Systems and methods for self-assembly of arrays of optoelectronic devices

Abstract

Systems and methods are presented for forming an array of optoelectronic devices. In an example, a method comprises providing an optical fiber, the optical fiber including a pair of cores displaced from one another in a transverse direction and a cladding; applying heat from a heat source at a first position along an axial direction of the optical fiber at a predetermined temperature profile predicted by an instability model based on a radius of the first core, a viscosity of the cladding, and a surface tension between the first core and the cladding; and moving the optical fiber along the axial direction of the optical fiber at a predetermined speed predicted by the instability model, such that a path of the optical fiber passes through the first position and such that a series of droplet pairs form at a second position along the axial direction of the optical fiber.

Description

Docket No.2024-024-02 (144578.00415) SYSTEMS AND METHODS FOR SELF-ASSEMBLY OF ARRAYS OF OPTOELECTRONIC DEVICES CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of and priority to U.S. provisional Application No.63 / 584,604, filed on September 22, 2023, and is related to U.S. Provisional Application No. 63 / 510,493, filed on June 27, 2023, the entire contents of which are each herein incorporated by reference. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under 2143467 awarded by National Science Foundation. The Government has certain rights in the invention. TECHNICAL FIELD

[0003] This disclosure relates to systems and methods for self-assembly of optoelectronic systems in fibers and for measuring an interface energy between two arbitrary melts. SUMMARY

[0004] Capillary breakup of cores is an approach to fabricating fiber-integrated optoelectronics and photonics. A physical understanding of this fluid-dynamic process is advantageous for yielding the desired solid-state fiber-embedded multimaterial architectures by design rather than by exploratory search. As described herein, the nonlinearly complex and, at times, even chaotic capillary breakup of multimaterial fiber cores becomes predictable when the fiber is exposed to the spatiotemporal temperature profile, imposing a viscosity modulation comparable to the breakup wavelength. The profile acts as a notch filter, allowing only a single wavelength out of the continuous spectrum to develop predictably, following Euler-Lagrange dynamics. Thus, the present disclosure enables designing the outcomes of the breakup useful for turning it into a technology for materializing fiber-embedded functional systems but also positions a multimaterial fiber as a universal physical simulator of capillary instability in viscous threads. 1 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0005] The present disclosure implements a model that predicts the architecture resulting from the capillary breakup of functional cores, such as metals and semiconductors, encapsulated in a silica fiber. The core breakup, resembling a dripping faucet, results from the capillary instability on the core-cladding interface invoked by feeding the fiber through a localized liquefaction zone, such as a hydrogen-oxygen flame, or a focal spot of a laser. The model defines the range of the fiber and process parameters, such as core diameter, interface energy between the cladding and the core, spatial temperature profile in the liquefaction zone, and the feed speed of the fiber through that zone, in which the breakup is strictly periodic, and happens at a predefined position, in spite of the fact that, in a general case, the breakup is chaotic, and neither breakup period no position can be defined. BRIEF DESCRIPTION OF THE DRAWINGS

[0006] Features, objects, and advantages of the present technology will become more readily apparent when consideration is given to the detailed description below. Such detailed description makes reference to the following drawings, wherein:

[0007] FIG.1 illustrates an example of liquid phase fiber manufacturing and post- processing according to various aspects of the present disclosure.

[0008] FIG.2 illustrates an example of capillary breakup through axial thermal gradient according to various aspects of the present disclosure.

[0009] FIG. 3 illustrates example Effective Capillary Number regimens according to various aspects of the present disclosure.

[0010] FIG.4 illustrates an example comparison between various data according to various aspects of the present disclosure.

[0011] FIG. 5 illustrates an example of evolution of the capillary number for β = 60 according to various aspects of the present disclosure.

[0012] FIG.6 illustrates concordance between various data according to various aspects of the present disclosure.

[0013] FIG. 7 illustrates an example extrapolation of AVG-IM compared to ab-initio solutions of the Navier-Stokes equations according to various aspects of the present disclosure. 2 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0014] FIG. 8 illustrates examples of obtaining the conditions for the in-fiber staggered assembly of Cu-Si Schottky diodes according to various aspects of the present disclosure.

[0015] FIG. 9 illustrates an example breakup in multicore fibers and its scaling to the nanometric regime according to various aspects of the present disclosure.

[0016] FIG.10 illustrates an example high-temperature breakup setup according to various aspects of the present disclosure.

[0017] FIG. 11 illustrates an example snapshot of a breakup experiment according to various aspects of the present disclosure.

[0018] FIG. 12 illustrates an example comparison of temperature vs. width according to various aspects of the present disclosure.

[0019] FIG. 13 illustrates an example graph of pinch-off time tpinch as function of temperature T according to various aspects of the present disclosure.

[0020] FIG. 14 illustrates an example verification of the installation of the ab-initio simulation code according to various aspects of the present disclosure.

[0021] FIG.15 illustrates an example short-term acceleration according to various aspects of the present disclosure.

[0022] FIG.16 illustrates an assembled preform for the fabrication of a silicon-core silica fiber according to various aspects of the present disclosure.

[0023] FIG. 17 illustrates an example fiber according to various aspects of the present disclosure.

[0024] FIG.18 illustrates an example extrapolation of AVG-IM to tapered fiber cores and claddings according to various aspects of the present disclosure.

[0025] FIG. 19 illustrates an example fiber according to various aspects of the present disclosure.

[0026] FIG. 20 illustrates an example fiber according to various aspects of the present disclosure.

[0027] FIG. 21 illustrates an example fiber according to various aspects of the present disclosure. 3 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0028] FIG. 22 illustrates a heuristic comparison of the flame resulting in the hydrogen flow of 0.4 l / m vs.1.2 l / m according to various aspects of the present disclosure.

[0029] FIG.23 illustrates an example method according to various aspects of the present disclosure.

[0030] FIG.24 illustrates an example method according to various aspects of the present disclosure. DETAILED DESCRIPTION

[0031] The present technology will now be described more fully with reference to the accompanying drawings, in which some, but not all, embodiments are shown. Indeed, the technology may be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather, these embodiments are provided so that this disclosure will satisfy applicable legal requirements.

[0032] Likewise, many modifications and other embodiments of the technology described herein will come to mind to one of skill in the art to which the invention pertains having the benefit of the teachings presented in the following descriptions and the associated drawings. Therefore, it is to be understood that the disclosure is not to be limited to the specific embodiments disclosed and that modifications and other embodiments are intended to be included within the scope of the disclosure. Although specific terms are employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of skill in the art to which the technology pertains.

[0033] The model disclosed herein (Axial Viscosity Gradient Instability Model or AVG- IM) predicts the range of parameters related to the fiber and its processing, in which the capillary breakup of the fiber cores resulting from feeding the fiber through the localized liquefaction zone yields predictable, and thus engineering multi-material architectures embedded in a fiber. The model is used as a design tool for the self-assembly of arrays of optoelectronic devices, such as Si- Cu Schottky diodes and pn Si diodes. The model allows measuring interface energy between any two core and cladding materials, given its known for one reference core-cladding combination. 4 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0034] Accordingly, the present disclosure provides ability to fabricate fiber embedded microelectronics, optoelectronic, and photonics embedded in a fibers by design, rather than trial- and error exploration. This is required for flexible and conformal fiber and fabric sensors and transducers arrays used in consumer wearables and apparel, biomedical and hazard management applications, physiological and environmental monitoring, optical communication, etc.

[0035] Monofilament fibers resulting from a thermal draw are a unique platform for implementing pervasive, functional devices and systems, from traditional fiber optics for telecommunication through fibers-devices for signal transduction and energy management to fibers with computational and data processing capabilities.

[0036] FIG. 1 illustrates an example of liquid phase fiber manufacturing and post- processing. In FIG.1, image (a) shows an example of thermal draw. In image (a), the notations are as follows: R-preform radius, r-fiber radius, R / r-draw scaling factor, vf-feed speed, vd-draw speed. Image (b) shows an example of system assembly by a spatially coherent material-selective capillary breakup in an externally induced time-varying temperature gradient. The two inner cores of the fiber are, for instance, p- and n-doped silicon, while the two outer cores high-temperature are metal, such as vanadium or platinum, which is liquid during the fiber draw at 2000 °C, while staying solid during the capillary breakup performed in a window between softening point of silica (~1650 °C) and melting point of metal (1768 °C for platinum, 1910 °C for vanadium). In that case, the process will result in an array of bi-spherical pn Si diodes contacted in parallel by metallic electrodes. In image (b), the notations are as follows: xbu-breakup location, λbu-breakup period, T(x,t)-spatiotemporal temperature profile. Images (c)-(g) show an example of breakup in Si-core single- and dual-core fibers. In particular, image (c) shows regular breakup in 4 μm thick core (scale bar 200 μm); image (d) shows coherent breakup in dual-core fiber with similar cores 2 μm thick (scale bar 200 μm); image (e) shows regular breakup in 0.35 μm thick core (scale bar 5 μm, the spheres appear nonspherical due to aberrations resulting from cylindrical fiber cladding); image (f) shows in-and-out of phase breakup in a dual-core fiber with dissimilar cores, 2 μm and 1 μm thick (scale bar 100 μm); and image (g) shows irregular breakup in 0.5 μm thick core (scale bar 100 μm). Herein, the x direction of image (b) is referred to as the axial or longitudinal direction, and a direction perpendicular to the x direction of image (b) is referred to as the transverse or cross- sectional direction. Fiber fabrication and post-processing combine pulling and shaping the cladding and materials it encapsulates in a molten, viscous state (FIG.1, image (a)). For instance, 5 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) it is a reflow of a viscous liquid, and as such, it is governed by nonlinear, often chaotic fluid dynamics. Nonetheless, it is required to deliver a predictable, ordered solid-state architecture, defining the functionality of the fiber-embedded device or system.

[0037] For instance, capillary instability is increasingly used to pattern the fiber (image (b)) along its axial, otherwise uniform out-of-the-draw, dimension. Either promoted globally by heating the entire fiber to a state of viscous liquid or locally by gradually feeding the fiber through a delineated liquefaction zone, the capillary instability often bears pronounced characteristics of chaos. In the former case, the breakup is polydisperse. The main breakup period fluctuates quite pronouncedly due to any material-morphology imperfection and is often accompanied by secondary breakups, known as satellites, analogous to bifurcations, accompanying the onset of chaos. In the latter case, the breakup resembles a dripping faucet, which is in itself a standard chaos demonstrator.

[0038] Other than ab-initio numerical simulations of Navier-Stokes equations, the mathematical approaches providing the physical intuition to the capillary instability dynamics in fibers typically fall into two categories. On the one hand, in a globally liquified fiber, the standard description is in terms of linear instability (“Tomotika”) analysis, where out of the continuous spectrum of all the possible instability wavelengths developing concurrently within an infinite viscous thread, one, called dominant, has the fastest development rate, and defines the eventual breakup period. On the other hand, the situation in the locally liquified fiber is better described by the Plateau-Rayleigh instability propagating along the semi-infinite viscous thread beginning its tip, according to the dynamics defined by the marginal stability criterion. Here, the dominant wavelength is the one that has the fastest rate of propagation. Both models yield a single predictable breakup period, while it is known from experiments that the real-world outcomes are far more complex.

[0039] Despite obscure physical intuition about the process, the capillary breakup may be used for the assembly of photonic devices and systems, including p-n diodes, heterojunctions, arrayed photodetectors, photonic gratings, and optical resonators. This is achieved under conditions of localized liquefaction by feeding the fiber through a hot zone of a hydrogen-oxygen flame or a high-intensity laser beam (FIG. 1, image (b)). The feed rate of the fiber through the liquefaction zone defines the breakup period, while the boundary of the liquefaction zone defines the pinch off-location of the droplets breaking off the cores. Starting from initially separate cores, 6 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) multi-spherical devices contacted in parallel can be assembled (FIG.1, images (b) and (d)) when the breakup is spatially coherent and monodisperse (FIG. 1, images (c)-(e)). It is believed, however, that in addition to some selected conditions where this is possible, in a general case, the breakup often is characterized by an incontrollable phase (FIG. 1, image (f)) or period (FIG. 1, image (g)).

[0040] Comparative experimental, numerical, and analytical studies provide significant insight into the behavior of capillary breakup. Some such studies show successful implementations of numerical methods for simulating capillary breakup. The development of a mathematical model capable of translating input conditions, such as the spatiotemporal viscosity profile, into breakup outcomes, such as the phase and period, in a one-to-one, quantitatively accurate fashion, yielding a physical intuition of the process is desirable to turning the molten-phase in-fiber device assembly into an engineerable technology.

[0041] Herein, the governing mechanisms underlying the breakup in fibers exposed to Axial Viscosity Gradient (AVG) are interpreted, as in localized liquefaction conditions described above, in terms of intuitive first principles, such as balance of forces and energy / momentum conservation laws, culminating in a mathematical model of the process, reduced to Euler-Lagrange dynamics. Subsequently, it is demonstrated that the physical intuition provided by the model enables steering the breakup towards predictable, spatially coherent, monodisperse outcomes by design rather than exploration.

[0042] Results

[0043] Physical Formulation and Mathematical Derivation of the Axial Viscosity Gradient Instability Model (AVG-IM)

[0044] In the AVG breakup, the fiber reliquefies locally, and capillary instability breaks up its initially continuous cylindrical cores into arrays of discrete spheres. Since a cylinder’s surface is larger than that of a sphere for the same volume of material, capillary instability is a process of surface energy minimization. Comparative mathematical descriptions of this process consider cylindrical liquid threads with axially uniform viscosity in which the instabilities develop at an accelerating rate from the initial white noise simultaneously at all possible spatial frequencies. Such competition between instability wavelengths translated to experimental conditions yields a breakup characterized by probabilistic rather than deterministic outcomes Conversely, it is observed that in AVG conditions, in which the fiber viscosity changes axially following the 7 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) temperature profile of the heating source, a specific wavelength out of the continuous spectrum can be amplified selectively, yielding a fully predictable breakup period.

[0045] FIG.2 illustrates an example of capillary breakup through an axial thermal gradient. Image (a) shows fiber being fed through the hot zone of a flame or laser. When the fiber is fed through the hot zone of the flame or laser, it is exposed to a temperature gradient T(x), which in turn produces a gradient in the fiber’s cladding viscosity, ηclad. As shown in image (b), as the fiber is fed through the flame, the temperature gradient induces capillary breakup of the core, which occurs at a fixed position xbu in regular intervals with period λ, depending on the feed speed vf. Image (c) is a detail on pinch-off position xbuand neck formation process. For small enough δ, the cladding viscosity may be approximated as a linear function. The core section reshaping into a growing droplet of melt at the core tip is overlayed over a fainted image of the original core, how it would look at a given time if it would not reshape. It demonstrates the time-lapse of the growing core-section length streamed into the growing droplet as the capillary instability develops between the pinch events.

[0046] Based on experimental observation, the present disclosure considers such an ordered breakup to be a steady-state process where a semi-infinite core of a fiber is fed into the flame at a speed ^^^(FIG. 2, image (a)). The fiber geometry remains roughly unchanged before reaching some axial position ^^^௨, at which the core material starts reshaping to form a droplet (FIG.2, image (b)), growing to a critical size and pinching off the core at ^^^௨, as is schematically demonstrated in FIG.2, image (c). The remainder of the pinching neck then retracts towards the detached droplet and the continuous core, and the process repeats.

[0047] One criterion for evaluating the breakup of a liquid thread ejected from an orifice, a process closely resembling the description above, is the Effective Capillary Number (Ca), as it represents the relative effects of viscous and surface tension forces, which determine the breakup behavior. FIG.3 illustrates examples of effective Ca regimes. Image (a) shows the jetting regime characterized by the formation of a fluid jet and pinch off away from the orifice (Ca >>1). Image (b) shows a dripping regime characterized by pinch off immediately after the orifice (Ca <<1). Image (c) shows a snapshot of AVG Capillary breakup experiment illustrating the formation of a jet and pinch-off at position xbu. Image (d) shows a graph of effective Ca as a function of position. In AVG breakup, the fluid thread switches from jetting to dripping regimens at the breakup location, characterized by Ca=1. 8 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0048] Qualitatively, if Ca ^ 1, the system is in a jetting regime since the inner fluid jet extends past the orifice to a length larger than the smallest possible droplet size in the time needed for it to pinch (FIG.3, image (a)). In the case of Ca <1, the system is in the dripping regime, and the pinching will occur immediately after the orifice (FIG.3, image (b)). The quantity Ca is defined in which Ca ൌ ^^^^^^^ / ^^^where t୮୧୬ୡ୦is the droplet pinch time, and t^is the growth time of the fluid thread before pinching happens. In multimaterial fibers, a core of radius ^^ is embedded in a cladding of viscosity ηୡ୪ୟ^, with surface tension γ between the materials. For the case where the cladding is orders of magnitude more viscous than the core, as is the case with a silica fiber with a silicon core, the pinching time is defined as ^^ఉ^ఎ^^^^^ൌ^^ೌ^ఊ . β is a function of the viscosity ratio between the cladding and core, whichto a constant towards very highviscosity ratios. The breakup wavelength is defined as product between the growth time at the breakup location ^^^௨and time ^^^௨, and the axial velocity v^of the core: ^^ ≡ ^^^^^^^௨, ^^^௨^^^^. Thus, the Effective Capillary Number can be expressed as Ca ൌ^^^^^^^ ^^ ^^ ^^^^^ௗ ^^^^^ ൌ^^ ^^ (1)Experimentally, the predictablepinches off with reaching the critical size at the same axial location ^^^௨. This can be formulated simply as ^^^^^^^ൌ ^^^, or Ca ൌ 1 at ^^^௨(FIG.3, images (c) and (d)).

[0049] One conclusion that stems from the capillary number analysis is that AVG breakup cannot be thought of in common terms of either dripping or jetting, but rather starts as a jetting process and switches to dripping, the deeper into the hot zone the core is fed, while the pinch happens at the regime-switching position, Ca ൌ 1.

[0050] The systems and methods herein may be implemented for a predictable breakup with a well-defined period ^^. Yet, obtaining such ^^ in AVG conditions is not only a matter of synchronizing the droplet growth and its pinch off. Since ηୡ୪ୟ^varies in space, Eq. (1), Ca ൌ௧^^^^^௧^ൌఉ^ ఎ^^ೌ^௩^ఊఒൌ 1, becomes a single equation of two unknowns, ^^^^^ௗand ^^, making ^^^௨indefinable. The second independent equation stems from the fact, overlooked in comparative examples, that the breakup location does not just exist but is also unique. At ^^^௨, the instability does not just synchronize the droplet’s pinch-off with its growth, but completes this 9 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) synchronization earlier than at any other location. In other words, not just that௧^^^^^௧^ฬ ൌ 1, ௫ୀ௫್ೠbut also௧^^^^^௧^ฬ ^ 1. I.e., Ca ൌ௧^^^^^assumes a minimum at ^^ , or, formally speaking, ௫ஷ௫ ௧^ ^௨್ೠ௫್ೠ: was a at௫್ೠ, to the one at ^^| result from reshaping a section of a core fed through ^ ^^ ^ δ^|್ೠby ^^ compared to the one fed through ^^|௫್ೠfor the same moment in time, as is demonstrated in FIG.2, image (c). On the other hand, the concurrent pinch-off at ^ ^^ െ δ^|௫್ೠwould result from asection of a core longer by δ. In other words, ^ ^^ ^^⁄ ^^ ^^ ^| ௗ௫್ೠ ൌ െ1. Noting that ௗ௫^௫ ൌ್ೠ^ డడ௫ ^ ௗ௫ௗఒ ∙ డడఒ^^௫ ൌ ^ డ െ డ ^^ , and assuming that ^^ is independent of temperature (or, at್ೠ డ௫ డఒ ௫್ೠon temperature than the cladding viscosity, which is validfor material combinations that are considered here), the two equations of the AVG-IM model may be written as: ^^ ^^ ^^^^^ௗ^ ^^^ ∙ ^^ Ca^ ^^, ^^^ ൌì^^^ ^^^ ൌ üHere, ^^൫ ൌ , ൌ ൈ ^^ ൌ 61812 K material constants of silica. Βthe viscosity contrast between the cladding and the core and for increasing contrast asymptotically converges to a constant. Β is the free parameter in the current model and has to be determined from ab-initio calculations of the full Navier-Stokes (N-S) equations. If rearranging Eq. (2) and (3) to formulate it in terms of ^^௩^^ ^^, ^^^ ≡ ^^^െ ^^^^^^^rather than Ca ൌ t୮୧୬ୡ୦ / t^, as initially formulated, it can be seen that ^^௩^^^^, ^^^|௫್ೠ ൌ 0^4^QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) , which can be derived using purely geometrical arguments (see Supplementary Note 1, below): Equivalent to Eq. (2), Eq. (4) simply means that the droplet grows as long as it has not pinched off; Equivalent to Eq. (3), Eq. (5) means that the pinch-off location is unique because, at any other location, the droplet would need longer to pinch-off than it is allowed to grow.

[0052] ^^௩^^ ^^, ^^^ can be rewritten as ^^௩^^ ^^, ^^^|௫್ೠ ൌ ^^^ െ ^^^^^^^ ൌ ^ ^^⁄ ^^^ െ^^ ^^ ^^^^^ௗ⁄ ^^ ^ห௫್ೠ , which multiplied by a constant ^^ ൌ 2 ^^ ^^ ^^ ௩^ ^^, results in ^^ ^^ ^^ ^^, ^^^ ൌ^2 ^^ ^^ ^^ ^^ െ 2 ^^ ^^ଶ^^ ^^^^^ௗ^^^^ห௫್ೠ. Note that: 2 ^^ ^^ ^^ ^^|௫್ೠis the energy stored in the fluid thread of length ^PE^ stored in the pinching-off section of a thread before theof capillary instability in it; 2 ^^ ^^ଶ^^ ^^^^^ௗ^^^ห௫್ೠis the work invested in overcoming the viscosity forces to cause the pinch- ^KE) imparted to the pinching-offsection of a thread by the progression of the instability towards the breakup. In other words, ^^ ^^௩^^^^, ^^^is, by definition, the Lagrangian ^^ ൌ PE െ KE of the system. In fact, the AVG-IM is equivalent to a description of the system in terms of Euler-Lagrange mechanics (discussed in more detail in Methods and Supplementary Note 2, below).

[0053] The model is subject to validity limitations (VLs) It can be seen (see Methods) that the model is valid when: VL1: The breakup period ^^|௫್ೠis comparable to the width of the hot- zone boundary ^^, and VL2: The pinch locationis within the hot-zone boundary ^^. The boundary w is the region of the hot zone which exhibits a significant temperature gradient. In other words, the pinch time ^^^^^^^is comparable to the fiber dwell time in the hot zone boundary ^^ / ^^^. Validity limitations (VL1+VL2) explain the heuristic observation that the width of the liquified region boundary and the feed speed are the user-controlled “tuning knobs” for the breakup period (λ|^^౫^ w) and the pinch-off location ( ^^^௨~ ^^^∙ ^^^^^^^^ w^.

[0054] Corroboration of the AVG-IM Against Experimental Results and Ab-Initio Calculations

[0055] To validate the AVG-IM, it is compared to the experimental breakup in silicon-core silica fibers. The silica-silicon interface is well studied in the microelectronics realm, with the interface tension ^^ ൌ 0.3 ^^ / ^^ଶmeasured experimentally (see Supplementary Note 3). ^^ is obtained by corroborating AVG-IM against the ab-initio Navier-Stokes equation solution. 11 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0056] An exponential function of the form below can describe the shape of the temperature profile at the hot-zone boundary: ^^^ ^^^ ൌ ^^^^௫െ ^ ^^^^௫െ ^^ௌ^^ ^^ି௫ / ௪(6) where ^^^^௫is the maximal and ^^ defines the profile width.first is purely heuristic: such an exponent is a monotonic function asymptotically converging to a maximal value ^^^^௫following a growth region of a width ^^. This fits a qualitative behavior of the flame: it assumes its maximal temperature ^^^^௫in its center following a monotonic growth within its boundary of a width ^^. The second argument is mathematical: complex exponentials form a complete set, i.e., any function, without the loss of generality, can be approximated by a set of exponentials, and thus ^^^ ^^^, in its approximate form, is just a first-order decomposition of the real temperature profile into exponential components.

[0057] In comparative examples, no reliable technique for directly measuring the temperature profile in breakup experiments exists. One comparative example is a reconstruction of the temperature profile from the core emissivity, demonstrated in the recrystallization of silicon- germanium-core silica fibers. In laser recrystallization, the core is held at temperatures that do not significantly exceed the melting point, just by a few tens of degrees. Hence, the intensity range of the core’s optical emission signal is relatively low. In the breakup process, on the other hand, the range of temperatures to be captured is of the order of one thousand degrees (Figure 4). This necessitates a custom-developed image-capturing setup with a high dynamic range and, simultaneously, low noise, so the details, such as the dip in emissivity signal due to the melting of silicon (see Supplementary Note 4), are reliably captured.

[0058] Starting from ^^ = 20, a monotonic parametric scan of ^^ is performed, ending at 165. For each chosen value of ^^, the temperature profile parameters ^^^^௫and ^^ were determined through an algorithm that minimizes the difference between the AVG-IM predictions for ^^|௫^௨and ^^^௨and the experimentally obtained data. In all cases, the maximal temperatureby the fiber in the simulations does not surpass the boiling point of silica.

[0059] The algorithm results in a 2D map (see Supplementary Note 5) of the model-to- experiment fit error as a function of the temperature profile parameters, yielding a single combination of ^^^^௫and ^^ minimizing it. FIG.4 illustrates a comparison between experimental 12 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) data, AVG-IM, and ab-initio simulation, assuming β values from 20 to 165 (From top to bottom, in rows). Results for β = 50 and 70 are highlighted by a red dashed box. The improved fit between the model-derived and simulation-derived xbu(the light-gray background highlights the best-fit graphs) signifies the range where the optimized β falls. From left to right: columns 1 and 2 show experimental results (green circles) for the breakup period λ|xbu and xbu, compared to the fitted model (solid blue line); column 3 shows the temperature profile obtained by fitting AVG-IM to the experimental values and implemented in the ab-initio simulations; and columns 4 and 5 show ab-initio simulation results (red circles) for λ|xbu and xbu, and model predictions. Thus, FIG. 4 shows the experimental data and the AVG-IM fit resulting from this process (FIG. 4, columns 1 and 2). The determined temperature profile (FIG. 4, column 3) was then used in ab-initio simulation of breakup. Finally, the results of ab-initio calculations were corroborated against the model predictions (FIG.4, columns 4 and 5), with optimal correspondence for β ൌ 60 േ 10.

[0060] Here, the level-set method implemented in the simulations introduces a simplification in the time propagation of the Navier-Stokes equations, as a result of which the mass – and consequently volume – is not conserved. For this reason, the breakup period data from the simulations may be trusted qualitatively, while the breakup location data can be trusted quantitatively. FIG.5 illustrates an example of the evolution of Ca for β = 60. In image (a), Ca is shown as a function of the distance from the melting point of Si, for selected feed speeds vf(in μm / sec), graphically representing Eq. (2a). Image (b) shows the derivative of Ca for the same feed- speed set, graphically representing Eq. (2b). Image (c), inset in image (b), shows the absolute value of dCa / dx in logarithmic scale to illustrate better the extreme points corresponding to the breakup position xbu.

[0061] As the capillary instability develops, the lack of mass conservation results in the core thinning out in addition to pinching, causing the pinch process to happen faster, at a higher rate. Assuming a steady-state process, the period of such a simulated breakup is expected to be shorter than experimentally observed. At the same time, to even start reshaping into a droplet that eventually pinches off, the core needs to reach the viscosity where Ca=1 (FIG. 5, image (a)), allowing such a reshaping. Such viscosity conditions translate one-to-one into the appropriate breakup location (FIG.5, images (b) and (c)), unimpacted by the lack of volume conservation.

[0062] Choosing ^^ ൌ 60, AVG-IM quantitatively reproduces the entire set of the experimental data and the pinch-off positions calculated from the first principles. This is illustrated 13 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) in FIG.6, which shows that experiments, AVG-IM, and ab-initio simulations are in accord for β = 60 ± 10. Image (a) shows experimental data, ab-initio simulation result, and prediction for the breakup wavelength λbu. Image (b) shows experimental data, ab-initio simulation result, and prediction for the breakup location xbu. In images (a) and (b), results are shown from the AVG-IM assuming β = 60 ± 10, for an exponential temperature profile with Tmax = 2475 ± 25 K and w = 1150 ± 50 μm. The shaded region around the model prediction indicates the error (one and two σ). It can be seen that the ab-initio corroborated analytical model fits the experimental results quantitatively, unlike in comparative methods. Images (c) and (d) show optical micrographs of the breakup, with image (c) being a microscope image of fiber post-breakup indicating the breakup period resulting from AVG breakup for the respective feed speeds; and image (d) being snapshots of the AVG breakup experiment at different feed speeds, where the breakup location can be identified as the distance of the pinch-off from the melting position of silicon (indicated by the dashed vertical line). The experiments were performed in 280 μm thick silica fiber with 4 μm Si core by feeding it through a hydrogen-oxygen flame resulting from a flow of 0.30 l / min of H2, 0.10 l / min of O2.

[0063] Using COMSOL, ^^ was evaluated for the breakup in a slightly different system - a stationary fiber experiencing increasing local liquefaction (see Supplementary Note 6). Comparing measured and simulated pinch-off times, ^^ was found to be approximately 65.

[0064] Assessment of AVG-IM General Applicability Limits. Examples of Practical Uses of AVG-IM in Fiber-Device Engineering.

[0065] Generally, parameters defining the breakup behavior divide into two groups: the first relates to the fiber, while the second – to the liquefaction zone it is fed through. The fiber- related properties are the dimensions and viscosities of the core and cladding and the interface energy between the two (five parameters). The relevant liquefaction zone properties are its maximal temperature and the width of its boundary (two parameters).

[0066] There are two Model Simplifications (MSs) that AVG-IM makes, narrowing the space of parameters:

[0067] MS1: The cladding-to-air interface of the fiber is not significantly prone to capillary instability; thus, the cladding can be considered of an infinitely large diameter. Thus, the model is valid for tapered-cladding geometries. It has been verified experimentally (see Supplementary Note 9) that AVG-IM is valid for adiabatically tapered geometries of both core and cladding. For 14 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) non-adiabatic tapers, i.e., when ^^ ൌ ^^^ ^^^ on a scale comparable to the breakup wavelength, themodel can be extended to the tapered geometries by replacing Ca in Eq. (2) and (3) with a modifiedCa௧^^^^^ ^^, ^^^ ≡ ఉ^^௫^ ఎ^^ೌ^^௫^ ௩^ఊఒ .cores are solid crystalline materials that, upon melting, abruptlyinviscid liquid; thus, core-cladding viscosity contrast is considered negligible. It holds for a wide variety of core materials relevant for silica-fiber embedded optoelectronics and photonics, including but not limited to semiconducting, metallic, magnetic, optically nonlinear, superconducting, thermoelectric, ferroelectric, and piezoelectric. For fibers where ηୡ୭୰^ / ηୡ୪ୟ^is finite, such as polymer-cladding chalcogenide / polymer-core, Ca in ആ^^^^Eq. (2) and (3) may be replaced with ^^aആ^^ೌ^^ ^^, ^^^ ≡ ^^ ∙ ^^^ / ^2 ^^ ∙ ^^ ^^^ ^^, ^^^^^ௗ^ ^^^, ^^^^^^^ ^^^^^,where in^λ, ηୡ୪ୟ^, ηୡ୭୰^^ is theആ^^^^Caആ^^ೌ^^ ^^, ^^^ converges towards Ca^ ^^, ^^^ for ^^^^^^ / ^^^^^ௗ → 0, as is demonstrated in Methods.

[0069] MS1 and MS2 under an assumption that the cladding material is uniform, and thus the spatiotemporal temperature profile fully defines the viscosity of the cladding, leaves only four parameters for which the AVG-IM validity is verified: the maximal temperature of the liquefaction zone ^^^^௫, its boundary width ^^, the core-cladding interface energy ^^, and the core radius ^^. To validate AVG-IM in the subspace spanned by these parameters, their heuristic, behavioral trends, and relevance limits are defined and then scanned one by one within those limits to compare the AVG-IM predictions resulting from each scan to the Navier-Stokes equations solved ab-initio for the accuracy of extrapolation.

[0070] This may begin with examining the liquefaction zone properties. The maximal temperature of the liquefaction zone ^^^^௫resulting from the hydrogen-oxygen flame would vary with the flame oxygenation (H2:O2 flow rate ratio), the overall gas-flow rate, and the geometry of the torch orifice by hundreds of degrees, while its boundary width ^^ would vary on a sub- millimeter to a millimeter scale. For the liquefaction zone resulting from a laser beam, ^^^^௫and ^^ would vary with the beam intensity, size of the focal spot, and beam profile. In line with those scenarios and assuming the temperature profile defined by Eq. (6), FIG. 7 illustrates a broad extrapolation of AVG-IM compared to ab-initio solutions of the Navier-Stokes Equations. Image (a) presents a validation of AVG-IM for varying maximal temperature of the liquefaction zone Tmax. Image (b) presents a validation of AVG-IM for varying liquefaction zone boundary width w. 15 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) Images (c) and (d) present a validation of AVG-IM for complex liquefaction boundary geometry; in particular, image (c) shows pinch position xbu resulting from the single- and double- exponential temperature profiles, T1and T2, and image (d) shows temperature profiles, T1and T2. In these images, Δxbu(T1) is xbu(T1) in the range of 10 < vf < 90 μm / s(light grey rectangle), and Δxbu(T2) is xbu(T2) in the range of 10 < vf < 90 μm / s (light green rectangle). ΔT1 and ΔT2 are the temperature intervals corresponding to Δxbu(T1) and Δxbu(T2), respectively. Image (e) shows a validation of AVG-IM for varying core-cladding interface energy γ. Image (f) shows a validation of AVG-IM for varying core radius a.

[0071] Thus, FIG. 7, image (a) demonstrates the influence of the ^^^^௫on the pinch-off location ^^^௨calculated from ab-initio and from AVG-IM for ^^ ൌ 60 in a window of temperatures around that of the etalon experiment in FIG.6 and for the same fiber properties. Similarly, FIG.7 image (b) demonstrates the influence of the liquefaction zone boundary width ^^ on the ab-initio- and AVG-IM-calculated ^^^௨. In some physical situations, Eq. (6) may not describe the accurate temperature profile. FIG. 7, image (c) demonstrates an example of AVG-IM validation for the temperature profile that cannot be approximated by the first-order decomposition in exponentials, i.e., the single-exponential. The next in its complexity is the second-order decomposition, i.e., the double-exponential. In FIG.7, images (c) and (d), the ^^^௨resulting from the double-exponential profile ^^^^ ^^^ ൌ ^^^^௫ െ ^ ^^^^௫ െ ^^ௌ^^^ ^^ ^^ି௫ / ௪భ^ ^1 െ ^^^ ^^ି௫ / ௪మ^(7)is corroborated,ଶ ^௨resulting from the single exponential profile Tଶ^x^ ൌ T ି^ / ^୫ୟ^ െ ^T୫ୟ^ െ Tୗ୧^e , where ^^ ൌ1100 ^^ ^^. ^^^^௫ ൌ 2450 ^^ for both profiles. For the reasoning behind the specific choice of ^^^and Tଶ, see Methods.

[0072] FIG. 7, images (c) and (d) overlay the pinch-off positions ^^^௨^ ^^^^ and ^^^௨^ ^^ଶ^ against ^^^and ^^ଶin the way that allows backtracking of the temperatures corresponding to the pinch positions defined by each feed speed for both temperature profiles. From this backtracking, it is apparent that the fiber exposed to ^^ଶhas overall shorter ^^^௨, i.e., it pinched sooner than the one exposed for ^^^, even though the fiber exposed to ^^^entered sooner the soft silica’s temperature regime (>1900 K), where the instabilities can develop. Additionally, for a given feed speed range, while the corresponding pinch-off location range ∆ ^^^௨^ ^^^,ଶ^ depends on the temperature profile 16 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) geometry ^^^,ଶoverall, the range of temperatures ∆ ^^^,ଶdefining the ∆ ^^^௨^ ^^^,ଶ^, within a reasonable approximation, is invariant of the overall shape of ^^^,ଶ.

[0073] In other words, for a given fiber undergoing a predictable AVG breakup at a given feed speed, the pinch-off location is defined by the local viscosity η^x^|^^౫at its vicinity only, regardless of the fluid dynamic history that the core has experienced prior to reaching this location. One can think of η^x^|^^౫as a phase filter for the breakup. Additionally, from Eq. (2) and (3) it follows that ^^^ ^^^| ப^^^^௫್ೠis defined by the viscosity gradientப^^ ^ only, since for a given feed ^౫speed Eq. (2) and (3) have only two inputs: the claddingη^x^ and its gradientப^^^^ப^. Thus,ப^^^^is the notch filter for the breakup wavelength mentioned in the abstractப^.This local nature of the predictable AVG breakup relaxes the requirements for the temperature profiles reconstructed by the AVG-IM from the experimental results, such as those in FIG.6, since they need to represent the real-world temperature accurately in a limited range of ∆ ^^, defined by the experimentally attainable feed speeds, while a typical ∆ ^^ is a hundred to two hundred degrees, as in FIG.7, image (d).

[0075] Finally, FIG. 7, image (e) examines the influence of the core-cladding interface tension ^^ on AVG-IM in correspondence to the ab-initio predictions. Note that once the cladding viscosity is set, a choice of ^^ emulates the choice of core material for AVG-IM under the simplifications MS1 and MS2. FIG. 7, image (e) implies that for the core materials with very dissimilar ^^’s encapsulated in the same cladding and experiencing the breakup within the same temperature profile, the pinch-off locations will be increasingly separated axially for the increasing feed speed. This separation can exceed millimeters, becoming comparable to the overall liquefaction zone width ^^, especially if the heat source is well-localized, such as a laser spot.

[0076] This last result, in combination with the ability to select the phase and the period of the breakup independently, culminates in the proposition of the following “staggered” scenario for the scalable assembly of in-fiber devices. Consider two cores, ^^^and ^^ଶ, with roughly similar radii but made of dissimilar materials, such that ^^^^ ^^ଶwithin a single cladding. Consider two heat sources ^^^and ^^ଶwith maximal temperatures ^^^^^௫^ ^^^ଶ^௫, boundary widths ^^^, ^^ଶ, and overall widths of ^^^, ^^ଶ, aligned along the fiber axis at a distance Δ from each other. If ^^^exists such that the pinch-off location of the first core ^^^is at ^^^^௨ ^ ^^^^ within ^^^, yet the pinch-off location of 17 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) the second core ^^ଶin the first (colder) heat source ^^^is at ^^^ଶ௨ ^ ^^^^ that exceeds its overall width ^^^, it will follow that ^^^will breakup periodically with the wavelength ^^|௫భ ್ೠ ^௩^^within the first source, while ^^ଶwill pass through intact. Now, ^^ଶcan be tuned, such that ^^ଶbreaks up in the second heat source ^^ଶwith a wavelength ^^|௫మ ್ೠ ^௩^^that is equal to ^^|௫భ ್ೠ ^௩^^. Next, the distance between the heat sources Δ can be tuned such that Δ ൌ ^^ ∙ ^^|௫್భ,ೠమ^௩^^, where ^^ is any positive integer. As a result, both cores will break up in-phase, with the same period in an axially staggered fashion. If the initial cores ^^^,ଶare spaced in the fiber cross-section closer than the sum of the spheres’ radii pinching off those cores, the spheres from the neighboring cores will get in contact in the breakup process and form an axial array of fused bi-spherical entities.

[0077] FIG.8 illustrates operations of obtaining the conditions for the in-fiber staggered assembly of Cu-Si Schottky diodes Image (a) illustrates the experimental breakup and its AVG- IM analysis of Si-core breakup in the H2 = 1.2 l / min flame. An AVG-IM fit yields Tmax = 2185 K and w = 1275 μm. Image (b) shows the experimental breakup and AVG-IM analysis of Si- and Cu- core fibers in the H2= 0.4 l / min flame. An AVG-IM fit to the Si data, knowing that the boundary width w is identical (see SI.X and Methods), yields Tmax = 2065 K. In images (a) and (b), pink circles highlight the experimental results with similar breakup wavelengths at identical feed speed; and purple stars signify the proposed conditions for the staggered device assembly in image (g). Images (c) and (d) shows examples of obtaining silica-copper interface energy γSiO2-Cu by extrapolating AVG-IM found for the Si core to the experimental results of the Cu-core breakup for the 2a = 6.3 μm core (image (c)), and for the 2a = 3.4 μm core (image (d)). Images (e) and (f) represent methods of interface energy measurement for melt M and silica γSiO2-M. The sessile-drop method of image (e) (a comparative example) obtains γSiO2-M = γSiO2-air – γM-air ∙ cos θc, given the silica-air interface energy γSiO2-airand melt-air interface energy γM-air. The fiber-AVG-IM of image (f) (set forth herein) obtained γSiO2-Mby extrapolating the AVG-IM-obtained temperature profile for the reference core material, Si, in this case, to the breakup results of the core material M in silica fiber. Image (g) shows staggered Cu-Si diodes assembly schematics and suggested initial conditions based on AVG-IM analysis of Si and Cu breakup results in images (a) and (b). For the breakup periods to match exactly, the flow of hydrogen at the hotter source can be finetuned around the initial setpoint of H2 = 1.2 l / min. The exact phase-matching between the two breakups can be 18 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) achieved by finetuning the distance between the two heat sources, moving the hotter source along the fiber axis further away from the colder one.

[0078] In the analysis presented in FIG.8, specific conditions are obtained for assembling Cu-Si Schottky diodes in a double-core fiber using such a coherently staggered multicore breakup. It should be noted from FIG.7, image (f) that ^^^௨has a weaker dependence on the core radius ^^ than on any other parameter of the model, and thus staggered in-fiber assembly may not be achieved for the same-material devices, such as Si pn-diodes, by simply choosing different radii for the p-doped and n-doped cores. For that case, a different “interference” scenario is set forth in FIG.9.

[0079] In FIG.9, image (a) is a cross-section of the double Si-core fiber (scalebar - 50 μm); inset is the zoom-in to the cores (scalebar – 5 μm). Image (b) shows schematics of the “interference” device assembly by AVG breakup. k⃗1and k⃗2are wavevectors for the thinner and thicker core breakup, respectively. Image (c) shows a set of breakup results of the fiber in image (a) in the flame resulting from frow rates of H2 at 0.45 l / min and O2 at 0.4 l / min at varying feed speed (scalebars – 100 μm). Image (d) presents an AVG-IM analysis of the results in image (c) using double-exponential profiles Ti= Timax– (Timax– TSi)(Ce-x / w1+ (1 – C)e-x / w2), where i = {1,2} are the core indices. The best fit to the thinner and thicker core results yields T1max = 2470 K and T2max = 2590 K, respectively. For both temperature profiles, w1 = 4150 μm, w2 = 65 μm, and C = 0.33. A T1fit (solid black line) for the 2a1= 0.7 μm breakup results (black bars), and its extrapolation (black dashed line) to the 2.0 breakup results (red dots), except for the 2 μm / sec breakup, is shown. T2 results in a good fit (solid red line) for the 2a2 = 2 μm breakup results.

[0080] Images (e)-(k) show the breakup of the 25 μm thick fiber scaled down from the fiber in image (a) (see Methods). The feed speed is 10 μm / sec, and the flame conditions are: H2flow of 0.17 l / min and O2 of 0.08 l / min. Image (e) is an optical micrograph (scalebars – 10 μm). The k⃗1 – k⃗2 are ≈ 0.3 Hz / μm quasiperiodic beat (white arrows) is apparent. Image (f) is a Scanning Electron Microscope (SEM) micrograph (scalebar – 5 μm) of the Si spheres released from the fiber in image (e). Image (g) is a SEM zoom in (scalebar – 1 μm) to a single beat period (green rectangle in image (f)). White arrows point to the contacted bi-spherical assemblies. The green arrow points to the smallest individual sphere observed. Image (h) shows a SEM zoom in and image (i) shows a Tunneling Electron Microscope (TEM) zoom in to the smallest sphere observed (scalebars – 50 nm). The SEM reveals surface porosity and TEM the core-shell structure 19 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) of the sphere. The shell of a sphere in fiber is formed due to the chemical reaction occurring at the Si-silica interface at elevated temperatures. It prevents the spheres in contact from merging into a single droplet, forming a mechanically isolating but electrically conducting membrane between the spheres. Image (j) shows a SEM of a typical bi-spherical assembly (scalebar – 100 nm). Image (k) shows a SEM of the smallest bi-spherical assembly observed (scalebar – 100 nm). Images (j) and (k) show that the spheres in the assemblies are physically fused.

[0081] Image (l) shows the onset of the VL2-violating breakup. Shown is the SEM micrograph of an incompletely broken 350 nm thick Si core quenched before entering the flame completely. Images (m)-(o) show the onset of predictable AVG breakup. Among these, image (m) shows the optical micrograph of the 2a2= 2 μm fiber broken up in H20.8 l / min flame at a feed speed of 10 μm / sec; image (n) shows the droplet forming at the core tip entering the flame, quenched before its pinch-off is complete (the SEM micrograph of the core section in the green rectangle in image (m) is shown; and image (o) shows the SEM micrograph of a sphere in the green square in image (m). All the exposed core sections and spheres were obtained by etching the silica cladding in hydrofluoric acid overnight and evaporating the residual acid.

[0082] Investigating the breakup of Si- and Cu-core fibers in a pure hydrogen flame for the ability to yield same-wavelength breakup in disparate-material cores (see Supplementary Note10), there are at least two sets of conditions generating very similar breakup wavelengths at ^^^ ൌ10 ^^ ^^ / ^^, which provides a starting point for achieving a staggered diode assembly scenario:^^|௫್ೠ ൌ 1220 േ 20 ^^ ^^, in 4 ^^ ^^ thick Si-core, @ H2 flow rate = 1.2 l / min; and ^^|௫್ೠ ൌ 1310 േin 3.4 ^^ ^^ thick Cu-core, @ H2 flow rate = 0.4 l / min.

[0083] Those exact conditions, though, may not lead to the desired outcome, because the2 ^^ ൌ 4 ^^ ^^ Si core at ^^^ൌ 10 ^^ ^^ / ^^ breaks in H2 flame of 0.4 l / min as well, with a longer ^^|௫್ೠൌ3400 േ 60 ^^ ^^, while to qualify for the staggered assembly scenario, the core with lower ^^ shouldpass the colder flame intact.

[0084] The set of conditions that resulted in the random breakup (see Supplementary Note 10, FIGS. 19-21), and thus cannot be represented as a point in FIG. 8, image (a), are: ^^|௫್ೠൌ ^^ ^^ ^^ ^^ ^^ ^^, in 0.5 ^^ ^^ thick Si-core, H2flow rate = 1.2 l / min, 2 ^^ ൌ 0.5 ^^ ^^, and ^^^ൌ 10 ^^ ^^ / ^^. As such, the limitation of AVG-IM validity can be seen to be violated when the feed speed becomes too slow. Note that the AVG-IM validity limitation VL2 is written as ^^^௨~ ^^^∙ ^^^^^^^^ w, defines a characteristic feed speed, 20 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) w^^^^^^^ଶ^ ^^, ^^^ ^^^^ ≡^^^^^^^ൌ ^^ ∙^^ ^^ ^^^ ^^^|௫್ೠୀ௪ (8), which is the feed speed pinchlocation ^^^௨occurring precisely the distance of the boundary width ^^ into the liquefaction zone. It is the feed speed at which the breakup is safely predictable. If the feed speed is much faster than ^^^^^ଶ, the core is expected to reach the uniform temperature too soon and fall into a Tomotika-like jetting regime.

[0085] Experimentally, it can be found that the opposite is also true: the feed speed being much slower than ^^^^^ଶresults in the core not experiencing a viscosity gradient high enough by the time of the and thus also falling into an effectively Tomotika-like jetting regime. Basedon the temperature profile obtained by AVG-IM for H2flow rate of 1.2 l / min, ^^^^^ଶൌ 67 ^^ ^^ / ^^ ≫^^ ൌ 10 ^^ ^^ / ^^ are calculated for 0.5 ^^ ^^ thick Si c ^^ଶ^ ore. For the^^^ ൌ8 ^^ ^^ / ^^ ^ ^^^ ൌ 10 ^^ ^^ / ^^ are calculated for the 4 ^^ ^^-thick Si core, which indeed sticks to theAVG-IM behavior, as can be seen in FIG. 8, image (a). Thus, ^^^^^ଶ^^^, ^^^ ^^^^^ ^^^is a reliable, practical AVG-IM validity criterion.

[0086] To extrapolate AVG-IM to the Cu-core fiber results in Figure 8b, the temperature profile are treated as set to the one found for ^^ଶൌ 0.4 ^^ / ^^ ^^ ^^ based on Si-core data and the core- cladding interface energy ^^ is used as the free fitting parameter. As is demonstrated in FIG. 8, images (c) and (d), the last step results in the best fit of AVG-IM to the Cu-core of 2 ^^ ൌ 3.4 ^^ ^^at ^^ௌ^ைమି^௨ ൌ 1.15 േ 0.07 ^^ ∙ ^^ିଶ, and best fit to the Cu-core of 2 ^^ ൌ 6.3 ^^ ^^ at ^^ௌ^ைమି^௨ ൌ1.21 േ 0.10 ^^ ∙ ^^ିଶ.

[0087] Two Cu-core results combined, ^^ௌ^ைమି^௨is measured to be 1.20 േ 0.12 ^^ ∙ ^^ିଶusing Si-core fiber as a reference and AVG-IM as the analysis framework. In fact, as is demonstrated in FIG. 8, images (e) and (f), this sets a method for measuring interface energy between two melts, more accurate than the comparative sessile-drop method. For a detailed comparison between the two measurement approaches, see Methods.

[0088] Here, the conditions for the staggered Cu-Si Schottky diodes assembly in a double- core fiber are described. AVG-IM for 2 ^^ ൌ 4 ^^ ^^ Si-core fiber @ ^^ଶൌ 0.4 ^^ / ^^ ^^ ^^ predicts breakup periods exceeding the width of the flame ^^ ൌ 7.2 ^^ ^^ (see Supplementary Note 10) for 21 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) the feed speeds exceeding 27 ^^ ^^ / ^^. Recalling that, ^^^௨for any dataset in this study was always larger than the corresponding ^^|௫್ೠ, it is clear that ^^^௨^ ^^ for ^^^^ 27 ^^ ^^ / ^^ as well.

[0089] Thus, if a double-core fiber with 2 ^^ ൌ 3.4 ^^ ^^ Cu core and 2 ^^ ൌ 4 ^^ ^^ Si core is fed at ^^^ൌ 30 ^^ ^^ / ^^ through a combination of two heat sources, the colder of which is the flame @ ^^ଶൌ 0.4 ^^ / ^^ ^^ ^^, and the hotter is the flame @ ^^ଶൌ 1.2 ^^ / ^^ ^^ ^^, the colder source first, the Cu core will break up in the colder source, while the Si core will pass intact and will break up later on in the hotter source, both with the breakup wavelength of ~2.4 mm. Those conditions, highlighted by purple stars in Figures 8a and 8b, provide a starting point which, after a slight finetuning of the synchronization between the two breakups, as is described in Figure 8g, will result in a staggered assembly of an array of fiber-embedded Cu-Si Schottky diodes.

[0090] As mentioned, a different “interference” approach is proposed for the devices’ assembly in multiple cores with identical core-cladding interface energy, such as Si pn-diodes, as described and analyzed in FIG.9. Investigating a breakup in a double core fiber in Figure 9a, in which the two cores with 2 ^^^ൌ 0.7 ^^ ^^, 2 ^^ଶൌ 2 ^^ ^^ were separated by a center-to-center gap of Γ ൌ 4.8 ^^ ^^, it is found that, unlike in fiber with identical cores (see FIG. 1, image (d)), each of the cores breaks up according to its individual wavevector ^^^^, ^^≡^ఒ|భ,మ^, and only occasionally the ್ೠspheres from the neighboring cores come in contact with eachat locations of constructive interference between ^^^^and ^^^^, with the beat wavevector of such an interference ^^ ൌ ^^^^െ ^^^^, as is depicted schematically in FIG. 9, image (b). This in-and-out-of-phase beat is evident at all feed speeds, as can be seen in FIG.9, image (c), showing the breakup of the double core from FIG. 9, image (a) in the flame resulting from flow rates of ^^ଶ@ 0.45 ^^ / ^^ ^^ ^^ and ^^ଶ@ 0.4 ^^ / ^^ ^^ ^^ at varying feed speed. The last is true, with the exception of ^^^ൌ 2 µ ^^ / ^^. At such a slow feed speed, the thin core broke up chaotically, likely due to the violation of VL2, i.e., ^^^≪ ^^^^^ଶ^ ^^^, ^^^ ^^^^.

[0091] Attempts at fitting AVG-IM to the results in FIG. 9, image (c) revealed that no temperature profile corresponds to the results observed for both cores concurrently. FIG.9, image (d), demonstrates the best AVG-IM fits resulting from the double-exponential temperature profiles ^^^ൌ ^^^^^௫– ^ ^^^^^௫െ ^^ௌ^^^ ^^ ^^ି௫ / ௪భ^ ^1 െ ^^^ ^^ି௫ / ௪మ^, where ^^ ൌ ^1,2^ are the core indices.results yield ^^^^^௫ ൌ2470 ^^ and ^^ଶ^^௫ ൌ 2590 ^^, respectively. ^^^ ൌ 4150 ^^ ^^, ^^ଶ ൌ 65 ^^ ^^, ^^ ൌ 0.33 for bothtemperature profiles. It is apparent from image (d) that the temperature profile ^^ଶ, fitting 22 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) well the thick core breakups, yields AVG-IM extrapolation that significantly undershoots those of the thinner core, and vice versa, the temperature profile ^^^, fitting well the thin core breakup periods, yields extrapolation of AVG-IM to the thick core that significantly overshoots the experimental results, with one notable exception – for the period resulting from ^^^ൌ 2 µ ^^ / ^^, which shoots up closer to AVG-IM prediction for ^^^.

[0092] This last exception, in combination with AVG-IM analysis, suggests that when the breakup in both cores is periodic, the instabilities developing in each core are mechanically coupled, drawing the breakup periods much closer together than would be expected for independent, decoupled breakups, with this coupling lost when one of the breakups falls into the random regime. When both cores are breaking up predictably, the periods are long enough for the sum of the sphere radii, which collect into them the core material of the entire period, to exceed the separation between the cores, i.e., ^^^^ ^^ଶ^ Γ, which makes the spheres touch and fuse into bi-spherical devices when in phase. contact is a blunt sign of mechanical coupling. Onthe contrary, when one of the cores breaks up randomly, its breakup period shortens. Due to the volume conservation argument, the spheres become much smaller and might lose that proximity to the spheres from the neighboring periodically breaking up core. Additionally, momentum conservation should hold like for any other mechanical system. In other words, breakups should have similar wavevectors, ^^^^^ ^^^^, to exchange energy efficiently, while ^^^^≫ ^^^^when one of the cores breaks up randomly.

[0093] Without seeking to be bound by any theory of the coupling mechanism, it can be said that in fibers with submicron core cross-sections (FIG.9, images (e)-(k)), the coupled breakup continues for tens of beat periods 1 / ^^ without signs of decay, resulting in 3-4 diode assemblies per bit period. FIG. 9, images (e) to (k) analyze the breakup at ^^^ൌ 10 ^^ ^^ / ^^, in the flame of ^^ଶൌ 0.17 ^^ / ^^ ^^ ^^ and ^^ଶൌ 0.08 ^^ / ^^ ^^ ^^, of the double-core fiber resulting from scaling down of the fiber in FIG. 9, image (a) to the cladding thickness of 25 µm (see Methods) such that the cores enter the submicron cross-section regime.

[0094] In thermal draw and breakup, a shell comprised of Si-silica nanocrystal mixture covering a sphere is formed, resulting from the chemical reaction occurring at the Si-silica interface at elevated temperatures. This shell of the third material (neither Si, nor silica) forms a mechanically isolating but electrically conducting membrane on the spheres. The presence of that shell is what allows the p- and n-doped Si droplets to come in contact, while maintaining the 23 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) junction, without merging into a uniformly doped single droplet. The SEM analysis of submicron spheres (FIG.9, image (h)) reveals the surface membrane porosity when the silica component is etched out. Augmenting the SEM, TEM (FIG.9, image (i)) shows the core-shell structure of the sphere. FIG. 9, images (j) and (k) show that the spheres in the assemblies are physically fused without merging. All the exposed spheres, sphere assemblies, and continuous core sections in FIG. 9, images (e) to (o) were obtained by etching the silica cladding in hydrofluoric acid overnight and evaporating the residual acid.

[0095] The breakup behavior in the scaled-down fiber in FIG. 9, image (e) cannot be analyzed directly using AVG-IM since it is happening deep in the regime, violating VL2 for both cores. From Eq. (8), ^^^^^ଶ^^^, ^^^ ^^^^is inversely proportional to ^^. In the scaled-down fibers, the cores are ^^ ^^ and 2 ^^ଶ^ 180 ^^ ^^, meaning that both cores are deeperinto the VL2-violating regimen than even the 2 ^^ ൌ 500 ^^ ^^ sample in FIG.8, image (a), while produced at an identical feed speed.

[0096] Experimentally, it is known that a submicron core of a single-core fiber (FIG. 1, image (e)) breaking up at identical conditions shows a clear Tomioka-like concurrent development of multiple instability periods in a long section of a core, if quenched before entering the flame completely, as can be seen in FIG.9, image (l). At the same time, an inspection of a comparable, halted-before-complete-breakup section of the core in the predictable AVG regime clearly shows a formation of one discrete droplet at the core tip, as is demonstrated in FIG.9, images (m) to (o).

[0097] The periodicity of the submicron breakup does not make it predictable, at least not in the AVG-IM sense. Based on qualitative experimental observations (see Methods), it can be seen that the submicron breakup is chaotic, yet with the reduction of the flame temperature concentrates towards a so-called fixed-point attractor, a well-concentrated preferred region in the otherwise chaotic phase space, resulting in a statistically predictable monodisperse breakup.

[0098] In fact, complex systems are rarely non-statistically predictable. In that sense, AVG-IM, obeying the linear Euler-Lagrange dynamics, thus mapping the user-engineered spatiotemporal conditions into the system dynamics in an exact, one-to-one, and not just probabilistic fashion, is a rare instance of “finding order in chaos.” Capable of heuristically decomposing a continuous spectrum of capillary instabilities into single-phase single-wavelength components, it is believed that the multimaterial fiber in combination with AVG-IM can be used 24 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) to physically simulate the otherwise computationally heavy (see Supplementary Note 7) Navier- Stokes equations of viscous multimaterial flow.

[0099] Methods

[0100] Fiber Draw

[0101] The silicon-core silica fiber in FIG.3 was drawn using a draw tower (Optogear OG- 510D) in two steps. For the first step, a five cm-long, two mm-thick Silicon rod, 5N purity (ESPI Metals) was sleeved in concentric silica tubes (Technical Glass Products) (see Supplementary Note8) and sealed under vacuum using a glass lathe (Optogear OG-440), resulting in a preform25 ^^ ^^ in diameter, which drawn at a temperature of 1950 ℃, feed speed of 1.28 ^^ ^^ / ^^ ^^ ^^ anddraws speed of 0.2 ^^ / ^^ ^^ ^^. The resulting cane was 2 ^^ ^^ in diameter, with a core 160 ^^ ^^ in diameter (Supplementary Note 8). It was re-sleeved in silica tubes and sealed under vacuum in theglass lathe to produce a preform, 25 ^^ ^^ in diameter. This new preform was then drawn at2000 ℃, with a feed speed of 6.25 ^^ ^^ / ^^ ^^ ^^ and draw speed of 10 ^^ / ^^ ^^ ^^, to obtain the final280 ^^ ^^-thick fiber, with a 4 ^^ ^^-thick Si core (see Supplementary Note 8). A conceptually similar fabrication procedure yielded an architecturally identical fiber in FIG.1, image (c) and FIG.6. The Si core fiber samples in Supplementary Note 9 with the 1.5 µ ^^ core and those in FIG.1, image (g) and FIG.8, image (a) with 0.5 µ ^^ core originate from the same draw as the samples in FIG. 1, image (c) and FIG.6, taken from the fiber section resulting from the draw stage where the Si- containing section of the preform enters the draw process, resulting in the adiabatic taper-in of the core over a few meters (~5 ^^), eventually stabilizing at a steady state for multiple tens of meters (~60 ^^) at the 4 µ ^^ core thickness.

[0102] The procedure for the fabrication of the double-core fibers in FIG. 1, images (d) and (f) and FIG. 9, images (a) and (c) is as follows. First, a preform containing two separate Si rods was prepared and drawn into a double-core fiber. One p-type Si rod (electron acceptors concentration ^^^ൌ 6.9 ൈ 10^଼  ^^ ^^ିଷ) and one n-type Si rod (electron donors concentration ^^^ൌ 1.6 ൈ 10^ଽ  ^^ ^^ିଷ) with a diameter of 2  ^^ ^^ were sleeved into a preform with a silica cladding of 12  ^^ ^^ and drawn at 2,130 ° ^^ with the feed speed ^^^ൌ 0.6  ^^ ^^ / ^^ ^^ ^^ and draw speed of up to ^^ௗൌ 0.6  ^^ / ^^ ^^ ^^. Then one of the sections, 11 ^^ ^^ long, of the resulting cane with an outer cladding diameter of 630 µ ^^ and containing two cores, each 110 µm thick separated by 200 µ ^^ center-to-center gap, was incorporated into a new 12 ^^ ^^ thick preform. That preform 25 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) was then drawn with the feed speed ^^^ൌ 1.2  ^^ ^^ / ^^ ^^ ^^ and the draw speed of ^^ௗൌ 2  ^^ / ^^ ^^ ^^ at the draw temperature of 2085 °C. This resulted in tens of meters of fiber with an outer cladding diameter of 280 µ ^^, with the pair of cores, the first with the thickness fluctuating slowly (on the scale of meters) around the average of 1.5 µ ^^ with the standard deviation of 0.4 µ ^^ in the range of 0.7 to 2.0 µ ^^, and the second with the thickness fluctuating around 1.8 µm with the standard deviation of 0.4 µm in the range of 1.2 to 2.4 µ ^^. The center-to-center separation between the two averaged at 5.0 µm with the standard deviation of 0.3 µ ^^ in the range of 4.6 to 5.4 µ ^^.

[0103] The Cu-core silica fibers used for the study in FIG.8 were produced by a redraw of 12 ^^ ^^ long 0.4 mm thick silica cane with 90 µ ^^ Cu core sleaved and vacuum sealed into aconcentric assembly of silica tubes of internal diameter of 0.5 ^^ ^^ and external diameter of13 ^^ ^^. The cane redraw was performed at 1.1 ^^ ^^ / ^^ ^^ ^^ preform feed speed into the furnace andmaximal fiber drawing capstan speed of 2 ^^ / ^^ ^^ ^^, in 2070 ℃ to 2090 ℃. The redraw processresulted in tens of meters ^~15 ^^) of fiber with a copper core of a thickness ranging from 0.9 to7 µ ^^. The cane used for the redraw resulted from a draw of 0.8 ^^ ^^ copper wire vacuum sealedin silica preform with an internal diameter of 1 ^^ ^^ and external diameter of 13 ^^ ^^. The draw of the cane was performed at 2100 ℃ furnace temperature, at 1.2 ^^ ^^ / ^^ ^^ ^^ preform feed speed into the furnace, and maximal fiber drawing capstan speed of 1.8 ^^ / ^^ ^^ ^^ .

[0104] AVG capillary breakup experiments

[0105] The breakup experiments were performed using a hydrogen-oxygen torch as a source of heat. Flows of each of the gases, ^^ଶand ^^ଶ, were controlled by two individual Omega FMA5400A_5500A digital mass-flow controllers calibrated to nitrogen. The mixture of burning gases were ejected through an orifice of a torch (National Torches Model 3H) 0.02” in diameter (MSOX-0 mini tip) to form the flame. The distance between the torch orifice and the fiber was 12 mm in the etalon experiment in FIG.6 but was adjusted up or down for lower and higher flames, respectively, for other experiments, wherever appropriate, such that the fiber is set at the tip of theflame, where the heat deposition is the most efficient. The feed speed ranged from 2 ^^ ^^ / ^^ to150 ^^ ^^ / ^^. A camera (Amscope MU900 (FIG. 3) / Motic Moticam1000 (FIG. 6)) registered thebreakup events. The movies collected were analyzed frame-by-frame to capture each individual droplet breakup. The breakup location was measured with respect to the melting position of the silicon core, which is identified as a dip in the captured image intensity (see Supplementary Fig.4 in Supplementary Note 4). The breakup period was calculated by multiplying the feed speed and 26 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) the time between consecutive breakups, as well as measured by optical inspection of the fibers post-breakup.

[0106] Ab-initio simulations

[0107] The Navier-Stokes ab-initio simulations were performed in Indiana University’s Big Red 3 and Big Red 200 supercomputers. The output was verified against the outputs obtained from running it at the jdj cluster at the Massachusetts Institute of Technology. Significant speedup was achieved without a change in outcome (Supplementary Note 7).

[0108] Proof of equivalence of AVG-IM to a description of the system in terms of Euler- Lagrange mechanics

[0109] Eq. (2) in its rearranged form, as Eq. (4), becomes simply ^^ ≡ PE െ KE ൌ 0.

[0110] The Lagrange equation of motionௗௗ௧^డ^ డ^^^ െడ^డ^ൌ 0 can be written as ௗௗ௧^^^ ^^ ^^^^^ௗ^^ ௫ ^ ^^ ൌ 0 , taking ^^ ൌ ^^, and noting that is the speed at which ್ೠthe length of the reshaping core section fed through the front ^^^௨grows (Figure 2c). Additionally, noting thatௗൌ ௗ௫ ௗ ൌ ^^ ௗ, thenௗఎ^^ೌ^^ ൌ െఊ,ఎ^^ೌ^ௗ௧ௗ௧ௗ௫ ^ ௗ௫ ௗ௫ ௫which from Eq. (2) equals to െ ^ . ್ೠ ^ఉ௩^ ఒ ௫್ೠSince ^^^^^ௗis aof ^^ only, the equation of motion can now be rewritten as ^^|௫್ೠൌ ^െఎ^^ೌ^^௫^డఎ^^ೌ^^௫^⁄ డ௫^^ , which is Eq. (3) in its explicit form. To see that, the constants canand thewhich Eq. (3) takes the form of ^డడఎ^^ೌ^^௫^^ డ௫െడఒ^ ^ఒ^^ ௫ ൌఒ∙

[0112] VL1: The model is valid only if the breakup period is comparable to the width of the hot-zone boundary: From Eq. (3) in the form ^^|ఎ௫^^ೌ^^௫^್ೠൌ ^െడఎ^^ೌ^^௫^⁄ డ௫^^ , the model is ௫್ೠinvalid for uniform-viscosity threads. If ^^^^^ௗ^^^^ൌ ^^ ^^ ^^ ^^ ^^, it follows that ^^ ^^^^^ௗ^ ^^^⁄^^ ^^ ൌ 0, and^^|௫್ೠൌ േ∞ is not physical. If, additionally, approximating viscosity and viscosity gradient by values within the hot-zone boundary, i.e., ^^^^^ௗ^ ^^^ ^ ^ ^^^^௫െ ^^^^^^ / 2 and ^^ ^^ ^ ^^^⁄ ^^ ^^ ^ ^ ^^^^௫െ ^^^^^^ / ^^, ^^|௫್ೠ^ ^^. In other words, AVG-IMperiods comparable to the width of the liquified region boundary. If ^^ ൌ 0, such as in dripping from an orifice, ^^|௫್ೠൌ 0 is not physical. 27 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0113] VL2: The model is valid only if the pinch location falls within the hot-zone boundary: From Eq. (2) it follows that ^^~ ^^^. Thus if ^^^ൌ 0, ^^|௫್ೠൌ 0 is not physical. Additionally, if ^^^is too large, the thread will reach the asymptotic temperature ^^^^௫before the pinch-off. The breakup then happens at uniform viscosity conditions, where the model does not hold (see (VL1)). In other words, the model is valid if the pinch-off location is within the liquified region boundary, meaning that ^^^^^^^^ ^^⁄ ^^^.

[0114] Fiber scaling

[0115] The fiber in Figure 9, image (e) resulted from scaling down the double-core fiber in Figure 9, image (a) (275 ^^ ^^-thick silica cladding with two Si cores, 2.0 and 0.7 ^^ ^^-thick, separated by 4.8 ^^ ^^ center-to-center) in three steps according to the procedure described in Supplementary Note 9. The fiber diameter reduction by a factor of√5 in each step resulted from ^^^ൌ 50 ^^ ^^ / ^^ ^^ ^^ and ^^ௗൌ 250 ^^ ^^ / ^^ ^^ ^^. The flame used as a heat source for the reduction procedure resulted from H2 flow of 0.25 l / m, 0.22 l / min, and 0.20 l / min for the first, second, and third scaling-down repetitions, respectively.

[0116] Extending AVG-IM to the general viscosity contrast regime

[0117] AVG-IM will hold true for the general core-cladding viscosity contrastఎ^^^^ఎ^^ೌ^, if Ca ആ^^^^in Eq. (2) and (3) is replaced with Caആ^^ೌ^^ ^^, ^^^ ≡ ^^ ∙ ^^^ / ^2 ^^ ∙ in^ ^^, ^^^^^ௗ^ ^^^, ^^^^^^^ ^^^^^, wherein^λ,ηୡ୪ୟ^, ηୡ୭୰^^ is the

[0118] To demonstrate this, one may look at in^^^, ^^^^^ௗ, ^^^^^^^, as is defined in Eq. (38) in Ref.23: in^^^, ^^^^^ௗ, ^^^^^^^ൌ ^ ^^ / 2 ^^ ^^^^^ௗ^ ∙ ^^, where ^^ ൌ ^^^ ^^ / ^^, ^^^^^^ / ^^^^^ௗ^ is a function of theηୡ୭୰^ / ηୡ୪ୟ^. ^^ → 1 when ^^^^^^ / ^^^^^ௗ→ 0 and ^^ / ^^|௫್ೠ→ 0. In the Tomotika model, λ|^^౫is called “thedominant wavelength.” It is the wavelength that maximizes the instability rate spectrumin^ ^^, ^^^^^ௗ, ^^^^^^^, and thus develops towards pinching faster than any other wavelength λ in thespectrum.

[0119] In silica-based optoelectronic and photonic fibers, heuristically, ηୡ୭୰^^x ^ 0^ ≪ ηୡ୪ୟ^^x ^ 0^, and any practical ^^|௫್ೠis of the order of tens to hundreds of microns, while ^^ ismicrometric or^^ ≪ ^^|௫್ೠ , resulting in ^^ ^ 1 and in^ ^^, ^^^^^ௗ, ^^^^^^^ ^^^ / 2 ^^ ^^^^^ௗ for λ in the vicinity of λ|^^౫ . Thus, in the limit where the simplification MS2 holds,28 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) ആ^^^^for λ in the vicinity of ^^|௫ആ್ೠ, Ca^^ೌ^^ ^^, ^^^ converges towards Ca^ ^^, ^^^ ൌ ^^ ^^ ^^^^^ௗ^ ^^^ ^^^ / ^ ^^ ^^^ defined in Eq. (1). ആ^^^^

[0120] Convergence of Caആ^^ೌ^^ ^^, ^^^ towards Ca^^^, ^^^for ^^^^^^ / ^^^^^ௗ→ 0 supports an educated guess that extrapolates the AVG-IM into to the general viscosity contrast.

[0121] Comparison of the methods for measuring the interface energy: sessile-drop method vs. fiber-AVG-IM method

[0122] Sessile-drop interface energy measurement of an arbitrary melt ^^ with silica, ^^ௌ^ைమିெ, shown in FIG.8, image (e) entails placing a droplet of melt ^^ on a planar SiO2 substrate, measuring the contact angle ^^^that the drop of melt forms with the SiO2surface, and calculating ^^ௌ^ைమିெaccording to ^^ௌ^ைమିெൌ ^^ௌ^ைమି^^^െ ^^ெି^^^∙ ^^ ^^ ^^ ^^^, while the silica-air interface energy ^^ௌ^ைమି^^^and melt-air interface energy ^^ெି^^^may be known from elsewhere.

[0123] To compare, the comparative sessile-drop method in FIG. 8, image (e) measures interface energy between solid substrate and melt using two reference interface energies, while the fiber-AVG-IM method in FIG. 8, image (f) measures interface energy between two melts using one reference interface energy. For the measurement of ^^ௌ^ைమି^௨the silica-silicon interface energy ^^ௌ^ைమିௌ^ൌ 0.30 ^^ ∙ ^^ିଶis used as a reference.

[0124] Each additional reference is a potential source of random error. Literature value for ^^ௌ^ைమି^௨is 1.04 ^^ ∙ ^^ିଶ, a 13% lower than the value found using the techniques set forth herein. Yet, ^^ௌ^ைమିௌ^is 0.24 ^^ ∙ ^^ିଶwhich is 20% lower than the value found using the techniques set forth herein. The discrepancy might arise primarily from the discrepancy in the reference values.

[0125] More so, in the sessile-drop method, to accurately measure the contact angle, the substrate stays planar and thus solid throughout the measurement, which can be challenging while in contact with the high-temperature melt. Additionally, openness to the ambient atmosphere frequently complicates temperature control. It can introduce unwanted chemical reactions affecting the conclusiveness of the sessile-drop method. At the same time, in the fiber-AVG-IM method, the measured melt can be fully encased in a thermally and chemically isolating cladding.

[0126] Guiding principles for the choice of ^^^^ ^^^ and ^^ଶ^ ^^^ in FIG.7, image (d) 29 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0127] ^^^^ ^^^ was constructed such that it reaches the softening point of silica (~1900 K) very rapidly, significantly faster than ^^ଶ^ ^^^, but attains ^^^^௫much slower than it, with the intention to examine the “globality” vs. “locality” of the nature of the predictable AVG capillary instability. On the one hand, to solve Navier-Stokes equations numerically, the viscosities of materials are defined globally, i.e., everywhere in the solution domain, while AVG-IM, in the way it is formulated, is fully defined by the local viscosities around the pinch-off location. In the former approach, the viscosities of melts are important everywhere in the solutions domain and are all considered to contribute to the development of the instability, while in the latter – only the viscosities in the vicinity of the actual pinch-off count.

[0128] A comparison of the breakup resulting from the two profiles reveals that the local scenario is more accurate, not only for AVG-IM, but also for ab-initio, which simplifies the numerical calculations and thus shrinks otherwise very extended calculation times.

[0129] AVG-IM application to the Si- and Cu-core breakup in a pure hydrogen flame

[0130] AVG-IM fitting to the set of the three Si-core data points collected for ^^ଶ ൌ1.2 ^^ / ^^ ^^ ^^, shown in FIG. 8, image (a), yields a temperature profile according to Eq. (6) with^^^^௫ൌ 2185^^^ and ^^ ൌ 1275 ^^ ^^.

[0131] To analyze the results for the Cu-core breakups in FIG. 8, image (b), (see Supplementary Note 10) that the gas flow of ^^ଶൌ 0.4 ^^ / ^^ ^^ ^^ results in geometrically similar but overall colder temperature profile than that of ^^ଶൌ 1.2 ^^ / ^^ ^^ ^^. Thus, to find the temperature profile for ^^ଶൌ 0.4 ^^ / ^^ ^^ ^^ ^^ is fixed at 1275 ^^ ^^, found for ^^ଶൌ 1.2 ^^ / ^^ ^^ ^^ data, and the ^^^^௫is lowered, until the best fit is obtained to the Si data point for the same temperature conditions as the Cu-core breakups under analysis. It is the 2 ^^ ൌ 4 ^^ ^^ Si-core data point in FIG.8, image (b). This results in the temperature profile with ^^^^௫ൌ 2065 ^^ for ^^ଶൌ 0.4 ^^ / ^^ ^^ ^^.

[0132] Assessment of the validity limits of AVG-IM based in Si-core breakup experiments

[0133] The set of conditions that resulted in the random breakup (see Supplementary Note 10, Figure 1g), and thus which cannot be represented as a point in FIG.8, image (a), are: ^^|௫್ೠൌ ^^ ^^ ^^ ^^ ^^ ^^: Si-core, H2flow rate = 1.2 l / min, 2 ^^ ൌ 0.5 ^^ ^^, and ^^^ൌ 10 ^^ ^^ / ^^ ^^ ^^.with another set of conditions in FIG. 8, image (a), yielding a predictable breakup, ^^|௫್ೠ ൌ 910 േ25 ^^ ^^: Si-core, H2 flow rate = 1.2 l / min, 2 ^^ ൌ 0.5 ^^ ^^, and ^^^ ൌ 50 ^^ ^^ / ^^ ^^ ^^, it allows access tothe validity-limitation of AVG-IM experimentally, violated when the feed speed becomes too 30 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) slow. Specifically, note that the AVG-IM validity limitation VL2, written as ^^^௨~ ^^^∙ ^^^^^^^^ w, defines a characteristic feed speed, ^^^ଶ^ ^^^ w ^^^^ ^^, ^^^ ^ ≡^^^^^^^ൌ ^^ ∙^^ ^^ ^^^ ^^^|௫್ೠୀ௪which is the feed speedradius, results in the pinch location ^^^௨occurring exactly the distance of the boundary width ^^ into the liquefaction zone. It defines the feed speed at which the breakup is safely predictable. If the feed speed is much faster than ^^^^^ଶ, the core is expected to reach the uniform temperature too soon and fall into a Tomotika- like jetting regime. Experimentally, it can be seen that the opposite is also true: the feed speed much slower than ^^^^^ଶresults in the core not experiencing a viscosity gradient high enough by the time of the breakup, and thus also falling into an effectively Tomotika-like jetting regime.

[0134] Once the temperature profile for ^^ଶൌ 1.2 ^^ / ^^ ^^ ^^ is obtained from AVG-IM fitting, it is calculated that ^^^^^ଶൌ 67 ^^ ^^ / ^^ ^^ ^^ for 2 ^^ ൌ 0.5 ^^ ^^. Knowing that ^^^ൌ 50 ^^ ^^ / ^^ ^^ ^^ yields a predictable breakup for that core radius, while for ^^^ൌ 10 ^^ ^^ / ^^ ^^ ^^ the breakup is random, one may assess that feed speeds equal or larger^^ଶof v^fall safely in the predictable breakup range, while feed speeds equal or smaller than 15% of v^^ଶ^ fall safely out of the predictable breakup range. For the comparison, calculating v^^ଶ^ ൌ 8 μm / sec for 2a ൌ 4 μm for the same temperature profile, it is verified that for the same feed speed ^^^ൌ 10 ^^ ^^ / ^^ ^^ ^^ that results in the random breakup in the core of 2 ^^ ൌ 0.5 ^^ ^^, the breakup is indeed expected to be predictable, in alignment with the findings in FIG.8, image (a), since ^^^ൌ 1.25 ∙ ^^^^^ଶin that case, high enough not to fall into an unpredictable range. For 2 ^^ ൌ 4 ^^ ^^ 75% of ^^^^^ଶcorresponds to ^^^ൌ 6 ^^ ^^ / ^^ ^^ ^^ and 15% of ^^^^ଶ ൌ 1.2 ^^ ^^ / ^^ ^^ ^^, the last of which is even below the slowest feed s ^^^^ peed ^^^ ൌby the experimental setup (see Supplementary Note 10).

[0135] Fixed attractor in the chaotic phase space of a submicron core breakup

[0136] It is qualitatively observed that once the core thickness 2 ^^ is small enough and the feed speed ^^^is slow enough for the breakup to fall into the random regime ^^^≪ ^^^^^ଶ^^^, ^^^ ^^^^, the instability becomes more and more periodic as the temperature of the liquefaction zone ^^^^௫is reduced. For the comparison, a random breakup in FIG.1, image (g) results from a relatively hot flame ^ ^^ଶൌ 1.2 ^^ / ^^ ^^ ^^), while that in FIG. 1, image (e) from a cold one ( ^^ଶൌ 31 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) 0.18 ^^ / ^^ ^^ ^^; ^^ଶൌ 0.07 ^^ / ^^ ^^ ^^). The second flame is an extreme case in which the heat deposited to the fiber by the flame is marginally sufficient to liquify the cladding. When ^^ଶflow in this flame was reduced by as small as 0.01 ^^ / ^^ ^^ ^^, the breakup in the core would stop occurring, and the core would pass the flame intact. In these marginally-liquefying conditions, the breakup becomes strictly periodic, approaching Plateau–Rayleigh limit of ^^ ൌ 1 / 2 ^^ ^^, which for submicron ^^ is technologically useful for the production of photonic gratings and optical metamaterials in fibers.

[0137] Though, the periodicity of the submicron breakup does not make it predictable, at least not in the AVG-IM sense. The submicron breakup is chaotic, yet with the reduction of the flame temperature concentrates towards a so-called fixed-point attractor, a well-concentrated preferred region in the otherwise chaotic phase space, resulting in a statistically monodisperse breakup.

[0138] Supplementary Notes

[0139] Supplementary Note 1. Model derivation – Geometrical Argument

[0140] This section describes the mathematical steps in the model derivation through a geometrical argument. As mentioned previously, the present disclosure seeks to describe the capillary breakup behavior in a regime that is not contemplated by traditional models, as described in the main text: namely of viscous fluids in a non-isothermal regime, where the fluid thread is fed through a hot zone, usually a flame or laser, which induces a propagating temperature gradient. Once the fiber is inside the hot zone of the flame, the Si core liquefies, and the silica cladding softens. While the core plus cladding system continues to move through the hot zone in the axial direction x, surface tension minimization drives the pinching-off of the core into droplets at an approximately constant pinching location ^^^௨for each feed speed (FIG. 2), as is observed experimentally. At this location, the core thread reshapes into a droplet and pinches off while a critical amount of material is fed into the droplet. This results in uniformly spaced spheres with a wavelength ^^.

[0141] To understand the process shown above, consider that the amount of core material that is fed into the droplet before it breaks off the core at the time ^^ = ^^^^^^^(pinch time, defined above) is equivalent to that stored in a section of a core of length ^^ being fed for a time ^^^(growth time). This relationship is mathematically expressed as: 32 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) ^^^^ ൌ^^ (S1)^, where ^^ may be a function of interface the derivative of the neck thickness around^^^௨.

[0142] FIG.2, image (c) shows an illustration of the pinch-off process: Once the length ^^ is fed through the pinching front at ^^^௨, a breakup occurs, and at this time, the neck thickness is zero. One can also define the pinching speed ^^^as the ratio between ^^ᇱand the pinch-off time ^^ where ^^ᇱis the difference between the thread radius ^^, and the neck half-thickness ^^^ / ଶat the front location.

[0143] In a position ^^^௨^ ^^, where ^^ is an infinitesimally small quantity: at the same moment as the pinching is occurring at ^^^௨, it has not yet occurred at ^^^௨^ ^^. Moreover, the material has begun to be fed through ^^^௨^ ^^ later than at ^^^௨by a factor of ^^ᇱൌ^^ൗ ^^^. In this way, an “excessive pinching time” ^^ାis defined as the pinching time ^^ atᇱ^^ : ^^ାൌ ^^^ ^^ െ ^^, ^^^௨ ^ ^^^ െ^^ െ ^^^^ (S2)Similarly, for the pinching front ^^^௨െ ^^: ^^ିൌ ^^^ ^^ ^ ^^, ^^^௨ െ ^^^ െ^^ ^ ^^^^ (S3)

[0144] By multiplyingthe locations ^^^௨േ ^^ is obtained, at the moment of pinch-off, which are expressed in Eq. (S2) and (S3): ା^^ ^ ^^ 1^^QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0145] As stated above, the derivative of the neck thickness at ^^^௨is smooth and equal to 0: ^^ ^^ ^^^ ^^ ^1 െ ^^ െ ^^ 1^^^ ^^^ ^^ െ ^^, ^^^௨ ^ ^^^^^ ൌ 0 (S6)Since ^^ ൌ ^^^^^, ^^^, the^^ ^^ ൌ^^ ^^ ^^ ^^ ^^^ ^^ ^^ ^^ ൌ ൬^^ ^^ െ^^ ^^^ ^^(S8)so Eq. (S7) can be rewritten^^^ ^^, ^^^௨^ ^ ^^ ൬^^ ^^െ^^ ^^^ ^^ ൌ 0 (S9)Substituting ^^ from Eq. (S1),1^^^^^ ^൬^^ ^^െ ^^^^ ^^^ ^^^ ^^, ^^^௨^ ൌ 0(S10)Thus, from Eq. (S1) and (S10), the following set of two equations in two variables are obtained, which result in the solution: ì൭^^െ^^^ ^^^^อ ൌ 0Solving this system,that ^^^^^ௗ≫ ^^^^^^, ^^ ∼^^ ^^ ^^^^^ௗ, where ^^ is a system-dependentmaterials. In some specific cases, for large viscosity contrast, ^^ converges asymptotically to a constant derived by solving the Marginal Stability Criterion for propagating Plateau-Rayleigh instability case, and to a different constant assuming the classical Tomotika's growth rate. 34 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0146] Feeding Eq. (S12) into (S11), and considering that, for silica, ^^^^^ௗൌ ^^^^^ௗ൫ ^^^ ^^^൯ ൌ ^^ ^^^ / ்^௫^, where ^^ ൌ 5.7 ൈ 10ି଼^^ ^^ ⋅ ^^ି^and ^^ ൌ 61812 ^^ are material as:^^ ^^^ / ்^௫^ì^^ ^^ ^^ ï^^^^െ^^^^ ൌ 0)

[0147]

[0148] Starting with ^^ ^^ ^^^^ ^^^^ ^^൬^^ ^^^^ െ^^ ^^ൌ 0 (S14)It holds that^^ ^^ ^^ ^^, ^^௩^^ ^^^ ^^^^^ ^^^ ^ െ^^ ^^ ^^ ^^, ^^^^ ^^ ൌ 0 (S15)with ^^^ ൌ ^^^and ^^ ൌ ^^.^^ ^^ ^^௩^^ ^^, ^^^ ^^^ ^ ^^௩^ ^^ ^^^^ ^ െ ^ ^^^^ ^^௩^^ ^^, ^^^^^ ^^ ^௩^ ^^ ^^^^ ^^^ ^^^^^^^ ^^^ ൌ 0 (S16)Asെ ൬ൌSinceడ^െ ൌ

[0149] It can be௩derivatives on ^^^^^^, ^^^: ^^ ௩^^^ ^^ ^^ ^^, ^^^^ൌ^^^^^^^^െ^^ ^^ ^^^^^ௗ^^And35 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) ^^ ^^௩^^ ^^, ^^^ ൌ^^ ^^^^^ ^^^ ^^ ^^^^^ௗ 1^^ ^^ ^ ^^ െ ^ ൌ (S21)^ ^^ ^^^Therefore,^^ ^^^^^ ^^ ^ ൌ^ (S22)^ ^ ^^

[0150] Supplementary

[0151] The value of the surface tension ^^ௌ^ିௌ^ைమbetween liquid silicon and silica was obtained as follows: In a comparative example, the surface tension ^^ௌ^of molten silicon is measured by the microgravity oscillating drop method and the following relation between ^^ௌ^(in mN / m) and the temperature T, is presented: ^^ௌ^ൌ 733 െ 0.062^ ^^ െ 1687^ (S23) for a temperature range of 733 K to 1890K.

[0152] In another comparative example, the surface tension of silica is ^^ௌ^ைమ ൌ280 ^^ ^^ / ^^ଶ . In a third comparative example, the wettability of silica by molten silicon isinvestigated through the sessile drop method, and the contact angle ^^ of silicon on silica was determined to be 92° at 1703 K. The work of adhesion W is then given by: ^^ ൌ ^^ௌ^^1 െ ^^ ^^ ^^ ^^^ൌ 708 ^^ ^^ / ^^ଶ(S24)With ^^ௌ^ൌ ^^ௌ^^ ^^ௌ^ைଶെ ^^ௌ^ିௌ^ைమ.

[0153] Thus, one can calculate ^^ௌ^ିௌ^ைమas ^^ௌ^ିௌ^ைమൌ ^^ௌ^^ ^^ௌ^ைమെ ^^ ൌ 0.298 ^^ / ^^ଶ ∼ 0.3 ^^ / ^^ଶ(S25)which is consistent with the values found in the comparative examples.

[0154] Supplementary Note 4. Capturing the breakup

[0155] A system for capturing the breakup is illustrated in FIGS.10 and 11.

[0156] FIG.10 illustrates a high-temperature breakup setup. Image (a) is a top-view of the setup, with main elements identified. The gas components are indicated in the red (hydrogen) and green (oxygen) boxes on the bottom of the picture., and are connected using a Twin Welding Hose (Nationaltorch CT-12-3AB). The torch (Nationaltorch 3H Hydrogen / Oxygen Torch 34-10099) is mounted on a Lab jack (Thorlabs L200) and a 1" Translation Stage with Standard Micrometer, 1 / 4"-20 Taps PT (Thorlabs PT1), and has a 0.02” stainless steel tip attached (Nationaltorch MSOX- 0 Part Number 34-10314-2). The fiber is secured on fiber holders (Newport ), which are mounted 36 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) on V-clamps (Thorlabs VC1) attached to pillar posts and holders (Thorlabs ½” optical holders series). Each post is then secured to 1” translation stages (Thorlabs PT1) equipped with a stepper motor (Thorlabs ZST225B), controlled individually by a set of K-Cube controllers (Thorlabs KST101), such that the translation speeds can be set independently. Image (b) is an inset of the fiber and torch tip during a breakup process, where the core of the fiber can be observed as continuous before it is fed through the torch, and broken up into spheres, afterwards. Image (c) is a picture of the setup installed on top of an optical table. A plexiglass cage was installed around the setup to minimize air flow disturbances on the torch and ensure its stability. A boom microscope (Amscope ZM-4TW3-FOD-9M) equipped with a digital camera (Amscope MU900 / Motic Moticam 1000) is used to capture the breakup experiments. Image (d) notes modes of operation: the setup can be used for capillary breakup experiments, when the feed speed vf is equal to the draw speed vd. When vf < vd, the system operates as a scaling setup, and when the stages are moving in opposite directions (for example vf = −vd), as a tapering setup.

[0157] FIG.11 illustrates how the breakup location is measured with respect to the melting position of silicon, which can be determined as a dip in the intensity profile (y-scale on the bottom graph is intensity in arbitrary units, x-scale is the number of pixels starting from right along the red line 580 pixels long on the image, along which the intensity profile was collected). To capture the dip in emissivity associated with the Si melting point, the dynamic range is compromised: it can be noted that the signal saturates around pixel 550, thus emissivity corresponding for Tmaxis not captured. At the same time, if the sensitivity of the camera is reduced, such that there is no saturation of the signal in the frame, the dip in the emissivity corresponding to the melting of silicon blends into the noise and may not be captured reliably.

[0158] Supplementary Note 5. Model fitting to Experimental data

[0159] To determine a set of temperature profile parameters to fit the model to the experimental results, a simple algorithm was developed: for any given combination of temperature profile parameters, the sum of the square difference (ssd) between the model prediction and the experimental results can be easily calculated. By sweeping the parameters ^^^^௫and ^^, a map is generated, which allows the identification of the parameters that yield the best model fit to the data. This was performed for each value of ^^ evaluated.

[0160] Supplementary Note 6. COMSOL evaluation of the numerical constant β: 37 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0161] Through COMSOL simulations, ^^ was evaluated by comparing the pinch-off time of the first sphere off the initially continuous room-temperature core when exposed at time=0 to a static localized heat source set to generate various temperatures T. This is illustrated in FIG. 12, which shows an example map of the sum of the ssd for λbuand +xbuwith respect to the experimental values, by sweeping the temperature parameters Tmaxand w. A 4 μm-thick silicon core surrounded by a silica cladding was considered, exposed to a reference temperature profile. The surface tension between silica and silicon was set to ^^ ൌ 0.3 N / m. The results, shown in FIG. 13, indicate good agreement between the COMSOL simulation results and the model predictions for ^^ ൌ 60 േ 10, with respect to the time needed for the first pinch-off in a stand-still initially continuous fiber core to occur while at time zero exposed to the hydrogen-oxygen flame. In FIG.13, an example graph of pinch-off time tpinchas function of temperature T a is shown for a 4 μm-thick silicon core surrounded by silica, under surface tension of 0.3 N / m. The results obtained from COMSOL simulations are compared to the AVG Instability model predictions under assumption of different β values, sowing best agreement for β = 60 ± 10.

[0162] Supplementary Note 7. Capillary breakup simulation dependence on the computational resource:

[0163] At jdj the simulations were set to run on two cores of a single CPU. At BR3 the simulations were set to run on 32 CPUs, two cores on each. The short-term acceleration is depicted in FIG. 14. FIG. 14 shows the ab-initio simulation code on Indiana University’s BR3 supercomputer and comparison with a simulation result obtained, for the same simulation parameters, at MIT’s jdj cluster. Simulations were run for a temperature profile of the form T = Tmax – (Tmax – TSi)e-x / w, with Tmax = 2410 K and x = 2735 μm. The long-term acceleration is such that for the breakup process to reach the steady state (~3 breakup periods are simulated), it would take 4-6 months on jdj, compared to 2-3 weeks on BR3 depending on the feed speed simulated – an order of magnitude faster, and fast enough to enable optimization calculations only in combination with AVG-IM model, as is described in FIG.15. In particular, FIG.15 shows a time propagation comparison between BR3 and jdj clusters under the simulation parameters described in Supplementary Note 6, as a function of simulation run time (days since submission). The data shown represents the simulation progress for vf = 30 μm / s from the start (time = 0) up to around the second breakup event (time ≈ 30 simulation units). One unit of time in the simulation is 38 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) equivalent to 2.06 s. In this example it can be seen how the simulations ran at BR3 achieved longer time propagations, i.e., more time is simulated) within shorter run times.

[0164] Supplementary Note 8. Si-core Fiber preform fabrication and draw results:

[0165] An assembled preform for the fabrication of a silicon-core fiber is illustrated in FIG.16 and 17. FIG.16, image (a) illustrates an assembled preform for the fabrication of a silicon- core silica fiber. A silicon rod is inserted in concentric silica tubes and sealed under vacuum with a glass lathe (top). The resulting first-draw cane is demonstrated next to a ruler with 1 mm ticks (bottom). This cane was stacked into a new preform, as is described in the Methods section and redrawn into a fiber on FIG.16, image (b). FIG.17, image (a) shows a SEM image of the silicon fiber cross-section, and image (b) shows details of the silicon core.

[0166] Supplementary Note 9. Determining the framework of parameters relevant to the AVG-IM:

[0167] AVG-IM-relevant parameters can be roughly divided into two groups: the first relates to the fiber, and the second relates to the liquefaction zone it is fed through. The fiber- related properties are core and cladding dimensions, viscosities, and the interface energy between the two (five parameters). The relevant liquefaction zone properties are its maximal temperature and the width of its boundary (two parameters).

[0168] The fiber-related properties are core and cladding dimensions, viscosities, and the interface energy between the two (five parameters). The relevant liquefaction zone properties are its maximal temperature and the width of its boundary (two parameters). A quantitatively conclusive examination of the general applicability of AVG-IM would necessitate an exhaustive numerical and experimental exploration of capillary instability in at least the seven-dimensional space spanned by the parameters above.

[0169] Instead, in this section, the roadmap for assessing the AVG-IM applicability to solving technology-relevant problems is outlined by presenting several specific use-case studies where the seven parameters above vary in the subspace relevant to the engineering of optoelectronic fiber devices and systems. To be technologically relevant, the variation of each parameter may stem from an experimentally realistic scenario.

[0170] Beginning with the two experimentally justifiable simplifications that AVG-IM leads to the following Model Simplifications: 39 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415)

[0171] Model Simplification MS1: The cladding-to-air interface of the fiber is not significantly prone to capillary instability; thus, the cladding can be considered an infinitely large diameter. This simplification is reasonable since the fiber cladding diameter is much wider than the diameters of the in-fiber cores considered in this study. Thus, ^^^^^^^, which is proportional to the liquid thread diameter under consideration, is significantly longer for the cladding than for the cores. Additionally, since the liquefaction zone is localized, and the feed speeds are finite, no section of the fiber dwells in the liquid state for long enough for the instability to develop significantly on the cladding-air interface. In other words, it is assumed that ^^^^^^^of the cladding is much larger than ^^^∙ ^^, where ^^ is the overall width of the liquefaction zone.

[0172] This simplification may be reconsidered case-by-case in scenarios where the cladding diameter is not axially uniform, such as tapered fibers. The experimental verification of its validity for the cladding dimensions of the etalon fiber from FIG.6 is demonstrated in FIG.18. In FIG.18, images (a) and (b) show the breakup of Si-core fibers in the flame resulting from H2 flow of 0.3 l / min and O2 flow of 0.2 l / min: Image (a) presents optical micrographs of the breakup in 280 μm thick silica fiber with 1.5 μm silicon core at varying feed speed; and image (b) presents optical micrographs of the breakup in 280 μm thick silica fiber with 4.0 μm silicon core scaled down in thickness by a factor of √5 once (top, cladding 125 μm, core 1.8 μm) and twice (bottom, cladding 56 μm, core 0.8 μm). Image (b) shows schematics of the scale-down procedure. Images (d) and (e) represent an AVG-IM applicability examination. In particular, image (d) is an AVG- IM fit to the 1.5 μm core breakup results yielding the temperature profile T(x) = Tmax – (Tmax – TSi)e-x / w, with Tmax= 3050 K and w = 2975 μm and image (e) is an extrapolation of AVG-IM to 1.8 μm and 0.8 core breakup compared to the experimental results. Di, ai∶ i = {0,1,2} are cladding diameter and core radius past the scaling step i.

[0173] Here, the breakup in the fiber in FIG.18, image (a) is investigated, which has the same cladding diameter ^^^ൌ 280 ^^ ^^ as that of the etalon fiber in FIG. 6, yet a significantly smaller core diameter 2 ^^ ൌ 1.5 ^^ ^^. Using AVG-IM interpolation (FIG. 18, image (d)), the temperature profile of the hot zone created by the flame was reconstructed. Using the same flame settings, a breakup of the fibers scaled down was performed, as is shown in Fig. 18c, from the etalon fiber such that the cladding dimensions of the scaled-down sections ^^^ൌ 125 µ ^^ and ^^ଶൌ 56 µ ^^ are significantly smaller than ^^^, yet the core dimensions 2 ^^^ൌ 1.8 µ ^^ and 2 ^^ଶൌ 40 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) 0.8 µ ^^ are comparable to 2 ^^ (FIG.18, image (b)). The fiber in FIG.18, image (b) resulted from scaling the etalon fiber (the one used for the experiments in FIG. 6, with a 280 ^^ ^^-thick silica cladding and a 4 ^^ ^^-thick Si core) in two steps according to the procedure schematically depicted in FIG.18, image (c).

[0174] Next, it was verified that AVG-IM extrapolates well to the results of the breakup in fibers with thinned-out claddings (FIG.18, image (e)), thus supporting experimentally that AVG- IM is independent of the cladding dimensions, i.e., MS1 holds. Even if MS1 holds, the cores of the tapered fibers are tapered too, i.e., ^^ ൌ ^^^ ^^^, thus AVG-IM may be verified for varying core dimensions. FIGS.6 and 18 present an experimental verification that AVG-IM describes well the breakup for four discrete core diameters – 4, 1.8, 1.5, and 0.8 µm, thus AVG-IM will hold for adiabatically tapered cores in this range of core dimensions. In the general tapered-fiber case, ^^ ൌ ^^^ ^^^ can be considered to be changing continuously and significantly within a single breakup period. It is expected that the central equation of AVG-IM, Eq. (2), will hold for such tapered cores if Ca in it is replaced with a modified Ca௧^^^^accordingly, i.e., Ca௧^^^^^ ^^, ^^^ ≡ఉ^^௫^ ఎ^^ೌ^^௫^ ௩^ఊఒ.

[0175] Model Simplification MS2: The fiber cores are solid thatabruptly change the aggregation state upon melting, becoming an inviscid liquid. This simplification results in the core-cladding viscosity contrast ^^^^^^ / ^^^^^ௗ→ 0 in the temperature range relevant for the development of capillary instability and allows consideration of only the cladding viscosity ^^^^^ௗ^ ^^^ past the melting point of the core ^^ ൌ 0 to define the AVG-IM. It holds true for a wide variety of core materials relevant for silica-fiber embedded optoelectronics and photonics, including but not limited to semiconducting, metallic, magnetic, optically non- linear, superconducting, thermoelectric, ferroelectric, and piezoelectric.

[0176] The situation is different for the polymer-fiber embedded devices and systems, where amorphous materials, such as chalcogenides or polymers different from this comprising the cladding, are used for the cores functionalizing the fiber optically or electronically. For such fibers ^^^^^^ / ^^^^^ௗis finite. Thus, it is expected that for the AVG-IM to hold true, Ca in Eq. (2) may be ആ^^^^replaced with a modified Caആ^^ೌ^for scenarios considering a finite ηୡ୭୰^ / ηୡ୪ୟ^, starting with a basis ആ^^^^point in ^^^ ≡ ^^ ∙ ^^ ∙ ^^ ^^^in^is41 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) ആ^^^^the Tomotika instability rate. This choice is guided by the fact that Caആ^^ೌ^^ ^^, ^^^ converges towardsCa^ ^^, ^^^ for ^^^^^^ / ^^^^^ௗ → 0, as is demonstrated in Methods.

[0177] Supplementary Note 10. Si vs. Cu breakup in a pure

[0178] Presented in this section is the investigation of Si- and Cu-core fibers in pure hydrogen flame for the ability to yield same-wavelength breakup in disparate-material cores, scanning the feed speed, core radius, and maximal temperature of the liquefaction zone. FIG. 8, images (a) and (b) show the subsets of the data set in Supplementary Note 10 that were collected in identical temperature profile conditions, and thus can be analyzed by AVG-IM, resulting from H2flow rate of 1.2 l / min and 0.4 l / min, respectively.

[0179] FIG.19 illustrates, in image (a), a fiber cross-section SEM: silica cladding of 288 μm, Si sore of 0.5 μm core; and in image (b), breakup results optical micrographs, side-view of the fiber: breakup conditions - H2flow of 1.2 l / min, varying feed speed. Red rectangle frames the data used for FIG.8, image (a). It can be seen that 0.5 μm core Silicon breakup at 10 μm / sec is chaotic, out of the Validity Limitations of AVG-IM. Elongated droplets result from the flame being too narrow for the full reshaping into a sphere to take place past the pinch-off before the droplet leaves the flame.

[0180] FIG. 20 illustrates, in image (a), fiber cross-section SEM: silica cladding of 282 μm, Si sore of 4 μm core; and in image (b), breakup results optical micrographs, side-view of the fiber: breakup conditions - feed speed 10 μm / sec, varying H2flow. Red rectangles frame the data used for FIG.8, images (a) and (b). Elongated droplets result from the flame being too narrow for the full reshaping into a sphere to take place past the pinch off before the droplet leaves the flame.

[0181] FIG.21 illustrates, in image (a) fiber cross-section SEMs: silica claddings of 294 and 299 μm with Cu cores of 6.3 and 3.4 μm cores, respectively; and in image (b), breakup results optical micrographs, side-view of the fiber: breakup conditions - feed speed 10 μm / sec, H2 flow of 0.4 l / min. The data is used for FIG.8, image (b). Elongated droplets result from the flame being too narrow for the full reshaping into a sphere to take place past the pinch before the droplet leaves the flame.

[0182] FIG. 22 illustrates a heuristic comparison of the flame resulting in the hydrogen flow of 0.4 l / m vs.1.2 l / m. The flame height grows with the increase in the gas flow rate. Yet, the fiber position in the flame for every consecutive breakup experiment is adjusted to stay at the flame 42 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) tip. Thus, the hot zone width and its geometry are roughly preserved. The heat deposition and, thus, the maximal temperature of the hot zone increases with the gas flow rate, resulting in increased emissivity of the core material.

[0183] From the results in FIGS.19-21, it can be seen that there are at least three sets of conditions yielding very similar breakup wavelengths: ^^|௫್ೠൌ 1380 േ 20 ^^ ^^: Si-core, H2 flow rate = 1.2 l / min, 2 ^^ ൌ 0.5 ^^ ^^, and ^^^ൌ 100 ^^ ^^ / ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^for a staggered diode assembly scenario because they yield a similar breakup period at the same feed speed.

[0185] Example Implementations

[0186] The above-described methodologies may be used to, among other things, form arrays of optoelectronic devices and / or to measure the interfacial tension between two melts. FIGS. 23 and 24 illustrate two such methods that implement the above methodologies.

[0187] In FIG. 23, a method 2300 of forming an array of optoelectronic devices is illustrated. The method 2300 includes an operation 2302 of providing an optical fiber. In examples, the optical fiber includes a first core, a second core displaced from the first core in a transverse direction (that is, perpendicular to an axial direction of the fiber), and a cladding. The first and second cores may be formed of a semiconductor material (which may be the same as or different from one another), and the first and second cores may have opposite conductivity types. In another example, the first core may be formed of a metal and the second core may be formed of a semiconductor (for example, when the method 2300 is used to form a series of Schottky diodes). The optical fiber is not limited to only two cores, and in some examples the optical fiber may further include a third core and a fourth core disposed such that the first and second cores are disposed between the third and fourth cores. Where third and forth cores are present, they may be formed of a metal (which may be the same metal or a different metal).

[0188] At operation 2304, heat is applied from a heat source. The heat may be applied at a first position along the axial direction of the optical fiber at a predetermined temperature profile, which may be predicted by an instability model. In such implementations, the instability model 43 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) may receive a radius of the first core, a viscosity of the cladding, and a surface tension between the first core and the cladding as arguments. In some examples, the instability model may further include a radius of the second core and a surface tension between the second core and the cladding as arguments. In one example, the temperature profile may have a maximum temperature that is greater than a melting point of the first and second cores. In another example, the maximum temperature may be greater than the melting point of the first core but less than the melting point of the second core, or vice versa. Where third and fourth cores are present, the maximum temperature may be less than a melting point of the third and fourth cores. In operation 2304, the heat may also be applied at a second position along the axial direction (e.g., via a second heat source) and a second predetermined temperature profile, which may also be predicted by the instability model.

[0189] At operation 2306, the optical fiber is moved along the axial direction thereof at a predetermined speed, for example as predicted by the instability model. The movement and speed may be such that a path of the optical fiber passes through the first position and such that a series of droplet pairs form at a second position along the axial direction of the optical fiber. The series of droplets may be separated by a period in the spatial dimension, and the temperature profile may have a hot zone with a sharp boundary. The length of the boundary may be approximately equal to the period. The second position may be located within the hot zone. Where heat is applied at both first and second positions, the movement may be such that the optical fiber passes through both positions. In such implementations, the second temperature profile may have a second hot zone with a sharp boundary, the length of which may be approximately equal to the second period. Additionally in such implementations, operation 2304 and / or operation 2306 may include tuning the second temperature profile, thereby to cause the second core to break into a second series of droplets respectively separated by a second period. The second period and the first period may be equal to one another. The distance between the first and second positions may additionally or alternatively be tuned (e.g., increased or decreased), thereby to cause the series of droplets to be in phase with one another.

[0190] In FIG.24, a method 2400 of measuring an interfacial tension between a first melt and a second melt is provided. The method 2400 includes an operation 2402 of providing an optical fiber. In examples, the optical fiber includes a core made of a first material (e.g., corresponding to 44 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) the first melt) and a cladding made of a second material (e.g., corresponding to the second melt). In one example, the second material is silica.

[0191] At operation 2404, heat is applied from a heat source. The heat may be applied at a first position along the axial direction of the optical fiber. In operation 2404, the optical fiber may be subjected to a maximum temperature that is greater than a melting point of the first material. Next, at operation 2406, the optical fiber is moved along the axial direction thereof at a predetermined speed, such that a path of the optical fiber passes through the first position and such that a series of droplet pairs form separated by a period along the axial direction of the optical fiber. This period is measured at operation 2408.

[0192] At operation 2410, several parameters are provided to an instability model that has been calibrated based on a surface tension between a reference material and the second material. In examples, the reference material is silicon. The parameters may include the predetermined speed, the period (as measured in operation 2408), a radius of the core, and a viscosity of the cladding. At operation 2412, the calibrated instability model measures the surface tension between the first material and the second material, which corresponds to the interfacial tension between the first melt and the second melt. This measured surface tension may be used to manufacture an electronic device, for example using the method 2300 of FIG.23.

[0193] The present disclosure has described one or more preferred embodiments. However, the invention has been presented by way of illustration and is not intended to be limited to the disclosed embodiments. It should be appreciated that many equivalents, alternatives, variations, and modifications, aside from those expressly stated, are possible and within the scope of the invention. 45 QB\144578.00415\92225795.1

Claims

Docket No.2024-024-02 (144578.00415) CLAIMS What is claimed is:

1. A method of forming an array of optoelectronic devices, the method comprising: providing an optical fiber, the optical fiber including a first core, a second core displaced from the first core in a transverse direction of the optical fiber, and a cladding; applying heat from a heat source at a first position along an axial direction of the optical fiber at a predetermined temperature profile predicted by an instability model, wherein the instability model receives a radius of the first core, a viscosity of the cladding, and a surface tension between the first core and the cladding as arguments; and moving the optical fiber along the axial direction of the optical fiber at a predetermined speed predicted by the instability model, such that a path of the optical fiber passes through the first position and such that a series of droplet pairs form at a second position along the axial direction of the optical fiber.

2. The method of claim 1, wherein the first core and the second core are formed of a semiconductor material, and wherein a conductivity type of the first core is opposite a conductivity type of the second core.

3. The method of claim 1 or claim 2, wherein the optical fiber further includes a third core and a fourth core, wherein the first core and the second core are disposed between the third core and the fourth core.

4. The method of claim 3, wherein the third core is formed of a first metal and the fourth core is formed of a second metal.

5. The method of claim 3 or claim 4, wherein a maximum temperature of the predetermined temperature profile is greater than a melting point of the first core and the second core, and is less than a melting point of the third core and the fourth core. 46 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) 6. The method of any one of claims 1 to 5, wherein the series of droplet pairs are separated by a period, the predetermined temperature profile has a hot zone with a sharp boundary, wherein the boundary has a length, and the period is approximately equal to the length.

7. The method of any one of claims 1 to 6, wherein the predetermined temperature profile has a hot zone, and the second position is within the hot zone.

8. The method of any one of claims 1 to 7, wherein both droplets of a respective droplet pair contact one another to form a pn diode.

9. A method of forming an array of optoelectronic devices, the method comprising: providing an optical fiber, the optical fiber including a first core, a second core displaced from the first core in a transverse direction of the optical fiber, and a cladding; applying heat from a first heat source at a first position along an axial direction of the optical fiber at a first predetermined temperature profile predicted by an instability model, wherein the instability model receives a radius of the first core, a radius of the second core, a viscosity of the cladding, a surface tension between the first core and the cladding, and a surface tension between the second core and the cladding as arguments; applying heat from a second heat source at a second position along the axial direction of the optical fiber at a second predetermined temperature profile predicted by the instability model; moving the optical fiber along the axial direction of the optical fiber at a predetermined speed predicted by the instability model, such that a path of the optical fiber passes through the first position and the second position; and tuning the second temperature profile, thereby to cause the first core to break into a first series of droplets respectively separated by a first period and to cause the second core to break into a second series of droplets respectively separated by a second period, wherein the first period is equal to the second period.

10. The method of claim 9, further comprising: tuning a distance between the first position and the second position, thereby to cause the first series of droplets to be in phase with the second series of droplets. 47 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) 11. The method of claim 10, wherein tuning the second temperature profile comprises increasing the distance.

12. The method of any one of claims 9 to 11, wherein the first core is formed of a first material and the second core is formed of a second material different from the first material.

13. The method of claim 12, wherein a maximum temperature of the first predetermined temperature profile is greater than a melting point of the first core, and is less than a melting point of the second core.

14. The method of claim 12, wherein a maximum temperature of the second predetermined temperature profile is greater than a melting point of the first core and a melting point of the second core.

15. The method of any one of claims 9 to 14, wherein the first series of droplets are separated by a first period, the first predetermined temperature profile has a first hot zone with a first sharp boundary, wherein the first boundary has a first length, and the first period is approximately equal to the first length.

16. The method of any one of claims 9 to 15, wherein the second series of droplets are separated by a second period, the second predetermined temperature profile has a second hot zone with a second sharp boundary, wherein the second boundary has a second length, and the second period is approximately equal to the second length.

17. The method of any one of claims 9 to 16, wherein the first predetermined temperature profile has a first hot zone, and the predetermined speed is such that the first core breaks into the first series of droplets at a position within the first hot zone. 48 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) 18. The method of any one of claims 9 to 17, wherein the second predetermined temperature profile has a second hot zone, and the predetermined speed is such that the second core breaks into the second series of droplets at a position within the second hot zone.

19. The method of any one of claims 9 to 18, wherein one of the first core and the second core is formed of a metal, and the other of the first core and the second core is formed of a semiconductor.

20. The method of claim 19, wherein respective ones of the first series of droplets contact corresponding ones of the second series of droplets, thereby to form a series of Schottky diodes.

21. A method of measuring an interfacial tension between a first melt and a second melt, the method comprising: providing an optical fiber, the optical fiber including a core made of a first material and a cladding made of a second material; applying heat from a heat source at a position along an axial direction of the optical fiber; moving the optical fiber along the axial direction of the optical fiber at a predetermined speed, such that a path of the optical fiber passes through the first position and such that a series of droplet pairs form separated by a period along the axial direction of the optical fiber; measuring the period; providing the predetermined speed, the period, a radius of the core, and a viscosity of the cladding to an instability model, wherein the instability model has been calibrated based on a surface tension between a reference material and the second material; and measuring, by the calibrated instability model, a surface tension between the first material and the second material.

22. The method of claim 21, wherein the second material is silica and the reference material is silicon. 49 QB\144578.00415\92225795.1Docket No.2024-024-02 (144578.00415) 23. The method of claim 21 or 22, wherein the operation of applying heat includes subjecting the optical fiber to a maximum temperature greater than a melting point of the first material.

24. The method of any one of claims 21 to 23, further comprising: manufacturing an electronic device based on the measured surface tension between the first material and the second material. 50 QB\144578.00415\92225795.1

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