Radiation-balanced fiber laser

Abstract

An apparatus and method for cooling an optical fiber, comprising impinging electromagnetic radiation from a laser on an optical fiber comprising a core, in which the electromagnetic radiation is substantially confined, and a cladding, in thermal communication with the core, configured to provide optically activated cooling of the core via the electromagnetic radiation from the laser.

Claims

1. A method of cooling an optical fiber, comprising impinging electromagnetic radiation from a laser on an optical fiber comprising: a core, in which the electromagnetic radiation is substantially confined; and a cladding, in thermal communication with the core, configured to provide optically activated cooling of the core via the electromagnetic radiation from the laser; wherein the cladding is a glass or a polymer and comprises a host material and an optically activated cooling material selected from the group of Yb:YLiF4, Yb:NaYF4, Yb:LuLiF4, Yb:KLuF4, and Yb:KYF4, wherein the optically activated cooling material is index matched to the host material.

2. The method of claim 1, wherein the electromagnetic radiation has a peak wavelength in a range of about 1020 nm to about 1064 nm.

3. The method of claim 1, wherein the optically activated cooling results from emission of upconverted, anti-Stokes photoluminescence in the cladding.

4. The method of claim 1, wherein the cladding is cooled greater than 15K.

5. The method of claim 1, wherein the optical fiber is a single-mode fiber.

6. The method of claim 1, wherein optical fiber is a portion of a core-pumped fiber-laser cavity.

7. An optical fiber configured to provide optically activated cooling, the optical fiber comprising: a core configured to substantially confine electromagnetic radiation from a laser; a cladding, in thermal communication with the core, configured to provide optically activated cooling of the core via the electromagnetic radiation from the laser; wherein the cladding is a glass or a polymer and comprises a host material and an optically activated cooling material selected from the group of Yb:YLiF4, Yb:NaYF4, Yb:LuLiF4, Yb:KLuF4, and Yb:KYF4, wherein the optically activated cooling material is index matched to the host material.

8. The optical fiber of claim 7, wherein the electromagnetic radiation has a peak wavelength in a range of about 1020 nm to about 1064 nm.

9. The optical fiber of claim 7, wherein the optically activated cooling results from emission of upconverted, anti-Stokes photoluminescence in the cladding.

10. The optical fiber of claim 7, wherein the cladding is cooled greater than 15K.

11. The optical fiber of claim 7, wherein the optical fiber is a single-mode fiber.

12. The optical fiber of claim 7, wherein optical fiber is a portion of a core-pumped fiber-laser cavity.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated as the same become better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings wherein:

(2) FIG. 1A is an optical microscope image of an example fiber in accordance with the present technology;

(3) FIG. 1B is a graph of an example Yb:YLF photoluminescence spectra at various laser irradiances in accordance with the present technology;

(4) FIG. 1C is a graph of an example integrated intensity ratio of P1 and P2 peaks with the corresponding calibrated temperature at each laser irradiance in accordance with the present technology;

(5) FIG. 2A is a schematic of an example fiber in accordance with the present technology;

(6) FIG. 2B is a cross-section of an example fiber in accordance with the present technology;

(7) FIG. 3A is a graph of the radial electric field in the core and cladding of an example fiber in accordance with the present technology;

(8) FIG. 3B is a graph of the radial distribution of the EM heat source within example fibers in accordance with the present technology;

(9) FIG. 4A is a graph of the propagation constant of an example fiber with various values of n.sub.cl in accordance with the present technology;

(10) FIG. 4B is a graph of the radial distribution of annular normalized E.Math.E* of an example fiber in accordance with the present technology;

(11) FIG. 4C is a graph of the longitudinal distribution of E.Math.E* of an example fiber in accordance with the present technology;

(12) FIG. 4D is the propagation constant of an example fiber with various values of kc in accordance with the present technology;

(13) FIG. 4E is the radial distribution of E.Math.E* of an example fiber in accordance with the present technology;

(14) FIG. 4F is the longitudinal distribution of E.Math.E* of an example fiber in accordance with the present technology;

(15) FIG. 5A is a graph of an example radial temperature distribution of example fibers in accordance with the present technology;

(16) FIG. 5B is a graph of an example average temperature for the whole volume of an example fiber for various amounts of Yb:YLF in accordance with the present technology;

(17) FIG. 6 is a graph of the radial temperature distribution of example fibers with various cladding glass refractive indices in accordance with the present technology; and

(18) FIG. 7 is a graph of the time-resolved temperature for various cladding thicknesses of example fibers in accordance with the present technology.

DETAILED DESCRIPTION

(19) The present technology demonstrates an approach to cooling an optical fiber >15K below room temperature using solid state refrigeration.

(20) In one aspect, an optical fiber is provided that is configured to provide optically activated cooling, the optical fiber comprising:

(21) a core configured to substantially confine electromagnetic radiation from a laser; and

(22) a cladding, in thermal communication with the core, configured to provide optically activated cooling of the core via the electromagnetic radiation from the laser.

(23) In one aspect, a method of cooling an optical fiber is provided. In one embodiment, the method includes impinging electromagnetic radiation from a laser on an optical fiber comprising:

(24) a core, in which the electromagnetic radiation is substantially confined; and

(25) a cladding, in thermal communication with the core, configured to provide optically activated cooling of the core via the electromagnetic radiation from the laser.

(26) FIG. 1A is an optical microscope image of an example fiber in accordance with the present technology. In some embodiments, the fiber is an optical fiber. In some embodiments, the optical fiber is a portion of a core-pumped fiber-laser cavity. In some embodiments, the fiber is an etched fiber. In some embodiments, the fiber is formed from glass. In some embodiments, microcrystals are attached to an etched fiber. In some embodiments, the microcrystals are formed from the group of Yb:YLiF.sub.4, Yb:NaYF.sub.4, Yb:LuLiF.sub.4, Yb:KLuF.sub.4, and Yb:KYF.sub.4. In some embodiments, such as the one shown in FIG. 1A, the optical fiber is a cladding-etched bare glass fiber with YLF crystals attached to it using a layer of commercially available fluoropolymer CYTOP with an ultra-low NIR absorption coefficient. The optical fiber in FIG. 1A is illuminated perpendicularly in the far field with a 1020 nm continuous-wave (CW) laser focused onto a crystal at a range of irradiances up to 1 MW/cm.sup.2.

(27) FIG. 1B is a graph of an example Yb:YLF photoluminescence spectra at various laser irradiances in accordance with the present technology. On the horizontal axis is wavelength. On the vertical axis is the normalized intensity. P1 and P2 are electronic transitions E.sub.6−E.sub.2 and E.sub.5−E.sub.2/E.sub.3, respectively. Yb:YLF photoluminescence spectra are shown at various laser irradiances, normalized to the P1 peak. At higher irradiance the P2 peak increases in intensity. The integration ranges for P1 and P2 are from 952 nm to 968 nm and 986 nm to 1000 nm, respectively. The intensity ratio of P1 and P2 bands in FIG. 1B is temperature dependent based on a Boltzmann distribution, and lower ratio values correspond to lower temperatures. PL spectra were normalized to the peak at 960 nm. These PL spectra arise from ytterbium's electronic transitions between .sup.2F.sub.5/2 and .sup.2F.sub.7/2 and the manifolds are illustrated. In some embodiments, a 1020 nm laser pumps a hole from energy level E.sub.5 energy level E.sub.4. The holes are then upconverted further by phonons within the .sup.2F.sub.7/2 manifold followed by anti-Stokes emission. The radiative relaxation following upconversion transports the heat extracted from the lattice to free space, resulting in cooling.

(28) In one embodiment, a 1020 nm laser from a fiber coupled single mode laser diode (QPhotonics, QFBGLD-1020-400) was focused to a diffraction limited spot (radius=1.2 μm) using a long working distance 50× objective (Mitutoyo, M Plan Apo), and photoluminescence (PL) was collected by the same objective. The PL spectra were recorded with a spectrometer (Ocean Optics, NIR512). A 1000 nm short-pass filter (Thorlabs, FESH1000) was used to filter the laser line. Ten spectra, collected for 100 ms each, were averaged to obtain the final PL spectrum. The temperature-calibrated PL spectra were obtained using a cryostat (Janis, ST500) in which the sample temperature was maintained at various points from 300 K to 350 K by a temperature controller (Lake Shore, 335) with resolution 0.01 K.

(29) FIG. 1C is a graph of an example integrated intensity ratio of P1 and P2 peaks with the corresponding calibrated temperature at each laser irradiance in accordance with the present technology. On the horizontal axis is irradiance. On the vertical axis is temperature. The integrated intensity ratio of P1 and P2 peaks are illustrated with the corresponding calibrated temperature at each laser irradiance. After calibrating the intensity ratio to the temperature, the Yb:YLF crystal decreases in temperature by 6.6 K. At each laser irradiance, a mean P1/P2 ratio was obtained by averaging six measurements and the error bars represent one standard deviation, which are smaller than 1% of the mean values. The hydrothermally synthesized laser cooling Yb:YLF nanocrystals can be mixed within the cladding material at various volumetric fractions. When the nanocrystals are embedded in the cladding near the core-shell interface, the evanescent field may optically pump the cooling materials and lower the local temperature through anti-Stokes photoluminescence. Rayleigh scattering can be reduced by index matching the composite host material with the index of YLF (n.sub.YLF˜1.47).

(30) FIG. 2A is a schematic of an example fiber in accordance with the present technology. In some embodiments, the optical fiber includes a core, an inner cladding, and an outer cladding. In some embodiments, the optical fiber is a portion of a core-pumped fiber-laser cavity. In some embodiments, the optical fiber is a single-mode fiber. In some embodiments, the optical fiber is a single-mode fiber with a step-index profile, which has the highest output laser quality. In some embodiments, a fiber laser is pumped through an inner core. In some embodiments, a first cladding can serve as the pump. In some embodiments, the optical fiber is a multi-mode fiber.

(31) For simplicity, the illustrated fiber includes a core of Yb-doped glass. In other embodiments, the core may be formed from the group of Yb:YLiF.sub.4, Yb:NaYF.sub.4, Yb:LuLiF.sub.4, Yb:KLuF.sub.4, and Yb:KYF.sub.4. In some embodiments, the core may be comprised of another material. The illustrated fiber also includes the inner region of the cladding composed of glass and YLF nanocrystals, and the outer region of the cladding, made of the same glass as the inner cladding region, but with no Yb:YLF. In some embodiments, the cladding comprises a host material and an optically activated cooling material. In some embodiments, the cladding is formed from a glass or a polymer. In some embodiments, the cladding may be comprised of other Yb materials such as those described previously.

(32) In operation, the core is configured to substantially confine electromagnetic radiation from a laser. In some embodiments, the electromagnetic radiation has a peak wavelength in the infrared range. In some embodiments, the wherein the electromagnetic radiation has a peak wavelength in a range of about 1020 nm to about 1064 nm. In some embodiments, the cladding is in thermal communication with the core.

(33) In operation, the optical fiber is configured to provide optically activated cooling of the core via the electromagnetic radiation from the laser. In some embodiments, the optically activated cooling results from an optically activated cooling material in the cladding. In some embodiments, the cooling material is selected from the group of Yb:YLiF.sub.4, Yb:NaYF.sub.4, Yb:LuLiF.sub.4, Yb:KLuF.sub.4, and Yb:KYF.sub.4. In some embodiments, high-quality 10% Yb:YLF laser-cooling materials are produced through hydrothermal synthesis.

(34) In one embodiment, the hydrothermal method used to synthesize single crystals of 10% Yb:YLF. Yttrium chloride hexahydrate (YCl.sub.3.Math.6H.sub.2O) and ytterbium chloride hexahydrate (YbCl.sub.3.Math.6H.sub.2O) were of 99.999% and 99.998% purity, respectively. Lithium fluoride (LiF), lithium hydroxide monohydrate (LiOH.Math.H.sub.2O), ammonium bifluoride (NH.sub.4HF.sub.2), and ethylenediaminetetraacetic acid (EDTA) were analytical grade and used directly in the synthesis without any purification. All chemicals were purchased from Sigma-Aldrich. For the synthesis of Yb:YLF, 0.585 g (2 mmol) of EDTA and 0.168 g (4 mmol) LiOH.Math.H.sub.2O were dissolved in 10 mL Millipore DI water and heated to approximately 80° C. while stirring. After the EDTA was dissolved, 1.8 mL of 1.0 M YCl.sub.3 and 0.2 mL of 1.0 M YbCl.sub.3 were added and continuously stirred for 1 hour. This mixture is denoted as solution A. Subsequently, 0.105 g (4 mmol) of LiF and 0.34 g (8 mmol) of NH.sub.4HF.sub.2 were separately dissolved in 5 mL Millipore DI water and heated to approximately 70° C. while stirring for 1 hour. This solution is denoted as solution B. After stirring, solution B was added dropwise into solution A while stirring to form a homogeneous white suspension. After 30 minutes, the mixture was transferred to a 23 mL Teflon-lined autoclave (Parr 4747 Nickel Autoclave Teflon liner assembly) and heated to 180° C. for 72 hours in an oven (Thermo Scientific Heratherm General Protocol Oven, 65 L). After the autoclave cooled to room temperature the Yb:YLF particles were sonicated and centrifuged at 4000 rpm with ethanol and Millipore DI water for three times respectively. The final white powder was then dried at 60° C. for 12 hours followed by calcination at 300° C. for 2 hours inside a quartz tube in a furnace (Lindberg blue).

(35) FIG. 2B is a cross-section of an example fiber in accordance with the present technology. In some embodiments, as illustrated in FIG. 2B, the core material is Yb-doped glass with a small diameter. In one embodiment, the core diameter is 1.38 μm. In some embodiments, the Yb-doped glass amplifies the laser signal and generates heat inside the core. In some embodiments, the cladding consists of a thin inner region doped with YLF nanocrystals, and an outer region free of nanocrystals. In some embodiments, the cladding has a lower optical refractive index than the core. In FIG. 2B, R.sub.co and R.sub.cl are the core radius and total cladding radius, which are 0.69 μm and 62.5 μm, respectively. Additional fiber parameters are listed in Table 1.

(36) TABLE-US-00001 TABLE 1 Fiber parameters Parameters Values Core radius R.sub.co 0.69 μm Composite cladding radius 5 μm Total cladding radius R.sub.cl 62.5 μm Fiber length L 1 m Wavelength λ 1020 nm Core refractive index N.sub.co  1.52 + 3.88E−8i Cladding glass refractive index N.sub.matrix 1.4705 + 1.003E−8i Yb:YLF refractive index N.sub.YLF 1.4705 + 5.346E−6i Glass thermal conductivity κ 1.5 W/(m .Math. K) Air thermal conductivity κ.sub.s 0.02624 W/(m .Math. K) Pumping irradiance 100 kW/(m.sup.2) Room temperature T.sub.∞ 298 K

(37) In operation, the optical fiber cladding is cooled by impinging electromagnetic radiation from a laser on an optical fiber. In some embodiments, the cladding is cooled greater than 15K. In some embodiments, the optically activated cooling results from emission of upconverted, anti-Stokes photoluminescence in the cladding.

(38) The optical fiber in FIG. 2B is illustrated as cylindrical, but in other embodiments, the optical fiber may take other shapes. In one embodiment, the cylindrical optical fiber the temperature is described in three dimensions by the energy equation:

(39) $\begin{array}{cc}\rho \hat{c}=\frac{\partial T}{\partial t}=\kappa \left(\frac{1}{r}\frac{\partial }{\partial t}\left(r\frac{\partial T}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}T}{\partial {\varphi }^2}+\frac{{\partial }^{2}T}{\partial z^2}\right)+Q^m,& \left(1\right)\end{array}$

(40) in which ρ is the mass density of the fiber [kg/m.sup.3], c{circumflex over ( )} is the heat capacity [J/(kg.Math.K)], K is the thermal conductivity [W/(m.Math.K)], T is the temperature [K], Q′″ is the rate of heat generation (or depletion) per unit volume due to electromagnetic (EM) heating or cooling [J/(m.sup.3.Math.s)]. In some embodiments, the thermal properties of the core and cladding are the same because they are similar glass materials. In some embodiments, the thermal properties of the core and cladding are different. In some embodiments, the heat source is found to be nearly independent of the angular coordinate, (p. Consequently, the angular conduction term in Eq. 1 can be neglected.

(41) The energy equation is written in a dimensionless form:

(42) $\begin{array}{cc}\frac{\partial \Theta }{\partial \tau }=\frac{1}{\xi }\frac{\partial }{\partial \xi }\left(\xi \frac{\partial \Theta }{\partial \xi }\right)+{\left(\frac{R_{\infty }}{L}\right)}^2\frac{{\partial }^2\Theta }{\partial {\zeta }^2}+S\left(\zeta ,\varphi ,\zeta \right),& \left(2\right)\end{array}$

(43) by introducing the dimensionless vaniables.

(44) $\begin{array}{cc}\Theta =\frac{T-T_{\infty }}{T_{\infty }},\xi =\frac{\tau }{R_{\infty }},\zeta =\frac{z}{L},\tau =\frac{\kappa }{\rho \zeta }\frac{t}{R_{\infty }^2},S=\frac{R_{\infty }^2{Q^{\prime }}^{″}}{\kappa T_{\infty }}.& \left(3\right)\end{array}$

(45) Here Θ is the dimensionless temperature and T∞ is the temperature of the surrounding medium, which is also the initial temperature of the example fiber.

(46) The solution is in a product form

(47) $\begin{array}{cc}\Theta \left(\xi ,\zeta ,\tau \right)=\underset{m}{.Math.}\underset{n}{.Math.}{A}_{\mathrm{mn}}\left(\tau \right)X_m\left(\xi \right)Z_n\left(\zeta \right),& \left(4\right)\end{array}$
where
Xm(ξ)=J.sub.0(γmξ), m=1, 2, 3, . . .  (5)
Zn(ζ)=cos(nπζ), p=0, 1, 2, 3, . . .  (6)

(48) J.sub.0 in Eq. 5 is the zero-order Bessel function. The eigenvalues γ.sub.m satisfy the boundary condition

(49) $\begin{array}{cc}\frac{d{X}_m\left(\xi \right)}{d\xi }{.Math.}_{{\xi }_s}=-\mathrm{Bi}-X_m\left({\xi }_s\right).& \left(7\right)\end{array}$

(50) in which ξ.sub.s=R.sub.cl/R.sub.co, and the Biot number is defined by

(51) $\begin{array}{cc}\mathrm{Bi}=\frac{\mathrm{Nu}.Math.\kappa s.Math.\mathrm{Rco}}{2\kappa R_{\mathrm{cl}}}& \left(8\right)\end{array}$

(52) The Nusselt number (Nu) is given by

(53) $\begin{array}{cc}\mathrm{Nu}=\frac{2R_{\mathrm{cl}}h}{{\kappa }_s},& \left(9\right)\end{array}$

(54) where h is the heat transfer coefficient for transport between the fiber and the surroundings, and κ.sub.s is the thermal conductivity of the surroundings. For heat transfer from a cylinder to a stagnant medium Nu=0.32. The time-dependent coefficients A.sub.mn(τ) in Eq. 4 are given by

(55) $\begin{array}{cc}{A}_{\mathrm{mn}}\left(\tau \right)=\frac{1-e^{\left(-{\lambda }_{\mathrm{mn}}^2\tau \right)}}{{\lambda }_{\mathrm{nm}}^2}\stackrel{{\xi }_{\zeta }}{\underset{0}{\int }}\stackrel{1}{\underset{0}{\int }}\frac{S\left({\xi }^{\prime },\varphi ,{\zeta }^{\prime }\right){\zeta }^{\prime }{X}_m\left({\zeta }^{\prime }\right)Z_n\left({\zeta }^{\prime }\right)d{\xi }^{\prime }d{\zeta }^{\prime }}{{.Math.X_m.Math.}^2{.Math.Z_n.Math.}^2},& \left(10\right)\end{array}$

(56) in which

(57) $\begin{array}{cc}{\lambda }_{\mathrm{mn}}^2={\gamma }_m^2+{\left(\frac{R_{\omega }}{L}\right)}^2{\left(n\pi \right)}^2,& \left(11\right)\end{array}$

(58) and the norms are

(59) 0$\begin{array}{cc}{.Math.X_m.Math.}^2\stackrel{{\xi }_{\zeta }}{\underset{0}{\int }}{\xi }^{\prime }{.Math.J_0\left({\gamma }_m{\xi }^{\prime }\right).Math.}^{2}d{\xi }^{\prime },& \left(12\right)\end{array}$$\begin{array}{cc}{.Math.Z_n.Math.}^2=\underset{0}{\overset{1}{\int }}{\cos }^2\left(n\pi {\zeta }^{\prime }\right)d{\zeta }^{\prime }.& \left(13\right)\end{array}$

(60) The steady state solution is obtained by taking the limit as τ.fwdarw.∞ to give

(61) $\begin{array}{cc}{A}_{\mathrm{mn}}=\frac{\stackrel{{\xi }_{\zeta }}{\underset{0}{\int }}\underset{0}{\overset{1}{\int }}S\left({\zeta }^{\prime },\varphi ,{\zeta }^{\prime }\right){\zeta }^{\prime }{X}_m\left({\zeta }^{\prime }\right)Z_n\left({\zeta }^{\prime }\right)d{\zeta }^{\prime }d{\zeta }^{\prime }}{{\lambda }_{\mathrm{mn}}^2{.Math.X_m.Math.}^2{.Math.Z_n.Math.}^2},& \left(14\right)\end{array}$

(62) FIG. 3A-3B show the EM-field within a single-mode guided step index fiber. The volumetric rate of heat generated inside the fiber is related to the EM-field associated with the laser irradiation, which is given by the equation:

(63) $\begin{array}{cc}{Q}^m\left(r,\varphi ,z\right)=\frac{1}{2}{\sigma }_i\left(E.Math.E^{*}\right),& \left(15\right)\end{array}$

(64) where E and E* are the internal electric-field vector and its complex conjugate, respectively, and σ.sub.i is the effective conductivity of the fiber core or cladding material. In some embodiments, during lasing there are two wavelengths propagating through the fiber—the pump wavelength and the lasing wavelength. For simplicity, these two wavelengths are not distinguished from one another. In one embodiment, only one high-power fundamental mode was considered as the EM heat source. Since Yb:YLF can be refrigerated optically in a wide excitation wavelength range (1005 nm-1065 nm, at room temperature), in some embodiments, both pump and lasing wavelengths lie in this range and excite the cooling of Yb:YLF within the cladding. Because of this, in some embodiments, it is not necessary to calculate the heat sources for the pump and lasing wavelengths separately. In some embodiments, the propagating light wavelength is 1020 nm, under which excitation Yb:YLF has been predicted to reach its lowest temperature. In some embodiments, the anti-Stokes emission from YLF within the cladding is not reabsorbed and that Rayleigh scattering is negligible. Snyder and Love have tabulated the components of the electric vectors for optical fiber waveguides having a step-index profile which we use below.

(65) FIG. 3A is a graph of the radial electric field in the core and cladding of an example fiber in accordance with the present technology. The radial electric field in the core and cladding is illustrated for φ=90°. On the horizontal axis is the dimensionless distance. On the vertical axis is E.sub.r. E.sub.r along the radius is plotted as a black line. The graph shows the radial component of the electric field at z=0, φ=90° for a vertically polarized (along v=90°) laser propagating in the z direction. The highlighted discontinuity at the core-cladding interface is due to the boundary condition:

(66) The optical field extending into the cladding excites the YLF leading to local laser refrigeration. This evanescent field penetrates only a short distance into the cladding layer, so in some embodiments. the Yb:YLF nanocrystals only need to be incorporated in that narrow region. In other embodiments, the Yb:YLF nanocrystals, or other such cooling material may be incorporated in a broader region. In some embodiments, the thickness of this inner region of the cladding, which contains the cooling materials, is slightly greater than the evanescent penetration thickness. This inner cladding layer is composed YLF nanocrystals uniformly dispersed within a glass matrix. This glass matrix is not Yb-doped as the core glass, but it still absorbs slightly and generates heat. In some embodiments, the outer cladding without the nanocrystals serves only as a mechanical supporting layer to protect the core. Since there is no electric field in this region, only the heat generated inside the core and the inner cladding region are considered.

(67) FIG. 3B is a graph of the radial distribution of the EM heat source within example fibers in accordance with the present technology. On the horizontal axis is the dimensionless distance. On the vertical axis is the electromagnetic heat source. The radial distribution of the EM heat source within the fiber without Yb:YLF cladding (squares) and a fiber with 10% volume fraction of Yb:YLF mixed into the cladding (circles) is plotted. When there is no Yb:YLF cooling nanocrystals in the cladding, both the core and cladding layers act as heating sources. When Yb:YLF nanocrystals are incorporated in the cladding with a volume fraction of 10%, and the cooling efficiency of Yb:YLF is 2% (v=10%, η.sub.coolin=2%), the cladding layer acts as a cooling source. The heating in the cladding may be eliminated with a YLF volume fraction of 8%.

(68) In one embodiment, the effective electrical conductivity of the fiber core or cladding materials is given by

(69) $\begin{array}{cc}{\sigma }_i=\frac{4\pi \mathrm{Re}\left\{N_i\right\}\mathrm{Im}\left\{N_i\right\}}{\lambda \mu c},& \left(17\right)\end{array}$
in which N.sub.i=n.sub.i+ik.sub.i is the complex refractive index of the fiber core or cladding. n.sub.i=Re{N.sub.i} is the real part and k.sub.i=Im{N.sub.i} is the imaginary part of the refractive index. λ is the wavelength in free space, p is the magnetic permeability, and c is the velocity of light in vacuum. The heat source in the core is

(70) $\begin{array}{cc}{\stackrel{.}{Q}}_{\infty }^m\left(r,\varphi ,z\right)=\frac{1}{2}{\sigma }_{\infty }\left(E.Math.E^{*}\right)=\frac{2\pi n_{\mathrm{co}}{k}_{\mathrm{co}}}{\lambda \mu c}\left(E.Math.E^{*}\right).& \left(18\right)\end{array}$

(71) In some embodiments, the EM source in the cladding is composed of the heating associated from the matrix material and the cooling due to the Yb:YLF nanocrystals. The thermal source in the cladding is given by

(72) $\begin{array}{cc}{\stackrel{.}{Q}}_{\mathrm{cl}}^m\left(r,\varphi ,z\right)=\left(1-\upsilon \right)\frac{1}{2}{\sigma }_{\mathrm{cl}}^{\mathrm{matrix}}\frac{2\pi n_{\mathrm{co}}{k}_{\mathrm{co}}}{\lambda \mu c}\left(E.Math.E^{*}\right)+(-)\upsilon {\eta }_{\mathrm{cooling}}\frac{1}{2}{\sigma }_{\mathrm{cl}}^{\mathrm{YLF}}\frac{2\pi n_{\mathrm{co}}{k}_{\mathrm{co}}}{\lambda \mu c}\left(E.Math.E^{*}\right)& \left(19\right)\end{array}$

(73) where v is the volume fraction of Yb:YLF nanocrystals in the composite matrix, σ.sub.cl.sup.matrix and σ.sub.cl.sup.YLF are the effective conductivities for the matrix glass material and Yb:YLF nanocrystals in the cladding respectively, and η.sub.cooling is the cooling efficiency of the Yb:YLF. n.sub.YLF, k.sub.YLF and n.sub.matrix, k.sub.matrix are the real and imaginary terms of the refractive indices for Yb:YLF and the cladding glass materials. The competition of the heating and cooling processes in the cladding depends on the volume fraction of YLF in the cladding, the refractive indices of YLF and matrix materials, and also the overall cooling efficiency.

(74) FIG. 4A-4E show the effects of an example cladding refractive index on the EM-field. The refractive index is the most important material property for an optical fiber in that every waveguide parameter depends on it. In some embodiments, refractive indices of core and the fiber geometry are the same. For the core material, larger k results in higher absorption and increased heat generation. However, the effect of the cladding refractive index is less apparent. Varying both the real and imaginary components of the refractive index of the cladding impacts laser cooling. The core parameters were assumed to remain constant while changing the real or imaginary components of the cladding. For a single mode fiber, the single mode condition: v=(2π/λ)a(n.sub.co.sup.2−n.sub.cl.sup.2)<2405 is satisfied. Therefore, Δn=n.sub.co−n.sub.cl should be smaller than 0.12. The electric field, which is the solution of the Maxwell equations for the waveguide boundary condition, is given by
E(r,φ,z)=e(r,φ)e.sup.iβz,  (20)

(75) in which β is the propagation constant. This suggests that a smaller Δn is helpful for cooling because more energy flux is produced in the cladding layer to excite the composite cooling region adjacent to the core. However, there is a trade-off in that more pump power is required to achieve an equivalent output power from the core. Additionally, the coupling efficiency of the pump also needs to be considered.

(76) FIG. 4A is a graph of the propagation constant of an example fiber with various values of n.sub.cl in accordance with the present technology. One the horizontal axis is n.sub.cl. On the vertical axis on the left side is the β.sub.real. On the vertical axis on the right side is the β.sub.imaj. The propagation constants for fibers with various values of n.sub.cl is plotted. The real part of the propagation constant describes the rate of phase change with z, while the imaginary part constitutes the decay rate of the electric field.

(77) For small n.sub.cl, or a larger Δn, β.sub.real is smaller and the electric field is more confined within the core (as shown in FIG. 4B). β.sub.imaj decreases with increasing n.sub.cl, which implies a lower decay rate along the z-axis (as shown in FIG. 4C).

(78) FIG. 4B is a graph of the radial distribution of annular normalized E.Math.E* of an example fiber in accordance with the present technology. On the horizontal axis is the dimensionless distance. On the vertical axis is the normalized E.Math.E*. Various values of n.sub.cl are shown and distinguished by color. As the n.sub.cl value increases, the normalized E.Math.E* decreases as the dimensional distance increases. This implies that the electric field is more confined within the core.

(79) FIG. 4C is a graph of the longitudinal distribution of E.Math.E* of an example fiber in accordance with the present technology. On the horizontal axis is dimensionless distance. On the vertical axis is the normalized E.Math.E*. Color legends are the same as FIG. 4B. To compare the longitudinal decay rate, the maxima of the energy flux are normalized to 1. As the n.sub.cl value increases, the normalized E.Math.E reduces at a slower rate as the dimensionless distance increases.

(80) FIG. 4D is the propagation constant of an example fiber with various values of k.sub.cl in accordance with the present technology. On the horizontal axis is k.sub.cl. On the vertical axis is β.sub.real. When tuning the imaginary part of the cladding refractive index, β.sub.real does not change with k.sub.cl while the imaginary part of the propagation constant, β.sub.imag, increases with larger k.sub.cl. With the same β.sub.real, the radial electric field distributions are identical when k.sub.cl is varied. To reduce the energy decay along z, a smaller kc for the cladding layer is optimal.

(81) FIG. 4E is the radial distribution of E.Math.E* of an example fiber in accordance with the present technology. On the horizontal axis is dimensionless distance. On the vertical axis is E.Math.E. The propagation constant of the fiber is plotted with various values of k.sub.cl. As the dimensionless distance increases, the E.Math.E* of the fiber decreases steeply.

(82) FIG. 4F is the longitudinal distribution of E.Math.E* of an example fiber in accordance with the present technology. Color legends are the same as FIG. 4E. FIG. 4E and FIG. 4F are not normalized because E.Math.E* does not change with the fraction of Yb:YLF within the cladding leads to a decrease in the steady state temperature of the fiber. Temperature reductions of 12 K for a 5% Yb:YLF incorporated fiber and 19 K for a 10% Yb:YLF incorporated fiber (as shown in FIG. 5B) may occur. In some embodiments, a volume fraction of up to 20% YLF, would reduce temperatures even further.

(83) FIGS. 5A-5B show the temperature of the fibers with 0%, 5% and 10% Yb:YLF nanocrystals in the inner cladding layer. On the horizontal axis is dimensionless distance. On the vertical axis is temperature. In one embodiment, to evaluate the cooling performance of the nanocrystals in the fiber, the temperature distributions were calculated using the analytical solution presented above implemented using the Python programming language. The YLF volume fractions ranged from 0% to 10%. The other parameters used in the calculations are listed in Table 1. In some embodiments, to minimize the scattering in the inner layer of the cladding, the cladding glass is selected to match the real part of the refractive index of the Yb:YLF nanocrystals. This can be achieved by adjusting the concentration of dopants within the glass. In some embodiments the refractive indices of the Yb:YLF glass do not match. In some embodiments, effective medium theory for composite materials is used to calculate the effective refractive indices when the refractive indices do not match.

(84) FIG. 5A is a graph of an example radial temperature distribution of example fibers in accordance with the present technology. On the horizontal axis is the dimensionless distance. On the vertical axis is temperature. The radial temperature distribution of an example fiber containing no Yb:YLF, 5% Yb:YLF, and 10% Yb:YLF in the inner cladding is plotted. The plot shows the fiber temperature distribution from the center to the outer surface of the cladding. As the concentration of Yb:YLF in the inner cladding is increased, the radial temperature distribution decreased.

(85) FIG. 5B is a graph of an example average temperature for the whole volume of an example fiber for various amounts of Yb:YLF in accordance with the present technology. On the horizontal axis is YLF content, in the concentrations of no Yb:YLF, 5% Yb:YLF, and 10% Yb:YLF. On the vertical axis is fiber heating in change in temperature. As the concentration of Yb:YLF in the inner cladding increases, the average temperature for the whole volume of the fiber decreases. In some embodiments, the concentration of the cooling element, such as Yb:YLF can be increased to above 10%.

(86) FIG. 6 is a graph of the radial temperature distribution of example fibers with various cladding glass refractive indices in accordance with the present technology. On the horizontal axis is normalized distance. On the vertical axis is temperature. The temperature response to various cladding glass refractive indices (n.sub.matrix) when 10% of Yb:YLF nanocrystals are doped within the inner cladding are plotted.

(87) Effective medium theory may be used to calculate the effective refractive index of a composite material with 10% Yb:YLF—90% glass. The real and imaginary terms of the effective refractive index are defined as
n.sub.eff=[A+(A.sup.2+B.sup.2).sup.1/2].sup.1/2/2,  (21)
and
k.sub.eff=[−A+(A.sup.2+B.sup.2).sup.1/2].sup.1/2/2,  (22)

(88) In one embodiment, the core index N.sub.co and Yb:YLF nanocrystal index N.sub.YLF are constants listed in Table 1. The real refractive index of the cladding glass n.sub.matrix was the only variable. In some embodiments, only the change in real refractive index is considered because the imaginary refractive index had negligible effect to the radial electric field distribution according as shown in FIG. 4E. The effective cladding refractive indices calculated with the above equations are listed in Table 2.

(89) In one embodiment, the effective thermal conductivity of this inner cladding region layer was not considered because the fluoride thermal conductivity is similar to glass; and it is a very thin layer incorporated with nanocrystals compared to the entire cladding region without the nanocrystals. In this embodiment, it's reasonable to neglect the thermal conductivity variation for the heat transfer model, which considers the thermal properties of the entire fiber. However, in some embodiments, this thin layer is where the evanescent field is located. The optical properties of this region are important and the effective medium theory was applied to obtain a reliable EM-field distribution.

(90) TABLE-US-00002 TABLE 2 Waveguide parameters: effective refractive indices and propagation constants N.sub.matrix n.sub.eff k.sub.eff Propagation constant β 1.4100 1.4162 3.5638E−07 9.0490 + 6.3301E−07i 1.4280 1.4323 3.5262E−07 9.0703 + 7.0196E−07i 1.4460 1.4485 3.4883E−07 9.0962 + 7.9586E−07i 1.4640 1.4646 3.4499E−07 9.1288 + 9.4015E−07i 1.4820 1.4809 3.4110E−07 9.1716 + 1.1776E−06i 1.5000 1.4971 3.3783E−07 9.2325 + 1.6104E−06i

(91) To calculate the EM source and the resulting fiber temperature, the pumping beam was assumed to have the same transverse intensity profile as the guided mode. No matter how much the numerical aperture changes with the cladding index, the total power coupled into the beam is the same for fibers with different cladding indices. Thus the power density in the beginning of the fiber (z=0) should be normalized as

(92) $\begin{array}{cc}{\int }_0^{{\xi }_s}{\int }_0^{2\pi }E\left({\xi }^{\prime },{\varphi }^{\prime },0\right).Math.E^{*}\left({\xi }^{\prime },{\varphi }^{\prime },0\right)d{\varphi }^{\prime }d{\xi }^{\prime }=\mathrm{Irradiance}& \left(25\right)\end{array}$

(93) When the n.sub.matrix is high, the effective n.sub.cl is close to the n.sub.co. As plotted in FIG. 4B, more electric field power extends into the cladding layer. After the normalization, the power in the core is much lower than those with lower effective cladding indices, which means lower heat generated inside the core. This agrees with the calculated equilibrium temperature results in FIG. 6. When n.sub.matrix is small, the EM energy is confined more within the core due to the larger index difference between the core and cladding, and thus a higher temperature.

(94) FIG. 7 is a graph of the time-resolved temperature for various cladding thicknesses of example fibers in accordance with the present technology. On the horizontal axis is time in seconds. On the vertical axis is the maximum temperature. FIG. 7 shows the time dependent center temperature (=0) of the fiber as a function of cladding thickness. Clearly, as the cladding thickness increases, the overall temperature of the fiber increases. Longer times are required to reach steady state with increased cladding thickness. Ideally the cladding should be as small as possible, but sufficient thickness is needed to ensure satisfactory mechanical properties or a multi-mode pumped cladding configuration.

(95) As used herein, the term “about” indicates a value can vary plus or minus 5%.