Heat mitigation in gas-filled hollow-core fiber lasers using gas flow

Abstract

The disclosure provides a method for heat mitigation in gas-filled hollow-core fibers such as used for gas lasers and a determination of the temperature reduction in the fibers that affects the maximum output power at which gas lasers can operate. A temperature reduction can be achieved by using a differential pressure between the inlet and outlet of the gas-filled hollow-core fiber lasers. The method uses coupled modeling of a gas flow model adapted to compressibility principles rather than the traditional incompressibility principles. The results of the adapted compressible gas flow model are transferred to the heat transfer model and the results of the heat transfer model is transferred iteratively to the compressible gas flow model until there is a predetermined amount of convergence. The convergence approximates a self-consistent stationary solution of operability for the given conditions. The solution can result in manufacturing and operational improvements of hollow-core gas filled lasers.

Description

208614.00474 PCT PATENT APPLICATION TITLE OF THE INVENTION

[0001] Heat Mitigation in Gas-filled Hollow-Core Fiber Lasers using Gas Flow CROSS REFERENCE TO RELATED APPLICATIONS

[0002] This application claims the benefit of U.S. Provisional Application No.63 / 671,497, filed July 15, 2024, entitled “Heat Mitigation in Gas-filled Hollow-Core Fiber Lasers using Gas Flow”, and is incorporated herein by reference. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0003] Not applicable. REFERENCE TO APPENDIX

[0004] Not applicable. BACKGROUND OF THE INVENTION

[0005] Field of the Invention.

[0006] The disclosure generally relates to a model for heat mitigation fiber lasers. More specifically, the disclosure generally relates to a method for heat prediction and mitigation in gas-filled hollow-core fiber lasers.

[0007] Description of the Related Art.

[0008] Gas lasers have been studied extensively over the years for applications in sensing, medical, and defense fields. Gas lasers have larger damage thresholds compared to step- index fiber lasers since the light propagates in a gas core rather than a glass core. Compared with rare-earth-doped fiber lasers, gas lasers can lase at higher power levels due to the weaker nonlinearity of the gas medium. The gas acts as a laser gain medium to amplify the input power of light and the resulting light beam. Conventionally, the gas is generally contained and sealed within the laser. While step-index fiber lasers can have impressive powers on the order of kilowatts, gas lasers can reach megawatts. The invention of hollow- core fibers enables new gas-filled hollow-core fiber lasers and sensors, due to their ability to host gases for long interaction lengths and the micrometer-scale mode areas in the hollow- core fiber. Gas-filled hollow-core fibers are attracting much attention as they have remarkable linear and nonlinear properties and have been used in several critical applications including gas lasers, high-power fiber delivery, pulse shaping, and nonlinear optics. The high optical intensity that can be obtained in hollow-core fibers enables the study of the nonlinear interaction of light with gases, vapors, and plasmas—all of which can be 1 45789110208614.00474 PCT PATENT APPLICATION introduced into the hollow core. Early work on nonlinear optics in hollow-core fibers used bandgap or kagome fibers. More recent work uses negative curvature fibers due to their low loss. The small overlap between the guided optical field and cladding glass, which is on the order of 10–5, leads to a significant increase in the damage threshold beyond what is possible in hollow-core bandgap fibers. The fabrication technology has become mature with the introduction of commercial products. The relative simplicity of the negative curvature structure facilitates the fabrication of fiber devices using non-silica glasses, such as chalcogenide. Since the gas media can be easily changed, gas-filled hollow-core fiber lasers can lase over a wide range of emission wavelengths from UV to IR.

[0009] Due to the high output power obtained in gas fiber lasers, heat has become a major factor in limiting the output power from gas laser systems. To achieve higher output power, it is important to reduce the temperature in the optical fiber.

[0010] There remains a need for a method to lower the temperature in gas-filled hollow-core fibers, which can effectively lead to a higher maximum output power at which gas lasers can operate, and further a method and system to mitigate heat generation in the fibers to enable additional functionality of lasers including higher power. BRIEF SUMMARY OF THE INVENTION

[0011] The disclosure provides a method for heat mitigation in gas-filled hollow-core fibers, such as used for gas lasers, which can lead to a higher maximum output power at which gas lasers can operate. A temperature reduction can be achieved by using a differential pressure between the inlet and outlet of the gas-filled hollow-core fiber lasers. The method uses coupled modeling of a gas flow model adapted to compressibility principles rather than the traditional incompressibility principles that is coupled with a heat transfer model. The results of the adapted compressible gas flow model are transferred to the heat transfer model and the results of the heat transfer model is transferred iteratively to the compressible gas flow model until there is a predetermined amount of convergence. The convergence approximates a self-consistent stationary solution of operability for the given conditions. The solution can result in manufacturing improvements and operational parameters for a higher output power of hollow-core gas filled lasers.

[0012] The disclosure provides a method of determining heat mitigation in hollow-core fibers having gas using a coupled model, comprising: determining values for variables in a gas flow model; determining values for variables in a heat transfer model; solving remaining variables in the gas flow model; providing relevant gas flow variables from the gas flow model to the heat transfer model; solving the heat transfer model using the relevant gas flow 45789110208614.00474 PCT PATENT APPLICATION variables thereby coupling the heat transfer model with the gas flow model; providing relevant heat transfer variables from the heat transfer model to the gas flow model; solving the gas flow model using the relevant heat transfer variables thereby coupling the gas flow model with the heat transfer model; and repeating the steps of providing variables and solving the models until there is a convergence of the gas flow variables and the heat flow variables that are common to both models to determine the heat mitigation.

[0013] The disclosure also provides a method for mitigating heat in a hollow-core fiber of a gas laser, comprising: establishing an inlet and an outlet on the hollow-core fiber; flowing a gain medium gas into the inlet and out of the outlet; and allowing a transfer of heat out of the gas laser.

[0014] The disclosure further provides a method for increasing a maximum output power of gas lasers having a hollow-core fiber under steady state conditions, comprising: determining a temperature in the hollow-core fiber using a coupled model of a gas flow model and a heat transfer model for a given power; and adjusting the power until the temperature is determined from the coupled model to be the maximum value. BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0015] Figure 1 is a schematic illustration of a hollow core fiber used for the modeling herein.

[0016] Figure 2A is an illustrative graph of velocity profiles at the midpoint of the fiber as a function of the ratio of radial distance to the hollow-core diameter r / d using the conventional incompressible model.

[0017] Figure 2B is an illustrative graph of velocity profiles at the midpoint of the fiber as a function of the ratio of radial distance to the hollow-core diameter r / d using the inventors’ compressible model.

[0018] Figure 3A is an illustrative graph of pressure distribution for both the incompressible and compressible models at the fiber center in the longitudinal direction.

[0019] Figure 3B is an illustrative graph of density profiles for both the incompressible and compressible models at the fiber center in the longitudinal direction.

[0020] Figure 3C is an illustrative graph of velocity profiles for both the incompressible and compressible models at the fiber center in the longitudinal direction.

[0021] Figure 4 is a schematic illustration of the steady state coupled model of the invention.

[0022] Figure 5A is an illustrative graph showing a temperature increase ^T from the 3 45789110208614.00474 PCT PATENT APPLICATION ambient temperature of 293 K from the center of the fiber in the radial direction on the Y-axis relative to the longitudinal location in the fiber starting at the fiber inlet on the X-axis without gas flow through the fiber.

[0023] Figure 5B is an illustrative graph showing a temperature increase ^T from the ambient temperature of 293 K from the center of the fiber in the radial direction on the Y-axis relative to the longitudinal location in the fiber on the X-axis with gas flow through the fiber.

[0024] Figure 6A is an illustrative graph showing the temperature distribution on the Y-axis relative to the longitudinal location in the fiber starting at the fiber inlet on the X-axis at the fiber center for various examples of differential pressure ^P, resulting from various gas flows.

[0025] Figure 6B is an illustrative graph showing the maximum differential temperature ^T relative to differential pressure ^P using the results of Figure 6A.

[0026] Figure 7 is an illustrative graph showing density profiles along the Y-axis at the center of the fiber relative to the longitudinal position in the fiber starting at the fiber inlet on the X- axis with and without heat.

[0027] Figure 8A is an illustrative graph of relative temperature change ^T on the left side Y-axis and relative temperature reduction, Rtemp, as a percentage on the right side Y-axis at the center of the fiber relative to heat source power on the X-axis.

[0028] Figure 8B is an illustrative graph of relative temperature change ^T on the left side Y-axis and relative temperature reduction, Rtemp, as a percentage on the right side Y-axis at the center of the fiber relative to fiber length on the X-axis.

[0029] Figure 8C is an illustrative graph of relative temperature change ^T on the left side Y-axis and relative temperature reduction, Rtemp, as a percentage on the right side Y-axis at the center of the fiber relative to FWHM of the heat profile on the X-axis.

[0030] Figure 8D is an illustrative graph of relative temperature change ^T on the left side Y-axis and relative temperature reduction, Rtemp, as a percentage on the right side Y-axis at the center of the fiber relative to the ratio of location at which the heat profile is maximum to the fiber length (Lmax / L) on the X-axis. DETAILED DESCRIPTION

[0031] The Figures described above and the written description of specific structures and functions below are not presented to limit the scope of what Applicant has invented or the scope of the appended claims. Rather, the Figures and written description are provided to 45789110208614.00474 PCT PATENT APPLICATION teach any person skilled in the art how to make and use the inventions for which patent protection is sought. Those skilled in the art will appreciate that not all features of a commercial embodiment of the inventions are described or shown for the sake of clarity and understanding. Persons of skill in this art will also appreciate that the development of an actual commercial embodiment incorporating aspects of the present disclosure will require numerous implementation-specific decisions to achieve the developer’s ultimate goal for the commercial embodiment. Such implementation-specific decisions may include, and likely are not limited to, compliance with system-related, business-related, government-related, and other constraints, which may vary by specific implementation, location, or with time. While a developer’s efforts might be complex and time-consuming in an absolute sense, such efforts would be, nevertheless, a routine undertaking for those of ordinary skill in this art having benefit of this disclosure. It must be understood that the inventions disclosed and taught herein are susceptible to numerous and various modifications and alternative forms. The use of a singular term, such as, but not limited to, “a”, is not intended as limiting of the number of items. Further, the various methods and embodiments of the system can be included in combination with each other to produce variations of the disclosed methods and embodiments. Discussion of singular elements can include plural elements and vice-versa. References to at least one item may include one or more items. Also, various aspects of the embodiments could be used in conjunction with each other to accomplish the understood goals of the disclosure. Unless the context requires otherwise, the term "comprise" or variations such as "comprises" or "comprising”, should be understood to imply the inclusion of at least the stated element or step or group of elements or steps or equivalents thereof, and not the exclusion of a greater numerical quantity or any other element or step or group of elements or steps or equivalents thereof. The device or system may be used in a number of directions and orientations. The terms “top”, “up”, “uphole”, “bottom”, “down”, “downhole”, and like directional terms are used to indicate the direction relative to the figures and their illustrated orientation and are not absolute relative to a fixed datum such as the earth in commercial use. The term “coupled”, “coupling”, “coupler”, and like terms are used broadly herein and may include any method or device for securing, binding, bonding, fastening, attaching, joining, inserting therein, forming thereon or therein, communicating, or otherwise associating, for example, mechanically, magnetically, electrically, chemically, operably, directly or indirectly with intermediate elements, one or more pieces of members together and may further include without limitation integrally forming one functional member with another in a unitary fashion. The coupling may occur in any direction, including rotationally. The order of steps can occur in a variety of sequences unless otherwise specifically limited. The various steps described herein can be combined with other steps, interlineated with the stated steps, and / or split into multiple steps. Similarly, elements have been described 45789110208614.00474 PCT PATENT APPLICATION functionally and can be embodied as separate components or can be combined into components having multiple functions. Some elements are nominated by a device name for simplicity and would be understood to include a system of related components that are known to those with ordinary skill in the art and may not be specifically described. Various examples are provided in the description and figures that perform various functions and are non-limiting in shape, size, description, but serve as illustrative structures that can be varied as would be known to one with ordinary skill in the art given the teachings contained herein. As such, the use of the term “exemplary” is the adjective form of the noun “example” and likewise refers to an illustrative structure, and not necessarily a preferred embodiment. Element numbers with suffix letters, such as “A”, “B”, and so forth, or numbers with prime, double prime, and so forth, such as 1, 1’, 1’’, and so forth, are to designate different elements within a group of like elements having a similar structure or function, and corresponding element numbers without the letters are to generally refer to one or more of the like elements. Any element numbers in the claims that correspond to elements disclosed in the application are illustrative and not exclusive, as several embodiments are disclosed that use various element numbers for like elements.

[0032] The disclosure provides a method for heat mitigation in gas-filled hollow-core fibers, such as used for gas lasers, and a determination of the amount of heat in the fibers during operation that affects the maximum output power at which gas lasers can operate. The heat mitigation is accomplished by varying from the conventional sealed gain medium gas in the gas laser and instead flowing the gain medium gas through the hollow-tube, such as at velocities and pressures described herein, to enhance the heat mitigation generated by the laser compared to a sealed gas laser and thereby allow higher output power from the gas laser, since output power is often limited by heat dissipation. Stated differently, a temperature reduction can be achieved by using a differential pressure between the inlet and outlet of the gas-filled hollow-core fiber lasers. Further, the method uses coupled modeling of a gas flow model, adapted to compressibility principles rather than the traditional incompressibility principles, with a heat transfer model. The results of the adapted compressible gas flow model are transferred to the heat transfer model and the results of the heat transfer model is transferred iteratively to the compressible gas flow model with recalculated values until there is a predetermined amount of convergence. The convergence approximates a self-consistent stationary solution of operability for the given conditions that can then be used to calculate a maximum amount of power in a steady state condition for the gas laser. The solution can result in manufacturing improvements and operational parameters for a higher output power of hollow-core gas filled lasers.

[0033] Numerical models based on known Navier–Stokes equations can be used to study 45789110208614.00474 PCT PATENT APPLICATION pressure-driven gas flow in hollow-core fibers and provide helpful information such as the velocity profile. Many studies solve the Navier-Stokes equations using the assumption that the gas is incompressible. However, the inventors describe a method of gas flow within hollow-core fibers that is compressible that surpasses those models based on incompressible constraints for more realistic results. In addition, the method of the inventors uses a coupled model to determine the gas flow and heat transfer in gas-filled hollow-core fibers iteratively until a convergence occurs to identify a self-consistent stationary solution. The convergence identifies the gas flows and heat transfer conditions from a flowing gas gain media for better heat mitigation, which would allow for a higher power output of the laser for manufacturing and operational parameters. The coupled model of gas flow combined with heat transfer iteratively accounts for a gas flow in the gas-filled hollow-core fiber that will increase heat dissipation for heat mitigation, and a temperature variation that will change the gas density in the gas-filled hollow-core fiber and therefore gas flow for the self-consistent stationary solution.

[0034] The heat profile that is generated in hollow-core gas-filled fiber lasers is heretofore unknown. Simple estimates of the heat power that may be found in the literature vary over a wide range, spanning 1 mW to 50 W. The convergence approximates a self-consistent stationary solution of operability for the given conditions that can then be used to calculate a maximum amount of power in a steady state condition for the gas laser.

[0035] Gas Flow Model

[0036] Gas is modeled as being introduced into a hollow-core fiber with a high pressure at the inlet. The gas encounters ambient pressure at the outlet. The motion of the gas is described by the Navier-Stokes equations. The momentum equation in steady state form may be written as ^^^ ^ ^^^ ^ ^ ^ K^ ^^, (1)

[0037] where u is the fluid velocity, ^ is the density of the fluid, p is the pressure within thefluid, and K is viscous stress tensor. The viscous stress in turn may be written K^ ^^^^^^ ^ ^^^^ ^ ^^^^ ^ ^^I^^, where I is the identity matrix and ^ is the viscosity of the fluid.The gas density is calculated using the ideal gas law so that ^ ^ ^^^^^^^, where M is molarmass, R is the ideal gas constant, and T is temperature. Several prior studies of gas-filled fiber lasers solve the Navier-Stokes equations using the assumption that the gas is incompressible. In contrast, the inventors consider a compressible gas for the Navier-Stokesequations. For compressible flows, the continuity equation is written ^ ^ ^^^^ ^ ^. Bycontrast, for incompressible flow, the continuity equation becomes ^ ^ ^ ^ ^, because thedensity is constant. For illustration of the differences in results, Figures 2A—2B and Figures 45789110208614.00474 PCT PATENT APPLICATION 3A—3C are provided and discussed below. To show the impact of temperature changes, a simplified laminar flow model is assumed that is believed to be a balance between accuracy and computational efficiency.

[0038] The Navier-Stokes equations can be solved by commercially available simulation software, COMSOL Multiphysics. The geometry can use a simple cylindrical shape for axial symmetry to reduce the problem to two dimensions, reducing computational demands for the purposes herein. The cylindrical shape corresponds to a straight fiber. For the case when a fiber is bent, both the mode profile and the gas flow in the hollow core are altered, which will require three dimensional simulations.

[0039] Figure 1 is a schematic illustration of a hollow core fiber (2) used for the modeling herein. The parameter r denotes the radial distance from the center (4) of the fiber and d denotes the diameter of the air core. Within the air core (6), the ratio r / d varies from 0 to 0.5. Hollow-core fiber lasers in the mid-IR wavelength range typically have core diameters on the order of 100 micrometers. Experimental work has used hollow-core fiber with a length ranging from several tens of centimeters to tens of meters. For illustrative purposes of the model, the diameter varied at 100, 150, and 200 µm and a fiber length of 0.5 m is used, unless otherwise specified.

[0040] Figure 2A is an illustrative graph of velocity profiles at the midpoint of the fiber as a function of the ratio of radial distance to the hollow-core diameter r / d using the conventional incompressible model. Figure 2B is an illustrative graph of velocity profiles at the midpoint of the fiber as a function of the ratio of radial distance to the hollow-core diameter r / d using the inventors’ compressible model. In this simulation, the inventors used a no-slip wall boundary condition and air as a gas medium. The density ^ of air is calculated based on the ideal gas law. For air, the molar mass M is 29 g / mol, the ideal gas constant R is 8.32 J / (mol- K), and the viscosity of air µ is assumed at 1.8×10–5Pa^s.

[0041] The results show that gas flow velocity increases with increasing core diameter. The incompressible model produces velocities that are consistently higher than those produced by the compressible model at the midpoint of the fiber. The circles indicate the analytical solution of velocity profiles along transverse direction for the incompressible model for different core diameters, which matches well with numerical solutions.

[0042] Figure 3A is an illustrative graph of pressure distribution for both the incompressible and compressible models at the fiber center in the longitudinal direction. In the incompressible model, there is a linear decline in pressure from an inlet value of 5 45789110208614.00474 PCT PATENT APPLICATION atmospheres (“atm”) to an outlet pressure of 1 atm. The pressure for the compressible model remains consistently above that of the incompressible model along the fiber and is no longer linear. The red circles indicate the analytical solution for the pressure along the fiber length for the compressible model.

[0043] Figure 3B is an illustrative graph of density profiles for both the incompressible and compressible models at the fiber center in the longitudinal direction. In the incompressible model, density is uniform along the fiber’s longitudinal direction. In the compressible model,the density is a function of pressure, ^ ^ ^^^^^^^, being larger where the pressure is higher.The red circles indicate the analytical solution for the density along the fiber length for the compressible model.

[0044] Figure 3C is an illustrative graph of velocity profiles for both the incompressible and compressible models at the fiber center in the longitudinal direction. In the incompressible model, velocity remains constant, suggesting a rigid-body motion of the gas where fluid molecules exhibit no relative movement and maintain identical velocities. The analytical solution for the incompressible model produces a velocity of 112 m / s, which matches well with the black line in Fig. 3C. On the other hand, the compressible model produces an increasing velocity profile that reaches a maximum near the outlet. The gas velocity increases in the longitudinal direction due to the consistently higher pressure near the inlet compared to the outlet, driving the gas to increase its velocity as it moves along the hollow- core fiber. This study expands on prior work and further illustrates that it is essential to use the compressible model. The results are consistent with prior work that studied the gas concentration and the gas filling time in hollow-core fibers. The decision to use incompressible flow or compressible flow in modeling gas flow depends on several factors, including the Mach number of the flow, the desired level of accuracy, and the specific application. Since the compressible model offers a more realistic simulation of gas flow, which yields different results compared with the incompressible model, the compressible model is chosen for the following figures and related discussions.

[0045] Gas Flow and Heat Transfer Coupled Model

[0046] Fiber lasers can deliver high output powers. However, these high output powers can lead to excessive heat, which can impair laser performance or even damage the host fiber. Thus, there is an upper limit (sometimes predetermined) of temperature in which a gas laser can operate in a steady state condition. It is a unique feature for hollow-core fibers that thermal mitigation can be achieved by using gas flow. The inventors use a heat transfer model in a steady-state solution. The heat transfer equation in steady-state is given by ^^^ ^ ^^ ^ ^^^^ ^ ^, (2)45789110208614.00474 PCT PATENT APPLICATION

[0047] where ^ is the fluid velocity, T is the temperature, ^ is the density, C is the heat capacity, ^ is thermal conductivity, and Q is the heat density of a heat source. Through the heat transfer equation, the temperature distribution due to the heat generated in the lasing process can be determined, as well as the gas flow.

[0048] Figure 4 is a schematic illustration of the steady state coupled model of the invention. The inventors couple a plurality of independent models of gas flow and heat transfer to create a coupled model to study the gas flow and heat transfer simultaneously. When a temperature rises, the gas density decreases and viscosity increases, which affect the gas flow through the hollow-core fiber. Simultaneously, the gas flow transports the heat in the longitudinal direction. Hence, in the coupled model, the inventors use the gas flow model, and the calculated values of gas velocity u and gas pressure p are passed to the heat transfer model as its initial conditions. The heat transfer model then determines the temperature, which isthen passed back to the gas flow model to determine the gas density ^ ^ ^^^^^^^ andviscosity µ, which is a function of temperature T. This iterative process continues until it converges, such as converges around a value for density and / or temperature.

[0049] COMSOL Multiphysics can be used to find the solution of the heat equation. In the example of simulation, the inventors modeled the hollow core filled with air and the glass cladding, as shown in Figure 1 and thereby allowing a 2D axial symmetry in the analysis. For example, the diameter of the hollow core was 200 µm, which was surrounded by a glass layer. The fiber length was 0.5 m. The density of glass is 2203 kg / m3. The viscosity, heat capacity, and thermal conductivity for air is a function of temperature. The temperature at the outer boundary of the glass cladding is set to the ambient temperature of 293K for this example. At that temperature, the thermal conductivity of glass is 1.38 W / m-K. The heat capacity of glass is 703 J / kg-K. The heat source was assumed to have a Gaussian distribution along the fiber’s longitudinal direction, representing the heat generated in a gas- filled hollow-core fiber laser. This assumption is consistent with the temperature profiles in the optical fiber amplifiers using solid core fibers. The heat maximum can be set at the center and full-width half-maximum (FWHM) equal to quarter of the fiber length. However, it is shown below that the results remain qualitatively similar regardless of where the longitudinal heat profile has its maximum. The heat profile can be assumed to be a Gaussian-distribution in the transverse direction, due to the light amplification gain of a fundamental model. Also, a FWHM equal to half the core diameter can be used, which is consistent with the ratio of fundamental mode to the fiber core. The total heat power can be set equal to 5 W after integrating over the whole fiber spatial domain. The temperature variation within the air core is significantly larger than within the glass cladding because the thermal conductivity in air is 45789110208614.00474 PCT PATENT APPLICATION 70 times smaller than that in the glass. Although glass parameters are used in this heat transfer model, the subsequent discussion will only show the results of the temperature distribution within the air core.

[0050] The results demonstrate that gas flow within hollow-core fibers can lead to significant temperature reduction. With a fixed input pressure and laser beam profile, the optical beam will reach a steady state in the optical fiber, leading to a fixed heat profile. The inventors for purposes herein, assumed a fixed heat profile resulting from a steady state, but varying the initial gas pressure, so that the thermal mitigation can be modeled due to varying gas pressure.

[0051] Simulation Results for Various Conditions

[0052] The following graphs show the results from using the above coupled model of gas flow and heat transfer together under the stated conditions and set values.

[0053] Figure 5A is an illustrative graph showing a temperature increase ^T from the ambient temperature of 293 K from the center of the fiber in the radial direction on the Y-axis relative to the longitudinal location in the fiber starting at the fiber inlet on the X-axis without gas flow through the fiber. Figure 5B is an illustrative graph showing a temperature increase ^T from the ambient temperature of 293 K from the center of the fiber in the radial direction on the Y-axis relative to the longitudinal location in the fiber on the X-axis with gas flow through the fiber. Without gas flow as shown in Figure 5A, the maximum temperature is located in the middle of the fiber at the peak of heat profile. With gas flow as shown in Figure 5B, the temperature peak shifts in the direction of the gas flow (that is, further from the hollow fiber inlet) and the maximum temperature decreases.

[0054] Figure 6A is an illustrative graph showing the temperature distribution on the Y-axis relative to the longitudinal location in the fiber starting at the fiber inlet on the X-axis at the fiber center for various examples of differential pressure ^P, resulting from various gas flows. Figure 6B is an illustrative graph showing the maximum differential temperature ^T relative to differential pressure ^P using the results of Figure 6A. The differential pressure, as used herein, is the difference in pressure between the two ends of the fiber. The temperature distribution profile broadens as ^P increases, due to the air flow from the inlet to the outlet. Again, the shift results in an asymmetrical temperature distribution due to the air flow, deviating from the initial symmetrical profile with ^P = 0. A larger pressure differential leads to a larger shift of the peaks of the temperature curves and a lower maximum temperature. The maximum illustrative ^P = 10 atm can be readily available in the laboratory setting using a pressurized gas cylinder. As shown in Figure 6B, due to the gas flow, the maximum ^T 45789110208614.00474 PCT PATENT APPLICATION decreases more than 20% from 273 K to 208 K. The gas flow within the hollow core of the fiber enhances the heat dissipation and results in heat mitigation.

[0055] Figure 7 is an illustrative graph showing density profiles along the Y-axis at the center of the fiber relative to the longitudinal position in the fiber starting at the fiber inlet on the X- axis with and without heat. For illustration, the simulation can set ^P equal to 10 atm. The curve without heat is obtained from the gas flow model only. The curve with heat is obtained from the coupled model. For the curve with heat, there is a dip at about 0.3 m, corresponding to the high temperature point in Figure 6A. The heat changes the density significantly.

[0056] Figure 8A is an illustrative graph of relative temperature change ^T (also referred to as “differential temperature” herein) on the left side Y-axis and relative temperature reduction, Rtemp, as a percentage on the right side Y-axis at the center of the fiber relative to heat source power on the X-axis. The test was on a fiber having a length of 0.5 m, a FWHM of 0.125 m, and a Lmax / L ratio of 0.5 with variable power in W on the X-axis. Figure 8B is an illustrative graph of relative temperature change ^T on the left side Y-axis and relative temperature reduction, Rtemp, as a percentage on the right side Y-axis at the center of the fiber relative to fiber length on the X-axis. The test was on a fiber having a FWHM of 0.25 x fiber length, an applied power of 5W, and a Lmax / L ratio of 0.5 with variable fiber length on the X-axis. Figure 8C is an illustrative graph of relative temperature change ^T on the left side Y-axis and relative temperature reduction, Rtemp, as a percentage on the right side Y- axis at the center of the fiber relative to FWHM of the heat profile on the X-axis. The test was on a fiber having a length of 0.5 m, an applied power of 5W, and a Lmax / L ration of 0.5 with variable FWHM on the X-axis. Figure 8D is an illustrative graph of relative temperature change ^T on the left side Y-axis and relative temperature reduction,as a percentage on the right side Y-axis at the center of the fiber relative to the ratio of location at which the heat profile is maximum to the fiber length (Lmax / L) on the X-axis. The test was on a fiber having a length of 0.5 m, an applied power of 5W, a FWHM of 0.25 x fiber length, with a variable Lmax / L ratio on the X-axis. For purposes herein, relative temperature reduction, Rtemp, can be defined by the following equation:

[0057] whereand ^^^*+-,^.^^are the relative temperature increases from the ambient temperature with ^P = 0 atm and ^P = 10 atm, respectively.

[0058] Figure 8A indicates that relative temperature reduction, Rtemp, decreases from 40% to 20% with gas flow when heat power increases from 1 W to 10 W. Figure 8B indicates that the temperature reduction increases with a shorter fiber, because the heat in the hollow core 45789110208614.00474 PCT PATENT APPLICATION flows out more rapidly when the fiber length is short. Figure 8C indicates that when the heat is more concentrated, it leads to more temperature reduction with gas flow. Figure 8D indicates that the location at which the heat profile is maximum does not have much impact on ^T.

[0059] Conclusions

[0060] While the illustrative application of the described coupled model is in high-power hollow-core optical fiber lasers, the coupled model can apply to other sources of heat generation in hollow-core fibers, independent of the heat source. The coupled model can be used to study the impact of gas flow on the temperature change along the fiber in steady state. The coupled model of using a gas flow model and heat transfer model iteratively to convergence in a hollow-core fiber is able to determine the expected temperature increase over ambient as a differential temperature, an expected heat profile within the hollow-core fiber, and location at which the heat profile is maximum along the length of the hollow-core fiber for various fiber lengths, total heat power, FWHM of the heat profile, and the input pressure. The coupled model discovered that fractional temperature reduction increases when the fiber becomes shorter, the heat profile becomes more localized, and the total power increases, while the location of the heat maximum has little impact. As an example, when the input pressure is 10 atm with a heat power of 5 W, the fractional temperature reduction is significant under various conditions and can be as high as 20%.

[0061] The results indicate that hollow-core fiber termination methods that leave the end of the fiber unsealed to be able to provide gas flow through the hollow-core fiber may have significant advantages for power scaling of gas-filled fiber lasers. The results of the coupled model can be used to design a system, such as a laser system that can operate in steady state at or below a given heat threshold that the manufacturer can identify above which the heat causes damage to the system. With the heat mitigation from the gas flow, the manufacturer or user can increase power output to a level that causes an increase in heat predicted by the coupled model that is below the heat threshold and still avoid the expected damage that the coupled model predicts would occur from an excess power.

[0062] The invention has been described in the context of preferred and other embodiments and not every embodiment of the invention has been described. Obvious modifications and alterations to the described embodiments are available to those of ordinary skill in the art. The disclosed and undisclosed embodiments are not intended to limit or restrict the scope or applicability of the invention conceived of by the Applicant, but rather, in conformity with the patent laws, Applicant intends to protect fully all such modifications and improvements that come within the scope of the following claims. 45789110

Claims

208614.00474 PCT PATENT APPLICATION WHAT IS CLAIMED IS:

1. A method of determining heat mitigation in hollow-core fibers having gas using a coupled model, comprising: determining values for variables in a gas flow model; determining values for variables in a heat transfer model; solving remaining variables in the gas flow model; providing relevant gas flow variables from the gas flow model to the heat transfer model; solving the heat transfer model using the relevant gas flow variables thereby coupling the heat transfer model with the gas flow model; providing relevant heat transfer variables from the heat transfer model to the gas flow model; solving the gas flow model using the relevant heat transfer variables thereby coupling the gas flow model with the heat transfer model; and repeating the steps of providing variables and solving the models until there is a convergence of the gas flow variables and the heat flow variables that are common to both models to determine the heat mitigation.

2. The method of claim 1, wherein the gas flow model comprises Equation 1.

3. The method of claim 1, wherein the heat transfer model comprises Equation 2.

4. The method of claim 1, further comprising determining convergence when variables that are common to the models are within a predetermined percentage.

5. A method for mitigating heat in a hollow-core fiber of a gas laser, comprising: establishing an inlet and an outlet on the hollow-core fiber; flowing a gain medium gas into the inlet and out of the outlet; and allowing a transfer of heat out of the gas laser. 6 The method of claim 5, wherein the gain medium gas is recirculated from the outlet through a heat exchanger to cool the gain medium gas to the inlet.

7. The method of claim 5, further comprising establishing a differential pressure between the inlet and outlet of the hollow-core fiber laser.

8. A method for increasing a maximum output power of gas lasers having a hollow-core 45789110208614.00474 PCT PATENT APPLICATION fiber under steady state conditions, comprising: determining a temperature in the hollow-core fiber using a coupled model of a gas flow model and a heat transfer model for a given power; and adjusting the power until the temperature is determined from the coupled model to be the maximum value.

9. The method of claim 8, wherein the gas flow model comprises Equation 1.

10. The method of claim 8, wherein the heat transfer model comprises Equation 2.

11. The method of claim 8, wherein determining the amount of heat in the hollow-core fiber comprises: determining values for variables in a gas flow model; determining values for variables in a heat transfer model; solving remaining variables in the gas flow model; providing relevant gas flow variables from the gas flow model to the heat transfer model; solving the heat transfer model using the relevant gas flow variables thereby coupling the heat transfer model with the gas flow model; providing relevant heat transfer variables from the heat transfer model to the gas flow model; solving the gas flow model using the relevant heat transfer variables thereby coupling the gas flow model with the heat transfer model; and repeating the steps of providing variables and solving the models until there is a convergence of the gas flow variables and the heat flow variables that are common to both models. 45789110

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