph shows the advantages of method 500.

[0119] The data taken along the shorter line section 390 are shown in the graph of FIG. 28 which shows 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, namely, a curve 397 passing through the measured line section data, a simulation fit curve 398, and a fit result (or hypothesis) curve 399. This graph shows the advantages of method 500.

[0120] These examples show that the calculated preform parameters, particularly the refractive index step, are much more accurate by method 500. Method 500 is beneficial to customers of asymmetric preforms, including multi-core preforms, hollow-core fiber preforms, Panda preforms, or preforms for fiber lasers having non-circular interfaces.

[0121] While the foregoing has been illustrated and described with reference to particular specific embodiments and examples, the present disclosure is 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 without departing from the spirit of the present disclosure. For example, it is expressly intended that all broad ranges described in this document include all narrow ranges included within that broad range.

Claims

1. A method for determining the refractive index profile (RIP) of an object, the method comprising: (a) providing the object comprising an optical object having a longitudinal axis, at least one layer extending outwardly about the longitudinal axis, the at least one layer lacking perfect rotational symmetry in cross-section; (b) scanning the object with a light beam starting from a first projection angle and creating raw data representing the object through measurement data; (c) rotating the object and repeatedly repeating step (b) until all projection angles have been scanned and all measurement data have been created; (d) processing the measurement data to form a sinogram and proceeding to step (f) when step (c) is complete; (e) rotating the object and repeatedly repeating steps (b) and (d) until all projection angles have been scanned; (f) calculating a 2D RIP; (g) selecting a target line section within the 2D RIP; (h) applying a fitting procedure to the target line section; (i) determining refractive index steps / gradients and dimensions; comprising, wherein said step (h) of applying the fitting procedure to the selected line section of interest comprises extracting start parameters for the fitting procedure from the line section, defining a first fitting region, and determining whether the actual layer is stepped.

2. A method for determining the refractive index profile (RIP) of an object, the method comprising: (a) providing the object comprising an optical object having a longitudinal axis, at least one layer extending outwardly about the longitudinal axis, the at least one layer lacking perfect rotational symmetry in cross-section; (b) scanning the object with a light beam starting from a first projection angle and creating raw data representing the object through measurement data; (d) processing the measurement data to form a sinogram; (e) rotating the object and repeatedly repeating steps (b) and (d) until all projection angles have been scanned; (f) calculating a 2D RIP; Step (g) of selecting a target line segment within the 2D RIP; Step (h) of applying a fitting procedure to the target line segment; Step (i) of determining the refractive index step / gradient and dimensions; comprising: The step (h) of applying the fitting procedure to the selected line segment of the target includes extracting start parameters for the fitting procedure from the line segment, defining a first conformity region, and determining whether the actual layer is stepped. A method. **Claim 3** The method according to claim 1 or 2, wherein the step of calculating the 2D RIP is performed based on an inverse Radon transform. **Claim 4** The method according to claim 1 or 2, wherein the step of processing the measurement 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. **Claim 5** The raw data includes a raw image, and the method further includes improving the quality of the raw image by one or more of using a scanning camera having an "image ROI" function, collecting several images with different exposures, using a scanning camera having HDR, and using a filtering algorithm to remove OVD diffraction. The method according to claim 1 or 2. **Claim 6** The method according to claim 1 or 2, further comprising improving the quality of the sinogram by one or more of applying radius runout correction and interpolating the sinogram into a fine mesh. **Claim 7** The method according to claim 1 or 2, wherein the object prepared in step (a) is a fiber preform. **Claim 8** The method according to claim 7, further comprising using the determined refractive index step / gradient and dimensions to characterize the fiber preform and adapt the preform manufacturing process. **Claim 9** The method according to claim 1 or 2, further comprising determining whether all projection angles have been scanned before the step (f). **Claim 10** After the step (f), determine whether a significant RIP measurement artifact is present in the 2D RIP. If not, further include the steps of determining the geometric shape and refractive index step and ending the method. The method according to claim 1 or 2.

11. The fitting procedure further includes the steps of generating an analytical deflection curve when the actual layer is stepped, converting the analytical deflection curve into a deflection RIP, and calculating the accuracy of the deflection RIP by comparing it with the initial line section. The method according to claim 1 or 2.

12. After the step (i), further include a step (j) of determining whether all line sections of the object in the 2D RIP have been characterized. The method according to claim 1 or 2.

13. The method according to claim 12 further includes the step of repeatedly repeating steps (g), (h), (i), and (j) until all line sections of the object in the 2D RIP are characterized.

14. The method according to claim 13 further includes the step of calculating a refractive index profile compensated for measurement artifacts of the object.

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