HOLLOW-CORE OPTICAL FIBERS AND METHODS FOR PRODUCING THE SAME

Abstract

A method includes heating a hollow-core preform comprising an outer tube and an inner tube. The outer tube includes an inner radius r.sub.ocp and an outer radius R.sub.ocp. The inner tube includes an inner radius r.sub.cp and an outer radius R.sub.cp. The method further includes drawing a hollow-core optical fiber from the hollow-core preform at a draw tension T.sub.g in grams, thereby elongating the outer tube into an outer cladding of the hollow-core optical fiber and the inner tube to a capillary of the hollow-core optical fiber. The draw tension T.sub.g and/or a differential capillary pressure p.sub.c are selected at least in part based on a non-dimensional parameter

[00001]$X_1=\frac{3\left(R_{\mathrm{ocp}}^2-r_{\mathrm{ocp}}^2\right)R_{\mathrm{cp}}\left(p_c{r}_{\mathrm{cp}}-2{}_c\right)}{4{\mathrm{Tr}}_{\mathrm{cp}}\left(R_{\mathrm{cp}}-r_{\mathrm{cp}}\right)},$

where T is the draw tension in dynes and T=981T.sub.g, p.sub.c is in dynes/cm.sup.2, .sub.c in dyne/cm is a surface energy of a glass material forming the inner tube, and 0.5X.sub.10.75.

Claims

1. A method of producing a hollow-core optical fiber from a hollow-core preform, the method comprising: heating a hollow-core preform comprising an outer tube and an inner tube, wherein the outer tube comprises an inner surface defining an interior cavity and an inner radius r.sub.ocp and an outer surface defining an outer radius R.sub.ocp, wherein the inner tube comprises an inner surface defining an interior cavity and an inner radius r.sub.cp and an outer surface defining an outer radius R.sub.cp, wherein the inner tube is formed from a glass material; and drawing a hollow-core optical fiber from the hollow-core preform at a draw tension T.sub.g in grams, thereby elongating the outer tube of the hollow-core preform into an outer cladding of the hollow-core optical fiber and elongating the inner tube of the hollow-core preform into a capillary of the hollow-core optical fiber, wherein the draw tension T.sub.g and a differential capillary pressure p.sub.c are selected at least in part based on a non-dimensional parameter X.sub.1, wherein the differential capillary pressure p.sub.c is defined as a difference between a pressure inside the interior cavity of the inner tube of the hollow-core preform and a pressure inside the interior cavity of the outer tube of the hollow-core preform, wherein X.sub.1 is defined as: $X_1=\frac{3\left(R_{\mathrm{ocp}}^2-r_{\mathrm{ocp}}^2\right)R_{\mathrm{cp}}\left(p_c{r}_{\mathrm{cp}}-2{}_c\right)}{4T{r}_{\mathrm{cp}}\left(R_{\mathrm{cp}}-r_{\mathrm{cp}}\right)},$ where: T is the draw tension in dynes, and T=981T.sub.g; p.sub.c is in dynes/cm.sup.2; and .sub.c in dyne/cm is a surface energy of the glass material forming the inner tube; and wherein X.sub.1 is greater than or equal to 0.5 and less than or equal to 0.75.

2. The method of claim 1, wherein the non-dimensional parameter X.sub.1 is greater than or equal to 0 and less than or equal to 0.7.

3. The method of claim 1, wherein the non-dimensional parameter X.sub.1 is greater than or equal to 0.25 and less than or equal to 0.65.

4. The method of claim 1, wherein the draw tension T.sub.g and the differential capillary pressure p.sub.c are selected at least in part further based on a non-dimensional parameter X.sub.2, wherein X.sub.2 is defined as: $X_2=\frac{3\left(R_{\mathrm{ocp}}^2-r_{\mathrm{ocp}}^2\right)r_{\mathrm{cp}}\left(p_c{R}_{\mathrm{cp}}-2{}_c\right)}{4T{R}_{\mathrm{cp}}\left(R_{\mathrm{cp}}-r_{\mathrm{cp}}\right)},$ wherein X.sub.2 is greater than or equal to 0.35 and less than or equal to 0.6.

5. The method of claim 4, wherein the non-dimensional parameter X.sub.2 is greater than or equal to 0 and less than or equal to 0.55.

6. The method of claim 4, wherein the non-dimensional parameter X.sub.2 is greater than or equal to 0.18 and less than or equal to 0.5.

7. The method of claim 1, wherein: the hollow-core preform further comprises a nested tube in contact with an inner surface of the inner tube of the hollow-core preform, and wherein the nested tube comprises an inner surface defining an interior cavity and an inner radius r.sub.ncp and an outer surface defining an outer radius R.sub.ncp; the drawing further elongates the nested tube of the hollow-core preform to a nested capillary of the hollow-core optical fiber; and the draw tension T.sub.g and a differential capillary pressure p.sub.nc are selected at least in part based on a non-dimensional parameter X.sub.3, wherein the differential nested capillary pressure p.sub.nc is defined as a difference between a pressure inside the interior cavity of the nested tube of the hollow-core preform and a pressure inside the interior cavity of the outer tube of the hollow-core preform, wherein X.sub.3 is defined as: $X_3=\frac{3\left(R_{\mathrm{ocp}}^2-r_{\mathrm{ocp}}^2\right)R_{\mathrm{ncp}}\left(p_{\mathrm{nc}}{r}_{\mathrm{ncp}}-2{}_{\mathrm{nc}}\right)}{4T{r}_{\mathrm{ncp}}\left(R_{\mathrm{ncp}}-r_{\mathrm{ncp}}\right)},$ where .sub.nc is the surface energy of a glass material forming the nested tube; wherein X.sub.3 is greater than or equal to 0.5 and less than or equal to 0.75.

8. The method of claim 7, wherein the non-dimensional parameter X.sub.3 is greater than or equal to 0 and less than or equal to 0.7.

9. The method of claim 7, wherein the non-dimensional parameter X.sub.3 is greater than or equal to 0.25 and less than or equal to 0.65.

10. The method of claim 7, wherein the draw tension T.sub.g and the differential capillary pressure p.sub.nc are selected at least in part further based on a non-dimensional parameter X.sub.4, wherein X.sub.4 is defined as: $X_4=\frac{3\left(R_{\mathrm{ocp}}^2-r_{\mathrm{ocp}}^2\right)r_{\mathrm{ncp}}\left(p_{\mathrm{nc}}{R}_{\mathrm{ncp}}-2{}_{\mathrm{nc}}\right)}{4T{R}_{\mathrm{ncp}}\left(R_{\mathrm{ncp}}-r_{\mathrm{ncp}}\right)},$ wherein X.sub.4 is greater than or equal to 0.35 and less than or equal to 0.6.

11. The method of claim 10, wherein the non-dimensional parameter X.sub.4 is greater than or equal to 0 and less than or equal to 0.55.

12. The method of claim 10, wherein the non-dimensional parameter X.sub.4 is greater than or equal to 0.18 and less than or equal to 0.5.

13. The method of claim 7, wherein the differential nested capillary pressure p.sub.nc is greater than or equal to 2,000 dynes/cm.sup.2 and less than or equal to 50,000 dynes/cm.sup.2.

14. The method of claim 1, wherein: a fiber draw speed is greater than or equal to 1 m/s and less than or equal to 20 m/s; and/or a preform feed rate is greater than or equal to 5 mm/min and less than or equal to 100 mm/min.

15. The method of claim 1, wherein the outer tube of the hollow-core preform satisfies at least one of the following: wherein the outer radius R.sub.ocp of the outer tube of the hollow-core preform is greater than or equal to 7.5 mm and less than or equal to 50 mm; or wherein the inner radius r.sub.ocp of the outer tube of the hollow-core preform is greater than or equal to 2 mm and less than or equal to 10 mm.

16. The method of claim 1, the inner tube of the hollow-core preform satisfies at least one of: wherein the outer radius R.sub.c of the inner tube of the hollow-core preform is greater than or equal to 0.7 mm and less than or equal to 5 mm; or wherein the inner radius r.sub.c of the inner tube of the hollow-core preform is greater than or equal to 0.5 mm and less than or equal to 4 mm.

17. The method of claim 1, wherein the draw tension under which the optical fiber is drawn is between 50 g and 600 g.

18. The method of claim 1, wherein a differential core pressure is between 0.01 and 2 psig.

19. The method of claim 1, wherein the differential capillary pressure p.sub.c is greater than or equal to 5,000 dynes/cm.sup.2 and less than or equal to 100,000 dynes/cm.sup.2.

20. The method of claim 1, wherein the hollow-core optical fiber is drawn in a draw furnace having hot zone ranging between 3 cm and 50 cm.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0008] The accompanying drawings are included to provide a further understanding, and are incorporated in and constitute a part of this specification. The drawings illustrate one or more embodiment(s), and together with the Detailed Description serve to explain principles and operation of the various embodiments. As such, the disclosure will become more fully understood from the following Detailed Description, taken in conjunction with the accompanying figured.

[0009] FIGS. 1A and 1B schematically depict cross-sectional views of exemplary hollow-core optical fibers.

[0010] FIGS. 2A and 2B schematically depict cross-sectional views of exemplary hollow-core preforms for drawing the hollow-core optical fibers of FIG. 1A and FIG. 1B, respectively.

[0011] FIG. 3 schematically depicts an exemplary drawing system.

[0012] FIG. 4 schematically illustrates an implementation of a finite-analytic solution of evolution of preform outer tube inner and outer radii.

[0013] FIG. 5 is a plot showing inner diameters of capillaries drawn as a function of differential capillary pressures during the drawing process for exemplary preform dimensions and operating conditions.

[0014] FIG. 6 is another plot showing inner diameters of capillaries drawn as a function of differential capillary pressures during the drawing process for further exemplary preform dimensions and operating conditions.

[0015] FIG. 7 is another plot showing inner diameters of capillaries drawn as a function of differential capillary pressures during the drawing process for further exemplary preform dimensions and operating conditions.

[0016] FIG. 8 is another plot showing inner diameters of capillaries drawn as a function of differential capillary pressures during the drawing process for further exemplary preform dimensions and operating conditions.

[0017] FIG. 9 is a plot showing the evolution of inner diameters of capillaries from inner tubes of exemplary hollow-core preforms in the neck down region as a function of axial velocity in the neck down region, for different combinations of preform sizes, draw speeds, and differential capillary pressures.

DETAILED DESCRIPTION

[0018] The present disclosure is provided as an enabling teaching and can be understood more readily by reference to the following description, drawings, examples, and claims. To this end, those skilled in the relevant art will recognize and appreciate that many changes can be made to the various aspects of the embodiments described herein, while still obtaining the beneficial results. It will also be apparent that some of the desired benefits of the present embodiments can be obtained by selecting some of the features without utilizing other features. Accordingly, those who work in the art will recognize that many modifications and adaptations are possible and can even be desirable in certain circumstances and are a part of the present disclosure. Therefore, it is to be understood that this disclosure is not limited to the specific compositions, articles, devices, and methods disclosed unless otherwise specified. It is also to be understood that the terminology used herein is for the purposes of describing particular aspects only and is not intended to be limiting.

[0019] In this specification and in the claims which follow, greater than or equal to and are used interchangeably, less than or equal to and are used interchangeably, greater than and are used interchangeably, and less than and are used interchangeably. When a parameter is described as greater than or equal to (or simply, ) a value, the parameter may be greater than (>) the referenced value or equal to (=) the referenced value. Similarly, when a parameter is described as less than or equal to (or simply, ) a value, the parameter may be less than (<) the referenced value or equal to (=) the referenced value.

[0020] Directional terms as used hereinfor example up, down, right, left, front, back, top, bottomare made only with reference to the figures as drawn and are not intended to imply absolute orientation.

[0021] Various components described herein may be referred to as directly connected or indirectly connected. Components are directly connected when they are joined to one another with no intervening structure. Components may be joined by fusing, melting, welding, soldering, adhesives, or any other suitable attachment means. Components are indirectly connected when they are joined to one another with intervening structure. Examples of intervening structure include welding aids (e.g. frits, solders, fluxes), adhesives, and bonding materials. In some embodiments, components connected indirectly are connected only by a welding aid, adhesive, or bonding material. The term connected means directly connected or indirectly connected. Components directly connected to one another are said to be in direct contact with each other. Components indirectly connected to one another are said to be in indirect contact with each other. Components connected to one another are in direct or indirect contact with each other.

[0022] As used herein, the terms upstream and downstream refer to the relative positioning of unit operations with respect to the direction of flow of the process streams. A first unit operation of a system may be considered upstream of a second unit operation if process streams flowing through the system encounter the first unit operation before encountering the second unit operation. Likewise, a second unit operation may be considered downstream of the first unit operation if the process streams flowing through the system encounter the first unit operation before encountering the second unit operation.

[0023] As used herein, the term linear refers to relative distances/lengths between points. A linear distance/length may refer to a distance between two points along a straight line.

[0024] As used herein, the singular forms a, an and the include plural referents in addition to the single referent unless the context clearly dictates otherwise. Thus, for example, reference to a component includes aspects having one such component as well as two or more such components, unless the context clearly indicates otherwise.

[0025] Reference will now be made in detail to various embodiments. Whenever possible, the same reference numerals will be used throughout the drawings to refer to the same or like parts.

Hollow-Core Optical Fiber

[0026] FIG. 1A schematically illustrates an example of a hollow-core optical fiber 100. The hollow-core optical fiber 100 may include an outer cladding 110. The outer cladding 110 may include an inner surface 111 defining an interior cavity 115 and an inner radius r.sub.ocf of the outer cladding 110, and an outer surface 112 defining an outer radius R.sub.ocf of the outer cladding 110.

[0027] The hollow-core optical fiber 100 may further include two or more (e.g., two, three, four, five, six, or more) cladding elements, such as capillaries 120, inside the interior cavity 115 of the outer cladding 110. The capillaries 120 may be in contact with and/or attached to the inner surface 111 of the outer cladding 110. The capillaries 120 may not be in contact with each other and may be evenly spaced along the inner surface 111 of the outer cladding 110. Each of the capillaries 120 may include an inner surface 121 defining an interior cavity 125 and an inner radius r.sub.cf of each of the capillaries 120, and an outer surface 122 defining an outer radius R.sub.ef of each of the capillaries 120.

[0028] In some embodiments, the cladding elements of the hollow-core optical fiber 100 may also include nested capillaries 130 with each disposed inside a capillary 120 and in contact with and/or attached to an inner surface of the capillary 120. Each of the nested capillaries 130 may include an inner surface 131 defining an interior cavity 135 and an inner radius r.sub.acf of each of the nested capillaries 130, and an outer surface 132 defining an outer radius R.sub.ncf of each of the nested capillaries 130. In some embodiments, the hollow-core optical fiber 100 may not include nested capillaries 130, such as shown in FIG. 1B.

[0029] The cladding elements of the hollow-core optical fiber 100, e.g., the capillaries 120 and/or the nested capillaries 130, may surround and define a hollow core 140 of the hollow-core optical fiber 100. The hollow core 140 may be the central portion of the interior cavity 115 and may correspond to the region of the hollow-core optical fiber 100 in which optical signals may be primarily confined and propagate. In some embodiments, the outer cladding 110, the capillaries 120, and/or the nested capillaries 130 may include silica glass and/or silica-based glass (i.e., silica glass comprising one or more dopants).

Hollow-Core Preform

[0030] The hollow-core optical fiber 100 may be produced by drawing a hollow-core preform into fiber. FIGS. 2A and 2B schematically illustrate non-limiting examples of hollow-core preform 200 that may be utilized for producing the hollow-core optical fiber 100 shown in FIGS. 1A and 1B, respectively. In some embodiments, the hollow-core preform 200 may include an outer tube 210. The outer tube 210 may be drawn into the outer cladding 110 of the hollow-core optical fiber 100. The outer tube 210 may include an inner surface 211 defining an interior cavity 215 and an inner radius r.sub.ocp of the outer tube 210, and an outer surface 212 defining an outer radius R.sub.ocp of the outer tube 210.

[0031] In some embodiments, the hollow-core preform 200 may further include two or more (e.g., two, three, four, five, six, or more) inner tubes 220 inside the interior cavity 215 of the outer tube 210. In some embodiments, the inner tubes 220 may be in contact with and/or attached to the inner surface 211 of the outer tube 210. In some embodiments, the inner tubes 220 may not be in contact with each other and may be evenly spaced along the inner surface 211 of the outer tube 210. Each of the inner tubes 220 may be drawn into a capillary 120 of the hollow-core optical fiber 100. Each of the inner tubes 220 may include an inner surface 221 defining an interior cavity 225 and an inner radius r.sub.cp of each of the inner tubes 220, and an outer surface 222 defining an outer radius R.sub.cp of each of the inner tubes 220.

[0032] In some embodiments, such as shown in FIG. 2A, the hollow-core preform 200 may also include two or more nested tubes 230 with each disposed inside an inner tube 220 and in contact with and/or attached to an inner surface of the inner tube 220. Each of the nested tubes 230 may be drawn into a nested capillary 130 of the hollow-core optical fiber 100. Each of the nested tubes 230 may include an inner surface 231 defining an interior cavity 235 and an inner radius r.sub.ncp of each of the nested tubes 230, and an outer surface 232 defining an outer radius R.sub.ncp of each of the nested tubes 230. In some embodiments, the hollow-core preform 200 may not include nested tubes 230, such as shown in FIG. 2B.

[0033] The inner tubes 220 may surround and define a hollow section 240 that may be the central portion of the interior cavity 215 and correspond to the hollow core 140 of the hollow-core optical fiber 100 that may be drawn from the hollow-core preform 200. In some embodiments, the outer tube 210, the inner tubes 220, and/or the nested tubes 230 may include silica glass and/or silica-based glass (i.e., silica glass comprising one or more dopants).

Fiber Drawing System

[0034] FIG. 3 schematically depicts an example of a drawing system 300 that may be utilized to produce the hollow-core optical fiber 100 from the hollow-core preform 200.

[0035] In some embodiments, the drawing system 300 may include a draw furnace 308 configured for receiving and heating the hollow-core preform 200, thereby subjecting the hollow-core preform 200 to a draw temperature. After, and as, the hollow-core preform 200 is subjected to the draw temperature, the hollow-core preform 200 may be necked down, and the hollow-core optical fiber 100 may be drawn from the hollow-core preform 200. The draw furnace 308 may include a hot zone, which is defined as the axial span of the heating elements that are used to heat up the preform for drawing into fiber. An axial length of the hot zone may be greater than or equal to (i.e., ) 3 cm and less than or equal to (i.e., <) 50 cmincluding all sub-ranges or values therebetween. For example, in some embodiments, the axial length of the hot zone may be 3 cm and 50 cm, 3 cm and 40 cm, 3 cm and 30 cm, 3 cm and 20 cm, 3 cm and 10 cm, 10 cm and 50 cm, 10 cm and 40 cm, 10 cm and 30 cm, 10 cm and 20 cm, 20 cm and 50 cm, 20 cm and 40 cm, 20 cm and 30 cm, 30 cm and 50 cm, 30 cm and 40 cm, or 40 cm and 50 cm. In some embodiments, the axial length of the hot zone may be 3 cm, 5 cm, 10 cm, 15 cm, 20 cm, 25 cm, 30 cm, 35 cm, 40 cm, 45 cm, or greater. In some embodiments, the axial length of the hot zone may be 50 cm, 45 cm, 40 cm, 35 cm, 30 cm, 25 cm, 20 cm, 15 cm, 10 cm, 5 cm, or less.

[0036] In some embodiments, the drawing system 300 may further include a pressure control system 310. The pressure control system 310 may be coupled to and configured for pressuring the interior cavities of the outer tube 210, the inner tubes 220, and/or the nested tubes 230 (if present) of the hollow-core preform 200. In some embodiments, the pressure control system 310 may include any or all of a pressure sensor, a vacuum system, a gas pressure source or gas supply, and/or a controller for monitoring the pressure(s) within the various cavities mentioned above and using vacuum and/or gas pressure to maintain the pressure(s) at a desired value(s). In some embodiments, the pressure control system 310 may include a manifold connected to the gas pressure source or gas supply for supplying gas to the hollow-core preform 200. The pressure(s) within the cavities mentioned above may at least in part determine the dimensions, such as radii, of the outer cladding 110, the capillaries 120, and/or the nested capillaries 130 of the hollow-core optical fiber 100, as will be discussed in more detail below.

[0037] In some embodiments, the drawing system 300 may further include a cooling chamber 312 to cool the hollow-core optical fiber 100. In some embodiments, the drawing system 300 may further include a non-contact sensor 314 for measuring the dimension, e.g., diameter of the outer surface 112 of the outer cladding 110. In some embodiments, the drawing system 300 may further include a coating apparatus 316 for applying and/or curing one or more coatings over the outer cladding 110.

[0038] In some embodiments, the drawing system 300 may further include a tension assembly 318, such as tractor, for applying a draw tension to draw the hollow-core optical fiber 100 from the hollow-core preform 200. The draw tension may be controlled via a control apparatus 320 to at least in part maintain the dimension of the hollow-core optical fiber 100 at a predetermined set point. In some embodiments, the drawing system 300 may further include a feedhead 322 for winding the hollow-core optical fiber 100 onto a storage spool 324.

Evolution of Outer Tube to Outer Cladding

(a) Finite Analytical Solution of Evolution of Preform Outer Tube Dimensions During Drawing

[0039] As discussed above, during the drawing of the hollow-core optical fiber, the hollow-core preform may be necked down, and the dimensions, e.g., the inner diameter/radius and outer diameter/radius, of the outer tube in the neckdown region evolve as they are subject to forces of surface tension, core pressure, and draw tension. The neckdown region refers to the region of the hollow-core preform between the axial location where the dimension(s) of the hollow-core preform, e.g., the outer diameter of the hollow-core preform, begins to reduce and the axial location where the final dimension(s), e.g., the outer diameter of the hollow-core optical fiber, is reached.

[0040] The evolution of the outer radius, R.sub.oc, and the inner radius, r.sub.oc, of the outer tube in the neckdown region is given by the following relation:

[00003]$\begin{array}{cc}\frac{d}{\mathrm{dz}}\left(r_{\mathrm{oc}}^2{V}_z\right)=\frac{d}{\mathrm{dz}}\left(R_{\mathrm{oc}}^2{V}_z\right)=\frac{P_{\mathrm{core}}{r}_{\mathrm{oc}}^2{R}_{\mathrm{oc}}^2-{}_{\mathrm{oc}}{r}_{\mathrm{oc}}{R}_{\mathrm{oc}}\left(r_{\mathrm{oc}}+R_{\mathrm{oc}}\right)}{\left(R_{\mathrm{oc}}^2-r_{\mathrm{oc}}^2\right)}& \left[1\right]\end{array}$

where z is the axial distance in the neckdown region, V.sub.z is the axial velocity, P.sub.eore is the differential core pressure, .sub.oc is the surface energy of the glass material forming the outer tube (e.g., 300 dynes/cm for silica glass), and p is the glass viscosity. The surface energy can be measured using the method described in Glass Engineering Handbook, 2nd Edition, by Shand, E. B.; Greene, C. H.; Grant, J. A.; Armistead, W. H., the content of which is incorporated by reference herein. The differential core pressure P.sub.eore, which may also be referred to as the gauge pressure P.sub.core, is defined as the pressure difference between the pressure inside the interior cavity of the outer tube of the hollow-core preform and the atmospheric pressure around the exterior of the outer tube of the hollow-core preform.

[0041] The draw tension under which the hollow-core optical fiber may be drawn from the hollow-core preform is given as:

[00004]$\begin{array}{cc}T=3\frac{d\left(V_z\right)}{\mathrm{dz}}\left(R_{\mathrm{oc}}^2-r_{\mathrm{oc}}^2\right)& \left[2\right]\end{array}$

[0042] Using a Corrocco type transformation and using Equation [2]to transform Equation Eq. [1]with axial velocity as the independent variable, the following is obtained:

[00005]$\begin{array}{cc}\frac{d}{{\mathrm{dV}}_z}\left(r_{\mathrm{oc}}\right)=\frac{3\left(P_{\mathrm{core}}{r}_{\mathrm{oc}}{R}_{\mathrm{oc}}^2-R_{\mathrm{oc}}\left(R_{\mathrm{oc}}+r_{\mathrm{oc}}\right)\right)}{2{\mathrm{TV}}_z}-\frac{r_{\mathrm{oc}}}{2V_z}& \left[3a\right]\end{array}$$\begin{array}{cc}\frac{d}{{\mathrm{dV}}_z}\left(R_{\mathrm{oc}}\right)=\frac{3\left(P_{\mathrm{core}}{R}_{\mathrm{oc}}{r}_{\mathrm{oc}}^2-r_{\mathrm{oc}}\left(R_{\mathrm{oc}}+r_{\mathrm{oc}}\right)\right)}{2{\mathrm{TV}}_z}-\frac{R_{\mathrm{oc}}}{2V_z}& \left[3b\right]\end{array}$

[0043] To capture the evolution of the outer radius R.sub.oc and the inner radius r.sub.oc of the outer tube of the hollow-core preform in the neckdown region, a finite-analytic solution, as shown in FIG. 4, is implemented, where the analytic solution between the ith and the (i+1) the node is given as:

[00006]$\begin{array}{cc}{r}_{\mathrm{oc},j+1}=\left(\frac{3R_{\mathrm{oc},j}^2}{T+3R_{\mathrm{oc},j}-3P_{\mathrm{core}}{R}_{\mathrm{oc},j}^2}\right)+\left(r_{\mathrm{oc},j}-\frac{3R_{\mathrm{oc},j}^2}{T+3R_{\mathrm{oc},j}-3P_{\mathrm{core}}{R}_{\mathrm{oc},j}^2}\right){\left(\frac{V_{z,j+1}}{V_{z,j}}\right)}^{-\left(T+3R_{\mathrm{oc},j}-3P_{\mathrm{core}}{R}_{\mathrm{oc},j}^2\right)/2T}& \left[4a\right]\end{array}$$\begin{array}{cc}{R}_{\mathrm{oc},j+1}=\left(\frac{3r_{\mathrm{oc},j}^2}{T+3r_{\mathrm{oc},j}-3P_{\mathrm{core}}{r}_{\mathrm{oc},j}^2}\right)+\left(R_{\mathrm{oc},j}-\frac{3r_{\mathrm{oc},j}^2}{T+3r_{\mathrm{oc},j}-3P_{\mathrm{core}}{r}_{\mathrm{oc},j}^2}\right){\left(\frac{V_{z,j+1}}{V_{z,j}}\right)}^{-\left(T+3r_{\mathrm{oc},j}-3P_{\mathrm{core}}{r}_{\mathrm{oc},j}^2\right)/2T}& \left[4b\right]\end{array}$

which can be implemented with the mass balance equation as:

[00007]$\begin{array}{cc}{V}_{z,j+1}\left(R_{\mathrm{oc},j+1}^2-r_{\mathrm{oc},j+1}^2\right)=V_{z,j}\left(R_{\mathrm{oc},j}^2-r_{\mathrm{oc},j}^2\right)& \left[5\right]\end{array}$

(b) Exemplary Implementations of the Finite-Analytic Solution

[0044] The finite-analytic solution was implemented for different preform geometries that resulted in the same inner diameter of the outer cladding of the hollow-core fiber (2r.sub.ocf) of about 79.7 m, and the same outer diameter of the outer cladding of the hollow-core optical fiber (2R.sub.ocf) of about 249 m. However, it should be noted that the present disclosure is not limited to these specific fiber target dimensions (i.e., inner diameter of 79.7 m and outer diameter of 249 m), which are used for non-limiting illustrative purposes. The present disclosure may be implemented for achieving other target dimensions of the hollow-core optical fiber from different preform dimensions. For example, the solution described herein may be utilized for achieving any target inner diameter of the outer cladding of the hollow-core optical fiber that may be greater than or equal to (i.e., ) 45 m and less than or equal to (i.e., ) 145 m including all sub-ranges or values therebetween. Similarly, the solution described herein may also be utilized for achieving any target outer diameter of the outer cladding of the hollow-core optical fiber that may be greater than or equal to (i.e., ) 150 m and less than or equal to (i.e., ) 300 mincluding all sub-ranges or values therebetween.

[0045] Table 1 below shows that different combinations of pressure, draw tension, and preform geometries that result in substantially the same inner and outer diameters of the outer cladding of the hollow-core optical fiber. As shown in Table 1, the various outer diameters of the outer tube of the hollow-core preform (2R.sub.ocp) between 17 mm and 50 mm and the various inner diameters of the outer tube of the hollow-core preform (2r.sub.ocp) between 5 mm and 9 mm result in substantially the same outer cladding dimensions of the hollow-core optical fiber, with the fiber draw speeds ranging between 1 m/s and 10 m/s.

TABLE-US-00001 TABLE 1 Fiber Preform Preform Fiber Fiber Differential Preform Fiber Draw OD, ID, OD, ID, Draw Core Feed Draw Speed, 2 R.sub.ocp 2 r.sub.ocp 2 R.sub.ocf 2 r.sub.ocf Tension, Pressure, Rate, V.sub.p Rate, V.sub.f V.sub.f/60000 Ex (mm) (mm) (um) (um) T.sub.g (g) P.sub.core (psig) (mm/min) (mm/min) (m/sec) 1 17.25 5 248.5 79.72 100 0.014 12.5 60000 1.00 2 17.25 5 249.4 79.67 150 0.032 12.5 60000 1.00 3 17.25 5 249.85 79.64 200 0.05 12.5 60000 1.00 4 17.25 5 250.12 79.71 250 0.069 12.5 60000 1.00 5 17.25 5 250.32 79.69 300 0.087 12.5 60000 1.00 6 25.875 7 246.96 79.73 100 0.013 12.5 136858.2 2.28 7 25.875 7 248.625 79.625 150 0.027 12.5 136857.4 2.28 8 25.875 7 248.99 79.81 200 0.04 12.5 136857 2.28 9 25.875 7 249.3775 79.74 250 0.056 12.5 136856.8 2.28 10 25.875 7 249.63 79.69 300 0.07 12.5 136856.6 2.28 11 34.6 8 246.99 79.72 100 0.018 12.5 247134.8 4.12 12 34.6 8 248.719 79.72 150 0.0335 12.5 247132.8 4.12 13 34.6 8 249.6 79.89 200 0.049 12.5 247131.8 4.12 14 34.6 8 250.08 79.77 250 0.064 12.5 247130.2 4.12 15 34.6 8 250.04 79.688 300 0.079 12.5 247130.8 4.12 16 49.6 9 246.42 79.63 100 0.0157 12.5 512464.3 8.54 17 49.6 9 248.74 79.82 150 0.0287 12.5 512458.3 8.54 18 49.6 9 249.88 79.78 200 0.0415 12.5 512455.3 8.54 19 49.6 9 250.58 79.77 250 0.0543 12.5 512453.4 8.54 20 49.6 9 251.03 79.706 300 0.067 12.5 512452.2 8.54

(c) Approximate Solution of Preform Outer Tube Dimensions

[0046] Based on the example shown in Table 1, the following relations can be approximated for the differential core pressure P.sub.eore as a function of draw tension T.sub.g in grams:

[00008]$\begin{array}{cc}{P}_{\mathrm{core}}\left(\mathrm{psig}\right)1.87710^{-5}{T}_g^{1.46}& \left[6\right]\end{array}$

[0047] Further, the following relations can be approximated for the inner radius r.sub.ocp and the outer radius R.sub.ocp of the outer tube of the hollow-core preform:

[00009]$\begin{array}{cc}{r}_{\mathrm{ocp}}\frac{r_{\mathrm{ocf}}}{{\left(V_f/V_p\right)}^{-\left(T+3R_{*}{}_{\mathrm{oc}}-3P_{\mathrm{core}}{R}_{*}^2\right)/2T}}& \left[7a\right]\end{array}$$\begin{array}{cc}{R}_{\mathrm{ocp}}\frac{R_{\mathrm{ocf}}}{{\left(V_f/V_p\right)}^{-\left(T+3r_{*}{}_{\mathrm{oc}}-3P_{\mathrm{core}}{r}_{*}^2\right)/2T}}& \left[7b\right]\end{array}$

where r.sub.ocf and R.sub.ocf are the inner and outer radii, respectively, of the outer cladding of the hollow-core optical fiber, V.sub.p is the hollow-core preform feed rate, V.sub.f is the hollow-core fiber draw rate, T is the draw tension in dynes and T=T.sub.g*981, .sub.oc is the surface energy of the glass material forming the outer tube (taken to be 300 dynes/cm), and r* and R* are given as:

[00010]$\begin{array}{cc}{r}_{*}5\sqrt{r_{\mathrm{ocp}}{r}_{\mathrm{ocf}}}& \left[8a\right]\end{array}$$\begin{array}{cc}{R}_{*}5\sqrt{R_{\mathrm{ocp}}{R}_{\mathrm{ocf}}}& \left[8b\right]\end{array}$

[0048] Based on the approximate solutions above, the inventor has found that when the differential core pressure P.sub.core satisfies the following relation:

[00011]$\begin{array}{cc}0.8P^{*}<P_{\mathrm{core}}<1.2P^{*}& \left[9\right]\end{array}$

where P*=1.87710.sup.5T.sub.g.sup.1.46 consistent target inner radius r.sub.ocf and consistent target outer radius R.sub.ocf of the outer cladding of the hollow-core optical fiber can be achieved with preforms having outer tuber inner radius r.sub.ocp and outer tube outer radius R.sub.ocp that satisfy the following relations:

[00012]$\begin{array}{cc}0.9r_p^{*}<r_{\mathrm{ocp}}<1.1r_p^{*}& \left[10a\right]\end{array}$$\begin{array}{cc}0.95R_p^{*}<R_{\mathrm{ocp}}<1.05R_p^{*}& \left[10b\right]\end{array}$$\mathrm{where}:$$\begin{array}{cc}{r}_p^{*}=\frac{r_{\mathrm{ocf}}}{{\left(V_f/V_p\right)}^{-\left(T+3R_{*}-3P_{\mathrm{core}}{R}_{*}^2\right)/2T}}& \left[11a\right]\end{array}$$\begin{array}{cc}{R}_p^{*}=\frac{R_{\mathrm{ocf}}}{{\left(V_f/V_p\right)}^{-\left(T+3r_{*}-3P_{\mathrm{core}}{r}_{*}^2\right)/2T}}& \left[11b\right]\end{array}$$\mathrm{and}$$\begin{array}{cc}{r}_{*}=5\sqrt{r_p^{*}{r}_{\mathrm{ocf}}}& \left[12a\right]\end{array}$$\begin{array}{cc}{R}_{*}=5\sqrt{R_p^{*}{R}_{\mathrm{ocf}}}& \left[12b\right]\end{array}$

Evolution of Inner Tubes to Capillaries

(a) Evolution of Preform Inner Tube Dimensions During Drawing

[0049] The evolution of the inner radius r.sub.c, and outer radius, R.sub.c, of the inner tube in the neckdown region is described by the following relation:

[00013]$\begin{array}{cc}\frac{d}{\mathrm{dz}}\left(r_c^2{V}_z\right)=\frac{d}{\mathrm{dz}}\left(R_c^2{V}_z\right)=\left(p_c-\frac{{}_c\left(R_c+r_c\right)}{R_c{r}_c}\right)\frac{r_c^2{R}_c^2}{\left(R_c^2-r_c^2\right)}& \left[13\right]\end{array}$

where z is the axial distance in the neckdown region, V.sub.z is the axial velocity, p.sub.c is the differential capillary pressure in dynes/cm.sup.2, .sub.c is the surface energy of the glass material forming the inner tube (e.g., 300 dynes/cm for silica glass), and p is the glass viscosity. The differential capillary pressure p.sub.c is defined as the difference between the pressure inside the interior cavity of each inner tube and the pressure outside the inner tubes, i.e., the pressure inside the interior cavity of the outer tube of the hollow-core preform. Equation [13]for thin capillaries can be transformed with axial velocity in the neck down region, Vz, as the independent variables as:

[00014]$\begin{array}{cc}\frac{d}{{\mathrm{dV}}_z}\left(r_c\right)=\frac{3\left(R^2-r^2\right)R_c\left(p_c{r}_c-2{}_c\right)}{4{\mathrm{TV}}_z\left(R_c-r_c\right)}-\frac{r_c}{2V_z}& \left[14a\right]\end{array}$$\begin{array}{cc}\frac{d}{{\mathrm{dV}}_z}\left(R_c\right)=\frac{3\left(R^2-r^2\right)r_c\left(p_c{R}_c-2{}_c\right)}{4{\mathrm{TV}}_z\left(R_c-r_c\right)}-\frac{R_c}{2V_z}& \left[14b\right]\end{array}$

where T is the draw tension in dynes, and T=T.sub.g*981. [0050] (i) Non-Dimensional Parameters X.sub.1 and X.sub.2

[0051] For characterizing the evolution from the inner tubes of the hollow-core preform to the capillaries of the hollow-core optical fiber during the drawing process, two non-dimensional parameters based on the structure in the starting hollow-core preform are defined as:

[00015]$\begin{array}{cc}{X}_1=\frac{3\left(R_{\mathrm{ocp}}^2-r_{\mathrm{ocp}}^2\right)R_{\mathrm{cp}}\left(p_c{r}_{\mathrm{cp}}-2{}_c\right)}{4{\mathrm{Tr}}_{\mathrm{cp}}\left(R_{\mathrm{cp}}-r_{\mathrm{cp}}\right)}& \left[15\right]\end{array}$$\begin{array}{cc}{X}_2=\frac{3\left(R_{\mathrm{ocp}}^2-r_{\mathrm{ocp}}^2\right)r_{\mathrm{cp}}\left(p_c{R}_{\mathrm{cp}}-2{}_c\right)}{4T{R}_{\mathrm{cp}}\left(R_{\mathrm{cp}}-r_{\mathrm{cp}}\right)}& \left[16\right]\end{array}$

where R.sub.ocp is the outer radius of the outer tube of the starting hollow-core preform, r.sub.ocp is the inner radius of the outer tube of the starting hollow-core preform, R.sub.cp is the outer radius of the inner tube of the starting hollow-core preform, and r.sub.cp is the inner radius of the inner tube of the starting hollow-core preform, p.sub.c is the differential capillary pressure in dynes/cm.sup.2, .sub.c is the surface energy of the glass material forming the inner tube (e.g., 300 dynes/cm for silica glass), and T is the draw tension in dynes, and T=T.sub.g*981.

[0052] The inventor has found that smaller values of the non-dimensional parameters X.sub.1 and X.sub.2 may result in collapse of the capillaries. The inventor has also found that larger values of the non-dimensional parameters X.sub.1 and X.sub.2 may result in very large capillaries and process conditions the small changes of which, such as small change in pressure, can result in much more significant change in the capillary diameters as discussed in more detail below. The inventor has found that ranges of the non-dimensional parameters X.sub.1 and X.sub.2 may be selected (by, e.g., choosing appropriate incoming preform dimensions, maintaining appropriate differential capillary pressure levels, applying appropriate draw tension values, etc.) such that relatively small changes in process conditions would not significantly alter the dimensions of the capillaries significantly, thereby achieving target capillary dimensions with precise during scale-up manufacturing.

[0053] In some embodiments, the non-dimensional parameter X.sub.1 may be greater than or equal to (i.e., )0.5 and less than or equal to (i.e., ) 0.75including all sub-ranges or values therebetweenfor stable drawing of the capillaries of the hollow-core optical fibers. For example, in some embodiments, the non-dimensional parameter X.sub.1 may be 0 and 0.7 for stable drawing of the capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.1 may be 0.25 and 0.65 for stable drawing of the capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.1 may be 0.5 and 0.75, 0.5 and 0.5, 0.5 and 0.25, 0.5 and 0, 0.5 and <0.25, 0.25 and 0.75, 0.25 and 0.5, 0.25 and 0.25, 0.25 and 0, 0 and 0.75, 0 and 0.5, 0 and 0.25, 0.25 and 0.75, 0.25 and 0.5, or 0.5 and 0.75. In some embodiments, the non-dimensional parameter X.sub.1 may be greater than or equal to (i.e., ) 0.5, 0.4, 0.3, 0.2, 0.1, 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, or greater. In some embodiments, the non-dimensional parameter X.sub.1 may be less than or equal to (i.e., ) 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, 0.05, 0, 0.1, 0.2, 0.3, 0.4, or less.

[0054] In some embodiments, the non-dimensional parameter X.sub.2 may be greater than or equal to (i.e., ) 0.35 and less than or equal to (i.e., ) 0.6including all sub-ranges or values therebetweenfor stable drawing of the capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.2 may be 0 and 0.55 for stable drawing of the capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.2 may be 0.18 and 0.5 for stable drawing of the capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.2 may be 0.2 and 0.5 for stable drawing of the capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.2 may be 0.35 and 0.6, 0.35 and 0.4, 0.35 and 0.2, 0.35 and 0, 0.35 and <0.2, 0.2 and 0.6, 0.2 and 0.4, 0.2 and 0.2, 0.2 and <0, 0 and 0.6, 0 and 0.4, 0 and 0.2, 0.2 and 0.6, 0.2 and 0.4, or 0.4 and 0.6. In some embodiments, the non-dimensional parameter X.sub.2 may be greater than or equal to (i.e., ) 0.35, 0.3, 0.2, 0.1, 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, or greater. In some embodiments, the non-dimensional parameter X.sub.2 may be less than or equal to (i.e., ) 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, 0.05, 0, 0.1, 0.2, 0.3, or less.

(ii) Non-Dimensional Parameters X.SUB.3 .and X.SUB.4

[0055] For characterizing the evolution from the nested tubes of the hollow-core preform to the nested capillaries of the hollow-core optical fiber during the drawing process, two non-dimensional parameters based on the structure in the starting hollow-core preform are defined as:

[00016]$\begin{array}{cc}{X}_3=\frac{3\left(R_{\mathrm{ocp}}^2-r_{\mathrm{ocp}}^2\right)R_{\mathrm{ncp}}\left(p_{\mathrm{nc}}{r}_{\mathrm{ncp}}-2{}_{\mathrm{nc}}\right)}{4{\mathrm{Tr}}_{\mathrm{ncp}}\left(R_{\mathrm{ncp}}-r_{\mathrm{ncp}}\right)}& \left[17\right]\end{array}$$\begin{array}{cc}{X}_4=\frac{3\left(R_{\mathrm{ocp}}^2-r_{\mathrm{ocp}}^2\right)r_{\mathrm{ncp}}\left(p_{\mathrm{nc}}{R}_{\mathrm{ncp}}-2{}_{\mathrm{nc}}\right)}{4T{R}_{\mathrm{ncp}}\left(R_{\mathrm{ncp}}-r_{\mathrm{ncp}}\right)}& \left[18\right]\end{array}$

where R.sub.ocp is the outer radius of the outer tube of the starting hollow-core preform, r.sub.ocp is the inner radius of the outer tube of the starting hollow-core preform, R.sub.ncp is the outer radius of the nested tube of the starting hollow-core preform, and r.sub.ncp is the inner radius of the nested tube of the starting hollow-core preform, p.sub.nc is the differential nested capillary pressure in dynes/cm.sup.2, which is defined as the difference between the pressure inside the interior cavity of each nested tube and the pressure outside the nested tubes, i.e., the pressure inside the interior cavity of the inner tubes of the hollow-core preform, an is the surface energy of the glass material forming the nested tube (taken to be 300 dynes/cm), and T is the draw tension in dynes, and T=T.sub.g*981.

[0056] The inventor has found that smaller values of the non-dimensional parameters X.sub.3 and X.sub.4 may result in collapse of the nested capillaries. The inventor has also found that larger values of the non-dimensional parameters X.sub.3 and X.sub.4 may result in very large nested capillaries and process conditions the small changes of which, such as small change in pressure, can result in much more significant change in the capillary diameters as discussed in more detail below. The inventor has found that ranges of the non-dimensional parameters X.sub.3 and X.sub.4 may be selected (by, e.g., choosing appropriate incoming preform dimensions, maintaining appropriate differential nested capillary pressure levels, applying appropriate draw tension values, etc.) such that relatively small changes in process conditions would not significantly alter the dimensions of the nested capillaries significantly, thereby achieving target capillary dimensions with precise during scale-up manufacturing.

[0057] In some embodiments, the non-dimensional parameter X.sub.3 may be greater than or equal to (i.e., ) 0.5 and less than or equal to (i.e., ) 0.75including all sub-ranges or values therebetweenfor stable drawing of the nested capillaries of the hollow-core optical fibers. For example, in some embodiments, the non-dimensional parameter X.sub.3 may be 0 and 0.7 for stable drawing of the nested capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.3 may be 0.25 and 0.65 for stable drawing of the nested capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.3 may be 0.5 and 0.75, 0.5 and 0.5, 0.5 and 0.25, 0.5 and 0, 0.5 and <0.25, 0.25 and 0.75, 0.25 and 0.5, 0.25 and 0.25, 0.25 and 0, 0 and <0.75, 0 and 0.5, 0 and 0.25, 0.25 and 0.75, 0.25 and 0.5, or 0.5 and 0.75. In some embodiments, the non-dimensional parameter X.sub.3 may be greater than or equal to (i.e., ) 0.5, 0.4, 0.3, 0.2, 0.1, 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, or greater. In some embodiments, the non-dimensional parameter X.sub.3 may be less than or equal to (i.e., ) 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, 0.05, 0, 0.1, 0.2, 0.3, 0.4, or less.

[0058] In some embodiments, the non-dimensional parameter X.sub.4 may be greater than or equal to (i.e., ) 0.35 and less than or equal to (i.e., ) 0.6including all sub-ranges or values therebetweenfor stable drawing of the nested capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.4 may be 0 and 0.55 for stable drawing of the nested capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.4 may be 0.18 and 0.5 for stable drawing of the nested capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.4 may be 0.2 and 0.5 for stable drawing of the nested capillaries of the hollow-core optical fibers. In some embodiments, the non-dimensional parameter X.sub.4 may be 0.35 and 0.6, 0.35 and <0.4, 0.35 and 0.2, 0.35 and 0, 0.35 and <0.2, 0.2 and 0.6, 0.2 and 0.4, 0.2 and 0.2, 0.2 and 0, 0 and 0.6, 0 and 0.4, 0 and 0.2, 0.2 and 0.6, 0.2 and 0.4, or 0.4 and 0.6. In some embodiments, the non-dimensional parameter X.sub.4 may be greater than or equal to (i.e., ) 0.35, 0.3, 0.2, 0.1, 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, or greater. In some embodiments, the non-dimensional parameter X.sub.4 may be less than or equal to (i.e., ) 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, 0.05, 0, 0.1, 0.2, 0.3, or less.

[0059] The inventor has found that by selecting the appropriate preform dimensions (e.g., inner and outer radii of the outer tube, inner tubes, and/or nested tubes) and/or the drawing conditions (e.g., draw tension, differential capillary pressures, differential nested capillary pressures, etc.), appropriate values of non-dimensional parameters X.sub.1, X.sub.2, X.sub.3, and X.sub.4 with the various range described above may be obtained for achieving various target dimensions of the capillaries and/or nested capillaries.

(b) Exemplary Implementations

[0060] The below examples are intended to be exemplary and are not intended to limit the scope of the disclosure. Tables 2-5 show the inner and outer diameters of the capillaries of the hollow-core optical fiber drawn as a function of differential capillary pressure for different draw tensions for preforms having outer tube outer diameter ranging between 17 mm and 50 mm and fiber draw speeds ranging between 1 m/s and 10 m/s. In Tables 2-5 below, the differential capillary pressure is shown as Capillary dP in dynes/cm.sup.2.

TABLE-US-00002 TABLE 2-A Preform Outer Tube OD, 2 R.sub.ocp (mm) 17.25 Preform Outer Tube ID, 2 r.sub.ocp (mm) 5 Preform Inner Tube OD, 2 R.sub.cp (mm) 1.1 Preform Inner Tube ID, 2 r.sub.cp (mm) 0.85 Fiber Draw Speed (m/s) 1 Fiber Outer Cladding OD, 2 R.sub.ocf (m) 250 Fiber Outer Cladding ID, 2 r.sub.ocf (m) 79.7

TABLE-US-00003 TABLE 2-B1 Tension (g) 200 P.sub.core (psig) 0.05 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.423 0.326 3.026 6 5000 0.273 0.211 3.76 6.74 10000 0.123 0.095 4.728 7.706 15000 0.026 0.02 6.024 9.002 20000 0.176 0.136 7.851 10.829 25000 0.326 0.252 10.594 13.253 30000 0.476 0.367 15.114 18.09 35000 0.625 0.483 23.749 26.727 40000 0.775 0.599 45.68 48.611 45000 0.925 0.715 174.42 177.39

TABLE-US-00004 TABLE 2-B2 Tension (g) 300 P.sub.core (psig) 0.087 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.282 0.218 4.82 7.8 5000 0.182 0.14 5.49 8.472 10000 0.082 0.063 6.295 9.274 15000 0.017 0.0136 7.271 10.25 20000 0.117 0.091 8.483 11.461 25000 0.217 0.168 10.02 13.007 30000 0.317 0.245 12.03 15.01 35000 0.417 0.322 14.76 17.74 40000 0.517 0.399 18.66 21.69 45000 0.617 0.476 24.61 27.59 50000 0.717 0.554 34.69 37.67 55000 0.817 0.631 54.95 57.93 60000 0.917 0.708 112.71 115.69 65000 1.017 0.785 758.99 761.97

TABLE-US-00005 TABLE 2-B3 Tension (g) 400 P.sub.core (psig) 0.125 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.211 0.163 5.91 8.89 5000 0.136 0.105 6.495 9.472 10000 0.0617 0.0477 7.164 10.14 15000 0.0132 0.102 7.94 10.91 20000 0.088 0.068 8.848 11.826 25000 0.163 0.126 9.923 12.902 30000 0.238 0.184 11.216 14.19 35000 0.313 0.241 12.7957 15.774 40000 0.388 0.2998 14.764 17.74 45000 0.463 0.357 17.278 20.256 50000 0.538 0.415 20.584 23.567 55000 0.612 0.473 25.125 28.1 60000 0.687 0.531 31.68 34.65 65000 0.762 0.589 41.88 44.86 70000 0.837 0.647 59.7 62.68 75000 0.917 0.705 97.61 100.79 80000 0.987 0.763 222.43 225.41

TABLE-US-00006 TABLE 2-B4 Tension (g) 500 P.sub.core (psig) 0.16 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.169 0.1308 6.63 9.61 10000 0.049 0.038 7.713 10.69 20000 0.0705 0.0545 9.081 12.06 30000 0.19 0.147 10.862 13.841 40000 0.31 0.2399 13.264 16.24 50000 0.43 0.332 16.66 19.63 60000 0.55 0.425 21.78 24.76 70000 0.67 0.518 30.31 33.28 80000 0.79 0.61 46.89 49.873 90000 0.91 0.703 90.89 93.87 100000 1.03 0.796 411.773 414.751

TABLE-US-00007 TABLE 3-A Preform Outer Tube OD, 2 R.sub.ocp (mm) 28.875 Preform Outer Tube ID, 2 r.sub.ocp (mm) 7 Preform Inner Tube OD, 2 R.sub.cp (mm) 1.5 Preform Inner Tube ID, 2 r.sub.cp (mm) 1.2 Fiber Draw Speed (m/s) 2.28 Fiber Outer Cladding OD, 2 R.sub.ocf (m) 250 Fiber Outer Cladding ID, 2 r.sub.ocf (m) 79.7

TABLE-US-00008 TABLE 3-B1 Tension (g) 200 P.sub.core (psig) 0.04 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.775 0.62 0.245 2.57 2500 0.5812 0.465 0.606 2.93 5000 0.387 0.31 1.092 3.417 7500 0.193 0.155 1.776 4.1 10000 0 0 2.8 5.125 12500 0.193 0.155 4.473 6.798 15000 0.387 0.31 7.62 9.954 17500 0.5812 0.465 15.5055 17.63 20000 0.775 0.62 53.127 55.452

TABLE-US-00009 TABLE 3-B2 Tension (g) 300 P.sub.core (psig) 0.07 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.516 0.413 1.96 4.285 2500 0.38772 0.31 2.4 4.7671 5000 0.258 0.206 2.948 5.27 7500 0.12924 0.1033 3.641 5.966 10000 0 0 4.54 6.869 12500 0.1292 0.1033 5.766 8.091 15000 0.258 0.206 7.497 9.822 17500 0.3877 0.3101 10.166 12.4409 20000 0.516 0.413 14.476 16.801 22500 0.646 0.516 22.966 25.29 25000 0.775 0.62 45.4208 47.745 27500 0.9046 0.7235 202.946 205.27

TABLE-US-00010 TABLE 3-B3 Tension (g) 400 P.sub.core (psig) 0.1 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.387 0.31 3.193 5.519 5000 0.194 0.155 4.168 6.492 10000 0 0 5.57 7.896 15000 0.193 0.155 7.74 10.072 20000 0.387 0.31 11.5057 13.83 25000 0.581 0.465 19.32 21.65 30000 0.775 0.62 43.49 45.82 35000 0.969 0.775 618.93 621.25

TABLE-US-00011 TABLE 3-B4 Tension (g) 500 P.sub.core (psig) 0.128 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.31 0.248 4.089 6.414 5000 0.155 0.124 5.006 7.331 10000 0 0 6.238 8.56 15000 0.155 0.124 7.968 10.29 20000 0.31 0.248 10.55 12.88 25000 0.465 0.372 14.788 17.11 30000 0.62 0.496 22.8 25.12 35000 0.775 0.62 42.76 45.09 40000 0.93 0.744 151.699 154.02

TABLE-US-00012 TABLE 4-A Preform Outer Tube OD, 2 R.sub.ocp (mm) 34.6 Preform Outer Tube ID, 2 r.sub.ocp (mm) 8 Preform Inner Tube OD, 2 R.sub.cp (mm) 1.9 Preform Inner Tube ID, 2 r.sub.cp (mm) 1.4 Fiber Draw Speed (m/s) 4.12 Fiber Outer Cladding OD, 2 R.sub.ocf (m) 250 Fiber Outer Cladding ID, 2 r.sub.ocf (m) 79.7

TABLE-US-00013 TABLE 4-B1 Tension (g) 200 P.sub.core (psig) 0.049 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.9215 0.679 2000 0.7605 0.52 4000 0.4914 0.362 6000 0.276 0.203 0.3692 3.216 8000 0.061 0.045 1.161 4 10000 0.1535 0.1131 2.4803 5.3312 12000 0.3686 0.2716 5.033 7.9008 14000 0.5836 0.43 11.75 14.59 16000 0.7986 0.588 54.14 56.99

TABLE-US-00014 TABLE 4-B2 Tension (g) 300 P.sub.core (psig) 0.079 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.614 0.452 0.768 3.616 2000 0.471 0.347 1.1263 3.97 4000 0.327 0.241 1.5739 4.42 6000 0.184 0.135 2.148 4.995 8000 0.04 0.03 2.9018 5.74 10000 0.102 0.075 3.949 6.79 12000 0.245 0.1811 5.46 8.306 14000 0.3893 0.286 7.817 10.66 16000 0.532 0.392 11.929 14.77 18000 0.676 0.498 20.63 23.47 20000 0.819 0.603 48.62 51.46 22000 0.963 0.709 1343.37 1346.22

TABLE-US-00015 TABLE 4-B3 Tension (g) 400 P.sub.core (psig) 0.11 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.4611 0.339 1.925 4.772 2000 0.353 0.26 2.302 5.149 4000 0.249 0.181 2.752 5.59 6000 0.1383 0.101 3.297 6.144 8000 0.03 0.022 3.968 6.816 10000 0.0768 0.0566 4.813 7.66 12000 0.1844 0.135 5.903 8.752 14000 0.292 0.215 7.362 10.2 16000 0.399 0.294 9.394 12.242 18000 0.5072 0.373 12.4 15.24 20000 0.614 0.453 17.237 20.08 22000 0.722 0.532 26.13 28.98 24000 0.83 0.611 47.05 49.9 26000 0.937 0.69 138.97 141.82

TABLE-US-00016 TABLE 4-B4 Tension (g) 500 P.sub.core (psig) 0.14 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.368 0.271 2.78 5.63 2500 0.261 0.192 3.254 6.1 5000 0.153 0.113 3.82 6.66 7500 0.046 0.033 4.51 7.36 10000 0.061 0.045 5.376 8.223 12500 0.1691 0.1246 6.478 9.325 15000 0.276 0.203 7.92 10.77 17500 0.384 0.283 9.914 12.761 20000 0.491 0.362 12.774 15.622 22500 0.599 0.441 17.207 20.05 25000 0.707 0.521 24.88 27.72 27500 0.814 0.6 40.918 43.76 30000 0.922 0.679 91.22 94.07

TABLE-US-00017 TABLE 5-A Preform Outer Tube OD, 2 R.sub.ocp (mm) 49.6 Preform Outer Tube ID, 2 r.sub.ocp (mm) 9 Preform Inner Tube OD, 2 R.sub.cp (mm) 2 Preform Inner Tube ID, 2 r.sub.cp (mm) 1.5 Fiber Draw Speed (m/s) 8.54 Fiber Outer Cladding OD, 2 R.sub.ocf (m) 250 Fiber Outer Cladding ID, 2 r.sub.ocf (m) 79.7

TABLE-US-00018 TABLE 5-B1 Tension (g) 400 P.sub.core (psig) 0.09 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 2000 0.713 0.535 4000 0.475 0.356 5000 0.356 0.267 0.067 2.0117 6000 0.237 0.178 0.299 2.243 8000 0 0 0.98 2.924 10000 0.237 0.178 2.246 4.19 12000 0.475 0.356 5.24 7.19 14000 0.713 0.534 18.47 20.42 15000 0.832 0.624 81.57 83.52

TABLE-US-00019 TABLE 5-B2 Tension (g) 500 P.sub.core (psig) 0.14 Capillary Capillary Capillary dP ID in Fiber, OD in Fiber, (dynes/cm.sup.2) X.sub.1 X.sub.2 2 r.sub.cf (m) 2 R.sub.cf (m) 0 0.76 0.57 2000 0.57 0.427 0.23 2.175 4000 0.38 0.285 0.559 2.5504 5000 0.285 0.213 0.768 2.712 8000 0 0 1.6896 3.634 10000 0.19 0.142 2.76 4.702 12000 0.38 0.283 4.694 6.638 14000 0.57 0.427 9.029 10.97 15000 0.6657 0.499 13.92 15.86 16000 0.76 0.57 25.29 27.29 17500 0.903 0.677 271.47 273.41

[0061] The plots of FIGS. 5-8 show the inner diameter of the capillaries drawn as a function of the differential capillary pressure during the drawing process. The inventor has found that at relatively low differential capillary pressure, a small increase or decrease in the differential capillary pressure may not significantly affect the inner diameter of the capillaries drawn; at relatively high differential capillary pressure, even a small increase or decrease in the differential capillary pressure may result in drastic change in the inner diameter of the capillaries drawn.

[0062] FIG. 9 plots the evolution of the inner diameters of the capillaries from the inner tubes of the hollow-core preform in the neck down region as a function of axial velocity in the neck down region, for the different combinations of preform sizes, draw speeds, and differential capillary pressures outlined in Tables 2-5. As shown, the inner diameter may initially gradually increase as the differential capillary pressure may be the dominant factor affecting the inner diameter. As the tubes are continuously drawn down, the effects of surface tension on the inner diameter may become prominent, in combination with the differential capillary pressure causing the inner diameter to gradually decrease. Thus, the evolution of the inner diameter of the capillaries from the inner diameter of the inner tubers of the hollow-core preform is a complex process. The non-dimensional parameters X.sub.1, X.sub.2, X.sub.3, and/or X.sub.4 described herein provide simplified guidance for selecting the appropriate process condition ranges for drawing hollow-core optical fibers from preforms of different sizes to various desired target capillary dimensions.

Operating Parameters

[0063] The various methods and relations described herein may be implemented with a wide range of operating parameters for scaling up the drawing of hollow-core optical fibers from hollow-core preforms of various sizes while maintaining a tight control over target microstructure dimensions of the hollow-core optical fibers drawn.

Draw Tension

[0064] In some embodiments, the draw tension under which the hollow-core optical fiber may be drawn from a hollow-core preform may be greater than or equal to (i.e., ) 50 g and less than or equal to (i.e., ) 600 gincluding all sub-ranges or values therebetween. For example, in some embodiments, the draw tension may be 50 g and 600 g, 50 g and 550 g, 50 g and 500 g, 50 g and 450 g, 50 g and 400 g, 50 g and 350 g, 50 g and 300 g, 50 g and 250 g, 50 g and 200 g, 50 g and 150 g, 50 g and 100 g, 100 g and 600 g, 100 g and 550 g, 100 g and 500 g, 100 g and 450 g, 100 g and 400 g, 100 g and 350 g, 100 g and 300 g, 100 g and 250 g, 100 g and 200 g, 100 g and 150 g, 150 g and 600 g, 150 g and 550 g, 150 g and 500 g, 150 g and 450 g, 150 g and <400g, 150 g and 350 g, 150 g and 300 g, 150 g and 250 g, 150 g and 200 g, 200 g and 600 g, 200 g and 550 g, 200 g and 500 g, 200 g and 450 g, 200 g and <400 g, 200 g and 350 g, 200 g and 300 g, 200 g and 250 g, 250 g and 600 g, 250 g and 550 g, 250 g and 500 g, 250 g and 450 g, 250 g and 400 g, 250 g and <350 g, 250 g and 300 g, 300 g and 600 g, 300 g and 550 g, 300 g and 500 g, 300 g and 450 g, 300 g and 400 g, 300 g and 350 g, 350 g and 600 g, 350 g and <550 g, 350 g and 500 g, 350 g and 450 g, 350 g and 400 g, 400 g and 600 g, 400 g and 550 g, 400 g and 500 g, 400 g and 450 g, 450 g and 600 g, 450 g and 550 g, 450 g and 500 g, 500 g and 600 g, 550 g and 600 g, or 550 g and 600 g.

[0065] In some embodiments, the draw tension under which the hollow-core optical fiber may be drawn from a hollow-core preform may be greater than or equal to (i.e., ) 50 g, 70 g, 90 g, 110 g, 130 g, 150 g, 170 g, 190 g, 210 g, 230 g, 250 g, 270 g, 290 g, 310 g, 330 g, 350 g, 370 g, 390 g, 410 g, 430 g, 450 g, 470 g, 490 g, 510 g, 530 g, 550 g, 570 g, 590 g, or greater. In some embodiments, the draw tension under which the hollow-core optical fiber may be drawn from the hollow-core preform may be less than or equal to (i.e., ) 600 g, 580 g, 560 g, 540 g, 520 g, 500 g, 480 g, 460 g, 440 g, 420 g, 400 g, 380 g, 360 g, 340 g, 320 g, 300 g, 280 g, 260 g, 240 g, 220 g, 200 g, 180 g, 160 g, 140 g, 120 g, 100 g, 80 g, 60 g, or less. Draw Speed

[0066] In some embodiments, the hollow-core optical fiber may be drawn at a draw speed of greater than or equal to (i.e., ) 1 m/s and less than or equal to (i.e., ) 20 m/sincluding all sub-ranges or values therebetween. For example, in some embodiments, the draw speed may be 1 m/s and 20 m/s, 1 m/s and 15 m/s, 1 m/s and 10 m/s, 1 m/s and 5 m/s, 5 m/s and 20 m/s, 5 m/s and 15 m/s, 5 m/s and 10 m/s, 10 m/s and 20 m/s, 10 m/s and 15 m/s, or 15 m/s and 20 m/s. In some embodiments, the hollow-core optical fiber may be drawn at a draw speed of greater than or equal to (i.e., ) 1 m/s, 3 m/s, 5 m/s, 7 m/s, 9 m/s, 11 m/s, 13 m/s, 15 m/s, 17 m/s, 19 m/s, or greater.

Preform Feed Rate

[0067] In some embodiments, the hollow-core preform may be fed into the draw furnace at a preform feed rate V.sub.p that may be greater than or equal to (i.e., ) 5 mm/min and less than or equal to (i.e., ) 100 mm/minincluding all sub-ranges or values therebetween. For example, in some embodiments, the hollow-core preform may be fed into the draw furnace at a preform feed rate V.sub.p that may be 5 mm/min and 100 mm/min, 5 mm/min and 75 mm/min, 5 mm/min and 50 mm/min, 5 mm/min and 25 mm/min, 25 mm/min and 100 mm/min, 25 mm/min and 75 mm/min, 25 mm/min and 50 mm/min, 50 mm/min and 100 mm/min, 50 mm/min and 75 mm/min, or 75 mm/min and 100 mm/min. In some embodiments, the hollow-core preform may be fed into the draw furnace at a preform feed rate V.sub.p that may be 5 mm/min, 10 mm/min, 15 mm/min, 20 mm/min, 25 mm/min, 30 mm/min, 35 mm/min, 40 mm/min, 45 mm/min, 50 mm/min, 55 mm/min, 60 mm/min, 65 mm/min, 70 mm/min, 75 mm/min, 80 mm/min, 85 mm/min, 90 mm/min, 95 mm/min, or greater. In some embodiments, the hollow-core preform may be fed into the draw furnace at a preform feed rate V.sub.p that may be less than or equal to 100 mm/min, 95 mm/min, 90 mm/min, 85 mm/min, 80 mm/min, 75 mm/min, 70 mm/min, 65 mm/min, 60 mm/min, 55 mm/min, 50 mm/min, 45 mm/min, 40 mm/min, 35 mm/min, 30 mm/min, 25 mm/min, 20 mm/min, 15 mm/min, 10 mm/min, or less.

Differential Core Pressure P.SUB.core

[0068] In some embodiments, the differential core pressure P.sub.core may be greater than or equal to (i.e., ) 0.01 psig and less than or equal to (i.e., ) 2 psigincluding all sub-ranges or values therebetween. For example, in some embodiments, the differential core pressure P.sub.core may be may be 0.01 psig and 2 psig, 0.01 psig and 1.5, 0.01 psig and 1, 0.01 psig and <0.5, 0.01 psig and 0.1 psig, 0.1 psig and 2 psig, 0.1 psig and 1.5, 0.1 psig and 1, 0.1 psig and 0.5, 0.5 psig and 2 psig, 0.5 psig and 1.5, 0.5 psig and 1, 1 psig and 2 psig, 1 psig and 1.5, or 1.5 psig and 2 psig.

[0069] In some embodiments, the differential core pressure P.sub.core may be greater than or equal to (i.e., ) 0.01 psig, 0.05 psig, 0.1 psig, 0.2 psig, 0.3 psig, 0.4 psig, 0.5 psig, 0.6 psig, 0.7 psig, 0.8 psig, 0.9 psig, 1 psig, 1.1 psig, 1.2 psig, 1.3 psig, 1.4 psig, 1.5 psig, 1.6 psig, 1.7 psig, 1.8 psig, 1.9 psig, or greater. In some embodiments, the differential core pressure P.sub.core may be less than or equal to (i.e., ) 2 psig, 1.9 psig, 1.8 psig, 1.7 psig, 1.6 psig, 1.5 psig, 1.4 psig, 1.3 psig, 1.2 psig, 1.1 psig, 1 psig, 0.9 psig, 0.8 psig, 0.7 psig, 0.6 psig, 0.5 psig, 0.4 psig, 0.3 psig, 0.2 psig, 0.1 psig, 0.05 psig, or less.

[0070] Differential Capillary Pressure P.sub.c

[0071] In some embodiments, the differential capillary pressure P.sub.c may be greater than or equal to (i.e., ) 5,000 dynes/cm.sup.2 and less than or equal to (i.e., ) 100,000 dynes/cm.sup.2 including all sub-ranges or values therebetween. For example, in some embodiments, the differential capillary pressure P.sub.c may be 5,000 dynes/cm.sup.2 and 100,000 dynes/cm.sup.2, 5,000 dynes/cm.sup.2 and 75,000 dynes/cm.sup.2, 5,000 dynes/cm.sup.2 and 50,000 dynes/cm.sup.2, 5,000 dynes/cm.sup.2 and K 25,000 dynes/cm.sup.2, 5,000 dynes/cm.sup.2 and K 10,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2 and K 100,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2 and K 75,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2 and K 50,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2 and K 25,000 dynes/cm.sup.2, 25,000 dynes/cm.sup.2 and K 100,000 dynes/cm.sup.2, 25,000 dynes/cm.sup.2 and K 75,000 dynes/cm.sup.2, 25,000 dynes/cm.sup.2 and K 50,000 dynes/cm.sup.2, 50,000 dynes/cm.sup.2 and K 100,000 dynes/cm.sup.2, 50,000 dynes/cm.sup.2 and K 75,000 dynes/cm.sup.2, or 75,000 dynes/cm.sup.2 and K 100,000 dynes/cm.sup.2.

[0072] In some embodiments, the differential capillary pressure P.sub.c may be greater than or equal to (i.e., ) 5,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2, 15,000 dynes/cm.sup.2, 20,000 dynes/cm.sup.2, 25,000 dynes/cm.sup.2, 30,000 dynes/cm.sup.2, 35,000 dynes/cm.sup.2, 40,000 dynes/cm.sup.2, 45,000 dynes/cm.sup.2, 50,000 dynes/cm.sup.2, 55,000 dynes/cm.sup.2, 60,000 dynes/cm.sup.2, 65,000 dynes/cm.sup.2, 70,000 dynes/cm.sup.2, 75,000 dynes/cm.sup.2, 80,000 dynes/cm.sup.2, 85,000 dynes/cm.sup.2, 90,000 dynes/cm.sup.2, 95,000 dynes/cm.sup.2, or greater.

[0073] In some embodiments, the differential capillary pressure P.sub.c may be less than or equal to (i.e., ) 100,000 dynes/cm.sup.2, 95,000 dynes/cm.sup.2, 90,000 dynes/cm.sup.2, 85,000 dynes/cm.sup.2, 80,000 dynes/cm.sup.2, 75,000 dynes/cm.sup.2, 70,000 dynes/cm.sup.2, 65,000 dynes/cm.sup.2, 60,000 dynes/cm.sup.2, 55,000 dynes/cm.sup.2, 50,000 dynes/cm.sup.2, 45,000 dynes/cm.sup.2, 40,000 dynes/cm.sup.2, 35,000 dynes/cm.sup.2, 30,000 dynes/cm.sup.2, 25,000 dynes/cm.sup.2, 20,000 dynes/cm.sup.2, 15,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2, or less.

Differential Nested Capillary Pressure P.SUB.nc

[0074] In some embodiments, the differential nested capillary pressure P.sub.nc may be greater than or equal to (i.e., ) 2,000 dynes/cm.sup.2 and less than or equal to (i.e., ) 50,000 dynes/cm.sup.2-including all sub-ranges or values therebetween. For example, in some embodiments, the differential nested capillary pressure P.sub.nc may be 2,000 dynes/cm.sup.2 and K 50,000 dynes/cm.sup.2, 2,000 dynes/cm.sup.2 and K 40,000 dynes/cm.sup.2, 2,000 dynes/cm.sup.2 and K 30,000 dynes/cm.sup.2, 2,000 dynes/cm.sup.2 and K 20,000 dynes/cm.sup.2, 2,000 dynes/cm.sup.2 and K 10,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2 and K 50,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2 and K 40,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2 and K 30,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2 and K 20,000 dynes/cm.sup.2, 20,000 dynes/cm.sup.2 and K 50,000 dynes/cm.sup.2, 20,000 dynes/cm.sup.2 and K 40,000 dynes/cm.sup.2, 20,000 dynes/cm.sup.2 and K 30,000 dynes/cm.sup.2, 30,000 dynes/cm.sup.2 and 50,000 dynes/cm.sup.2, 30,000 dynes/cm.sup.2 and K 40,000 dynes/cm.sup.2, or 40,000 dynes/cm.sup.2 and K 50,000 dynes/cm.sup.2.

[0075] In some embodiments, the differential nested capillary pressure P.sub.nc may be greater than or equal to (i.e., ) 2,000 dynes/cm.sup.2, 5,000 dynes/cm.sup.2, 10,000 dynes/cm.sup.2, 15,000 dynes/cm.sup.2, 20,000 dynes/cm.sup.2, 25,000 dynes/cm.sup.2, 30,000 dynes/cm.sup.2, 35,000 dynes/cm.sup.2, 40,000 dynes/cm.sup.2, 45,000 dynes/cm.sup.2, or greater. In some embodiments, the differential nested capillary pressure P.sub.nc may be less than or equal to (i.e., ) 50,000 dynes/cm.sup.2, K 45,000 dynes/cm.sup.2, K 40,000 dynes/cm.sup.2, K 35,000 dynes/cm.sup.2, K 30,000 dynes/cm.sup.2, K 25,000 dynes/cm.sup.2, K 20,000 dynes/cm.sup.2, K 15,000 dynes/cm.sup.2, K 10,000 dynes/cm.sup.2, K 5,000 dynes/cm.sup.2, or less.

Preform Outer Tube Dimesions

[0076] In some embodiments, the inner diameter of the outer tube of the hollow-core preform (2r.sub.ocp) may be greater than or equal to (i.e., ) 4 mm and less than or equal to (i.e., ) 20 mmincluding all sub-ranges or values therebetween. For example, in some embodiments, the inner diameter of the outer tube of the hollow-core preform (2r.sub.ocp) may be 4 mm and K 20 mm, 4 mm and 16 mm, 4 mm and 12 mm, 4 mm and 8 mm, 8 mm and 20 mm, 8 mm and 16 mm, 8 mm and 12 mm, 12 mm and 20 mm, 12 mm and 16 mm, or 16 mm and 20 mm. In some embodiments, the inner diameter of the outer tube of the hollow-core preform (2r.sub.ocp) may be greater than or equal to (i.e., ) 4 mm, 5 mm, 6 mm, 7 mm, 8 mm, 9 mm, 10 mm, 11 mm, 12 mm, 13 mm, 14 mm, 15 mm, 16 mm, 17 mm, 18 mm, 19 mm, or greater. In some embodiments, the inner diameter of the outer tube of the hollow-core preform (2r.sub.ocp) may be less than or equal to (i.e., ) 20 mm, 19 mm, 18 mm, 17 mm, 16 mm, 15 mm, 14 mm, 13 mm, 12 mm, 11 mm, 10 mm, 9 mm, 8 mm, 7 mm, 6 mm, 5 mm, or less.

[0077] In some embodiments, the outer diameter of the outer tube of the hollow-core preform (2R.sub.ocp) may be greater than or equal to (i.e., ) 15 mm and less than or equal to (i.e., ) to 100 mmincluding all sub-ranges or values therebetween. For example, in some embodiments, the outer diameter of the outer tube of the hollow-core preform (2R.sub.ocp) may be 15 mm and 100 mm, 15 mm and 75 mm, 15 mm and 50 mm, 15 mm and 25 mm, 25 mm and 100 mm, 25 mm and 75 mm, 25 mm and 50 mm, 50 mm and 100 mm, 50 mm and 75 mm, or 75 mm and 100 mm. In some embodiments, the outer diameter of the outer tube of the hollow-core preform (2R.sub.ocp) may be greater than or equal to (i.e., ) 15 mm, 20 mm, 25 mm, 30 mm, 35 mm, 40 mm, 45 mm, 50 mm, 55 mm, 60 mm, 65 mm, 70 mm, 75 mm, 80 mm, 85 mm, 90 mm, 95 mm, or greater. In some embodiments, the outer diameter of the outer tube of the hollow-core preform (2R.sub.ocp) may be less than or equal to (i.e., ) 100 mm, K 95 mm, 90 mm, K 85 mm, 80 mm, K 75 mm, 70 mm, K 65 mm, K 60 mm, 55 mm, K 50 mm, K 45 mm, K 40 mm, K 35 mm, K 30 mm, K 25 mm, 20 mm, or less.

Preform Inner Tube Dimensions

[0078] In some embodiments, the inner diameter of the inner tube of the hollow-core preform (2r.sub.cp) may be greater than or equal to (i.e., ) 1 mm and less than or equal to (i.e., ) 8 mmincluding all sub-ranges or values therebetween. For example, in some embodiments, the inner diameter of the inner tube of the hollow-core preform (2r.sub.cp) may be 1 mm and 8 mm, 1 mm and 6 mm, 1 mm and 4 mm, 1 mm and 2 mm, 2 mm and 8 mm, 2 mm and 6 mm, 2 mm and K 4 mm, 4 mm and 8 mm, 4 mm and 6 mm, or 6 mm and K 8 mm. In some embodiments, the inner diameter of the inner tube of the hollow-core preform (2r.sub.cp) may be greater than or equal to (i.e., ) 1 mm, 1.5 mm, 2 mm, 2.5 mm, 3 mm, 3.5 mm, 4 mm, 4.5 mm, 5 mm, 5.5 mm, 6 mm, 6.5 mm, 7 mm, 7.5 mm, or greater. In some embodiments, the inner diameter of the inner tube of the hollow-core preform (2r.sub.cp) may be less than or equal to (i.e., ) 8 mm, 7.5 mm, 7 mm, 6.5 mm, 6 mm, 5.5 mm, 5 mm, 4.5 mm, 4 mm, 3.5 mm, 3 mm, 2.5 mm, 2 mm, 1.5 mm, or less.

[0079] In some embodiments, the outer diameter of the inner tube of the hollow-core preform (2R.sub.cp) may be greater than or equal to (i.e., ) 1.4 mm and less than or equal to (i.e., ) 10 mmincluding all sub-ranges or values therebetween. For example, in some embodiments, in some embodiments, the outer diameter of the inner tube of the hollow-core preform (2R.sub.cp) may be 1.4 mm and 10 mm, 1.4 mm and 8 mm, 1.4 mm and 6 mm, 1.4 mmand4 mm, or 1.4 m and 2 mm. In some embodiments, the outer diameter of the inner tube of the hollow-core preform (2R.sub.cp) may be greater than or equal to (i.e., ) 1.4 mm, 1.5 mm, 2 mm, 2.5 mm, 3 mm, 3.5 mm, 4 mm, 4.5 mm, 5 mm, 5.5 mm, 6 mm, 6.5 mm, 7 mm, 7.5 mm, 8 mm, 8.5 mm, 9 mm, 9.5 mm, or greater. In some embodiments, the outer diameter of the inner tube of the hollow-core preform (2R.sub.cp) may be less than or equal to (i.e., ) 10 mm, 9.5 mm, 9 mm, 8.5 mm, 8 mm, 7.5 mm, 7 mm, 6.5 mm, 6 mm, 5.5 mm, 5 mm, 4.5 mm, 4 mm, 3.5 mm, 3 mm, 2.5 mm, 2 mm, 1.5 mm, or less.

Preform Nested Tube Dimensions

[0080] In some embodiments, the inner diameter of the nested tube of the hollow-core preform (2r.sub.ncp) may be greater than or equal to (i.e., ) 0.5 mm and less than or equal to (i.e., ) 1.3 mmincluding all sub-ranges or values therebetween. For example, in some embodiments, the inner diameter of the nested tube of the hollow-core preform (2r.sub.ncp) may be 0.5 mm and <1.3 mm, 0.5 mm and 1.1 mm, 0.5 mm and 0.9 mm, 0.5 mm and 0.7 mm, 0.7 mm and 1.3 mm, 0.7 mm and 1.1 mm, 0.7 mm and 0.9 mm, 0.9 mm and 1.3 mm, 0.9 mm and 1.1 mm, or 1.1 mm and 1.3 mm. In some embodiments, the inner diameter of the nested tube of the hollow-core preform (2r.sub.ncp) may be greater than or equal to (i.e., ) 0.5 mm, 0.6 mm, 0.7 mm, 0.8 mm, 0.9 mm, 1 mm, 1.1 mm, 1.2 mm, or greater. In some embodiments, the inner diameter of the nested tube of the hollow-core preform (2rn.sub.ep) may be less than or equal to (i.e., ) 1.3 mm, 1.2 mm, 1.1 mm, 1 mm, 0.9 mm, 0.8 mm, 0.7 mm, 0.6 mm, or less.

[0081] In some embodiments, the outer diameter of the nested tube of the hollow-core preform (2R.sub.ncp) may be greater than or equal to (i.e., ) 0.6 mm and less than or equal to (i.e.,) to 1.5 mmincluding all sub-ranges or values therebetween. For example, in some embodiments, the outer diameter of the nested tube of the hollow-core preform (2R.sub.ncp) may be 0.6 mm and 1.5 mm, 0.6 mm and 1.3 mm, 0.6 mm and 1.1 mm, 0.6 mm and 0.9 mm, 0.6 mm and 0.7 mm, 0.8 mm and 1.5 mm, 0.8 mm and 1.3 mm, 0.8 mm and <1.1 mm, 0.8 mm and 0.9 mm, 1 mm and 1.5 mm, 1 mm and 1.3 mm, 1 mm and <1.1 mm, 1.2 mm and 1.5 mm, 1.2 mm and 1.3 mm, or 1.4 mm and 1.5 mm. In some embodiments, the outer diameter of the nested tube of the hollow-core preform (2R.sub.ncp) may be greater than or equal to (i.e., ) 0.6 mm, 0.7 mm, 0.8 mm, 0.9 mm, 1 mm, 1.1 mm, 1.2 mm, 1.3 mm, 1.4 mm, or greater. In some embodiments, the outer diameter of the nested tube of the hollow-core preform (2R.sub.ncp) may be less than or equal to (i.e., ) 1.5 mm, 1.4 mm, 1.3 mm, 1.2 mm, 1.1 mm, 1 mm, 0.9 mm, 0.8 mm, 0.7 mm, or less.

Fiber Outer Cladding Dimensions

[0082] In some embodiments, the inner diameter of the outer cladding of the hollow-core optical fiber (2r.sub.ocf) may be greater than or equal to (i.e., ) 45 m and less than or equal to (i.e., ) 135 mincluding all sub-ranges or values therebetween. For example, in some embodiments, the inner diameter of the outer cladding of the hollow-core optical fiber (2r.sub.ocf) may be 45 m and 135 m, 45 m and 115 m, 45 m and 95 m, 45 m and <75 m, 45 m and 55 m, 55 m and 135 m, 55 m and 115 m, 55 m and 95 m, 55 m and 75 m, 75 m and 135 m, 75 m and 115 m, 75 m and 95 m, 95 m and 135 m, 95 m and 115 m, or 115 m and 135 m. In some embodiments, the inner diameter of the outer cladding of the hollow-core optical fiber (2r.sub.ocf) may be greater than or equal to (i.e., ) 45 m, 50 mm, 55 mm, 60 mm, 65 mm, 70 mm, 75 mm, 80 mm, 85 mm, 90 mm, 95 mm, 100 mm, 105 mm, 110 mm, 115 mm, 120 mm, 125 mm, 130 m, or greater. In some embodiments, the inner diameter of the outer cladding of the hollow-core optical fiber (2r.sub.ocf) may be less than or equal to (i.e., ) 135 m, 130 m, 125 m, 120 m, 115 m, 110 m, 105 m, 100 m, 95 m, 90 m, 85 m, 80 m, 75 m, 70 m, 65 m, 60 m, 55 m, 50 m, or less.

[0083] In some embodiments, the outer diameter of the outer cladding of the hollow-core optical fiber (2R.sub.ocf) may be greater than or equal to (i.e., ) 150 m and less than or equal to (i.e., ) 300 mincluding all sub-ranges or values therebetween. For example, in some embodiments, the outer diameter of the outer cladding of the hollow-core optical fiber (2R.sub.ocf) may be 150 m and 300 m, 150 m and 250 m, 150 m and 200 m, 200 m and 300 m, 200 m and 250 m, or 250 m and 300 m. In some embodiments, the outer diameter of the outer cladding of the hollow-core optical fiber (2R.sub.ocf) may be greater than or equal to (i.e., ) 150 m, 160 m, 170 m, 180 m, 190 m, 200 m, 210 m, 220 m, 230 m, 240 m, 250 m, 260 m, 270 m, 280 m, 290 m, or greater. In some embodiments, the outer diameter of the outer cladding of the hollow-core optical fiber (2R.sub.ocf) may be less than or equal to (i.e., ) 300 m, 290 m, 280 m, 270 m, 260 m, 250 m, 240 m, 230 m, 220 m, 210 m, 200 m, 190 m, 180 m, 170 m, 160 m, or less.

Fiber Capillary Dimensions

[0084] In some embodiments, the inner diameter of the capillaries of the hollow-core optical fiber (2r.sub.cf) may be greater than or equal to (i.e., ) 10 m and less than or equal to (i.e., ) 40 mincluding all sub-ranges or values therebetween. For example, in some embodiments, the inner diameter of the capillaries of the hollow-core optical fiber (2r.sub.cf) may be 10 m and <40 m, 10 m and 30 m, 10 m and 20 m, 20 m and 40 m, 20 and 30 m, or 30 m and 40 m. In some embodiments, the inner diameter of the capillaries of the hollow-core optical fiber (2r.sub.cf) may be greater than or equal to (i.e., ) 10 m, 12.5 m, 15 m, 17.5 m, 20 m, 22.5 m, 25 m, 27.5 m, 30 m, 32.5 m, 35 m, 37.5 m, or greater. In some embodiments, the inner diameter of the capillaries of the hollow-core optical fiber (2r.sub.cf) may be less than or equal to (i.e., ) 40 m, 37.5 m, 35 m, 32.5 m, 30 m, 27.5 m, 25 m, 22.5 m, 20 m, 17.5 m, 15 m, 12.5 m, or less.

[0085] In some embodiments, the outer diameter of the capillaries of the hollow-core optical fiber (2R.sub.cf) may be greater than or equal to (i.e., ) 14 m and less than or equal to (i.e., ) 50 mincluding all sub-ranges or values therebetween. For example, in some embodiments, the outer diameter of the capillaries of the hollow-core optical fiber (2R.sub.cf) may be 14 m and <50 m, 14 m and 40 m, 14 m and 30 m, 14 m and 20 m, 20 m and 50 m, 20 m and 40 m, 20 m and 30 m, 30 m and 50 m, 30 m and 40 m, or 40 m and 50 m. In some embodiments, the outer diameter of the capillaries of the hollow-core optical fiber (2Ref) may be greater than or equal to (i.e., ) 14 m, 15 m, 17.5 m, 20 m, 22.5 m, 25 m, 27.5 m, 30 m, 32.5 m, 35 m, 37.5 m, 40 m, 42.5 m, 45 m, 47.5 m, or greater. In some embodiments, the outer diameter of the capillaries of the hollow-core optical fiber (2Ref) may be less than or equal to (i.e., ) 50 m, 47.5 m, 45 m, 42.5 m, 40 m, 37.5 m, 35 m, 32.5 m, 30 m, 27.5 m, 25 m, 22.5 m, 20 m, 17.5 m, 15 m, orless.

Fiber Nested Capillary Dimesions

[0086] In some embodiments, the inner diameter of the nested capillaries of the hollow-core optical fiber (2r.sub.ncf) may be greater than or equal to (i.e., ) 10 m and less than or equal to (i.e., ) 28 mincluding all sub-ranges or values therebetween. For example, in some embodiments, the inner diameter of the nested capillaries of the hollow-core optical fiber (2r.sub.ncf) may be 10 m and 28 m, 10 m and 24 m, 10 m and 20 m, 10 m and 16 m, 10 m and 12 m, 14 m and 28 m, 14 m and 24 m, 14 m and 20 m, 14 m and 16 m, 18 m and 28 m, 18 m and 24 m, 18 m and 20 m, 22 m and 28 m, 22 m and 24 m, or 26 m and 28 m. In some embodiments, the inner diameter of the nested capillaries of the hollow-core optical fiber (2r.sub.ncf) may be greater than or equal to (i.e., ) 10 m, 11 m, 12 m, 13 m, 14 m, 15 m, 16 m, 17 m, 18 m, 19 m, 20 m, 21 m, 22 m, 23 m, 24 m, 25 m, 26 m, 27 m, or greater. In some embodiments, the inner diameter of the nested capillaries of the hollow-core optical fiber (2r.sub.ncf) may be less than or equal to (i.e., ) 28 m, 27 m, 26 m, 25 m, 24 m, 23 m, 22 m, 21 m, 20 m, 19 m, 18 m, 17 m, 16 m, 15 m, 14 m, 13 m, 12 m, 11 m, or less.

[0087] In some embodiments, the outer diameter of the nested capillaries of the hollow-core optical fiber (2R.sub.ncf) may be greater than or equal to (i.e., ) 11 m and less than or equal to (i.e., ) 30 mincluding all sub-ranges or values therebetween. For example, in some embodiments, the outer diameter of the nested capillaries of the hollow-core optical fiber (2R.sub.ncf) may be 11 m and 30 m, 11 m and 25 m, 11 m and 20 m, 11 m and 15 m, 15 m and 30 m, 15 m and 25 m, 15 m and 20 m, 20 m and 30 m, 20 m and 25 m, or 25 m and 30 m. In some embodiments, the outer diameter of the nested capillaries of the hollow-core optical fiber (2R.sub.ncf) may be greater than or equal to (i.e., ) 11 m, 12 m, 13 m, 14 m, 15 m, 16 m, 17 m, 18 m, 19 m, 20 m, 21 m, 22 m, 23 m, 24 m, 25 m, 26 m, 27 m, 28 m, 29 m, or greater. In some embodiments, the outer diameter of the nested capillaries of the hollow-core optical fiber (2R.sub.ncf) may be less than or equal to (i.e., ) 30 m, 29 m, 28 m, 27 m, 26 m, 25 m, 24 m, 23 m, 22 m, 21 m, 20 m, 19 m, 18 m, 17 m, 16 m, 15 m, 14 m, 13 m, 12 m, or less.

[0088] It will be apparent to those skilled in the art that various modifications to the preferred embodiments of the disclosure as described herein can be made without departing from the spirit or scope of the disclosure as defined in the appended claims. Thus, the disclosure covers the modifications and variations provided they come within the scope of the appended claims and the equivalents thereto.