1. Introduction

In the past decade, there has been significant progress in average power scaling of fiber lasers. Currently, fiber lasers are used in a wide range of industrial processes from marking and engraving to heavy duty cutting and welding. There is still a strong interest in further average power scaling of fiber lasers, mostly driven by new applications in science [1,2] and defense [3]. A quantitative understanding of thermal lensing is becoming increasingly critical for any further average power scaling efforts. It is also important to understand the impact of thermal lensing on transverse mode instability (TMI) [4–6].

Considering the importance of thermal lensing, there are surprisingly few published general studies of this effect in optical fibers. Dawson et al. studied the effect of thermal lensing based on a parabolic thermal profile and an ABCD matrix to approximate Gaussian beam propagation [7]. The guidance of the original optical fiber is ignored in this study. The balance of wave guidance due to thermal lensing and diffraction leads to an approximate power limit for thermal lensing. Such analysis is invalid at low heat loads where the original fiber design plays a dominant role in waveguiding. Thermal-optical effects were studied numerically for a certain fiber amplifier arrangement in [8]. Thermal lensing and transition of an original single-mode fiber to a multimode fiber at high heat load were observed. No systematic study was performed. Recently the reduction of mode field diameter (MFD) at high average powers was experimentally characterized for a number of optical fibers [9]. The observed MFD reduction was much more pronounced in fibers with larger MFD at a similar heat load. One of these fibers was simulated using a 3D model for its MFD reduction at high thermal loads [10]. Similar MFD reduction was also observed in thermally guiding and index-anti-guiding fiber [11]. A theoretical analysis of optical modes at various thermal loads for this type of fibers was also conducted [12]. No general studies of thermal lensing were performed in these works.

In this work, we attempt to develop a full quantitative analysis of thermal lensing in optical fibers. We first study thermal lensing in optical fibers based on a perturbation method. Although this analysis is inaccurate for strong thermal lensing, it does provide an insight into the key parameters involved in thermal lensing. We then use the normalized thermal lensing parameter developed in the perturbation study in a more accurate numerical model. A cylindrical layered optical mode solver is used in the numerical study. A numerical method for solving a heat conduction equation with a spatially non-uniform heat load and cylindrical symmetry is developed. The parabolic thermal profile based on approximated uniform heating which was used frequently in the past is found to underestimate the core temperature. A simple criteria for thermal lensing is developed. The impact of thermal lensing on TMI is also studied and TMI is found to be generally suppressed by thermal lensing. This is largely due to the fact that thermal lensing pulls power in optical modes towards the center of the optical fiber. This pulling effect is stronger for the fundamental mode than for higher-order modes, consequently reducing the overlap between the fundamental mode and higher-order modes.

2. Perturbation analysis of thermal lensing

The perturbation method used is very similar to that described in [13]. The thermal-optic effect is introduced by

The temperature profile is approximated by the parabolic solution for uniform heating in the core with a heat load of Q₀ (w/m) and we have ignored the temperature change outside the core by setting ΔT = 0 for r>ρ, where ρ is the core radius. This is reasonable for multimode fibers commonly found in fiber lasers as the optical power is mostly in the core in these cases. It is however not a good approximation for single-mode fibers where there is significant optical power outside the core.

 

Fig. 1 Mode collapse under the influence of thermal lensing. Equation (3) is evaluated for normalized frequency V = 3-8. A numerical simulation is also performed for comparison for fibers with core diameters of 10μm, 20μm and 30μm and NA of 0.06. The results are plotted as MFD for near-field MFD and eMFD for effective MFD, i.e. 2(A_eff/π) ^½.

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The relative mode size is plotted against the normalized thermal lensing parameter ξQ₀w₀² in Fig. 1. It can be seen that the rate of change in the relative mode size is slow at small ξQ₀w₀², but accelerates at larger ξQ₀w₀². There is very little difference in the curves for different V values. The relative mode size can collapse to small values at high heat load and the rate of change slows down when ξQ₀w₀²>2. The most interesting observation is the fact that the effect of thermal lensing is fully characterized by the normalized thermal lensing parameter ξQ₀w₀², where ξ is fully determined by material properties and laser wavelength. This normalized thermal lensing parameter scales linearly with heat load Q₀ and also scales quadratically with MFD. This MFD dependence of thermal lensing was also shown in [7]. Since the total effect of thermal lensing is determined by the integrated effect seen by the entire mode, it is reasonable to expect it to scale with the mode area, i.e. w₀².

3. Numerical analysis of thermal lensing

For a more accurate study of the impact of thermal lensing, we need to conduct a numerical study. We will focus on the regime where the fundamental mode dominates and ignore any mode distortion due to bending. Since we are only dealing with optical modes with a cylindrical symmetry in this case, we will use an optical mode solver for an arbitrary refractive index profile with a cylindrical symmetry. We already have established such a vector optical mode solver for an earlier work [14,15]. The formalism is very similar to that described in [16].

We also need to deal with the temperature profile resulting from the heat load from an optical mode with cylindrical symmetry. In our analysis, we have assumed that heat load is proportional to the local mode intensity; consequently we have ignored the effect of gain saturation. Gain saturation will lead to a flattening out of heat load in the center of the core and therefore slightly weaken the effect of thermal lensing.

Since a simple parabolic solution exists for the temperature profile in the case of a spatially uniform heat load in a cylinder, it has been used in many previous thermal analysis of fiber lasers [7,17]. We need to have a more accurate numerical model for the temperature profile in a cylinder given an arbitrary cylindrical heat load profile. We start by dividing the area with the heat source into many fine layers so that we can assume each layer has a uniform heat load and we can then use the parabolic solution over each layer. This approach, however, does not have enough free parameters to ensure the solution has both continuity and continuity in the 1st order derivative at the boundary between layers, both required for a rigorous solution. We subsequently devised a scheme where we first divide the area with the heat load into many fine layers with equal thickness. We then divide each layer into two sub-layers, one sub-layer with the total heat load for the layer and one sub-layer without source. We can then use the known parabolic solution over the sub-layer with the source and the known logarithmic solution for the sub-layer without source. The relative thickness of the sub-layers can be determined by the continuity condition at the boundary between layers. This new scheme allows us to find an accurate solution while ensuring necessary continuities at all the boundaries. A detailed derivation is provided in Appendix B. In the simulations in this work, the region with the heat load is typically divided into 100-200 layers. This is found to be sufficient for a stable solution.

A comparison of the temperature profiles from the numerical model used in this study and the uniform heat load is shown in Fig. 2. The fiber has a NA of 0.06 and a core diameter of 20μm, operated at 1.03μm. We use a cladding diameter of 400μm throughout this study. The total heat load in the core is 64w/m in both cases and there is no heat load outside the core. The normalized heat load profile, i.e. the normalized mode intensity profile, is also shown. The numerical model used an iteration process described in the next paragraph to find the optical mode under heat load. It can be seen clearly that the solution in the cladding is the same for both cases and the uniform heat load model underestimates the temperature in the core. This is due to the fact that more heat is deposited near the core center than that accounted in the uniform heat load model.

 

Fig. 2 Comparison of the temperature profile from the numerical method used in this work and the parabolic solution for a uniform heat load. ΔT is set to be 0K at the cladding boundary. Core diameter is 20μm; cladding diameter is 400μm and NA is 0.06. The total heat load is 64w/m in both cases. The laser wavelength is at 1.03μm. The normalized heat load profile, i.e. normalized mode intensity profile, is also shown.

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In the following analysis of thermal lensing in an optical fiber, we first find the fundamental optical mode in the fiber without any heat load. We then apply one M^th of the heat load with a heat load profile equaling the fundamental optical mode intensity. The temperature profile is then calculated, which is then used to calculate the refractive index profile of the new fiber. The fundamental mode is then found for this new fiber. Two M^th of the heat load is then used with a heat load profile equaling the new fundamental mode intensity for the next iteration. This is repeated until total heat load is applied. This slow increase of heat load over many iterations is essential to keep the change in optical mode profile to a minimal for each iteration. Otherwise it can lead to numerical instabilities especially at large heat loads. We found M = 30 to be adequate to ensure stability and convergence for our analysis. This is used throughout this study.

The evolution of MFD (near-field MFD) and eMFD (effective MFD obtained from effective mode area) over the gradual increase of heat load for the case in Fig. 2 is shown in Fig. 3. The relative change in MFD and eMFD is shown on the right vertical axis. The reduction of mode size due to thermal lensing when heat load is gradually increased can be clearly seen. Also can be seen is that the mode size stabilizes at a constant value after the 31st run when the heat load is no longer increased. This demonstrates the convergence of the solution and stability of the numerical process.

 

Fig. 3 Typical run for the case in Fig. 2. Core diameter is 20μm and NA is 0.06. The total heat load is 64w/m. The laser wavelength is at 1.03μm. Relative changes are shown on the vertical axis on the right. The heat load is gradually increased over 30 steps.

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The refractive index profiles obtained by the numerical model for the 20μm-core fiber are shown in Fig. 4 for various total heat loads. The refractive index profile is truncated at a radius 20μm, i.e. twice the core radius in this case. In our following analysis of optical mode, we are only concerned with the part of the waveguide seen by the optical mode of interest. We typically truncate the refractive index profile to 2 to 5 times of the core radius depending on guiding strength of the waveguide. Using an unnecessarily large cladding radius in the analysis can lead to poor numerical stability in the optical mode solver in addition to longer computation time. Care is taken in each case to ensure that the truncation does not compromise accuracy.

 

Fig. 4 Refractive index profiles under various heat loads. Core diameter is 20μm and NA is 0.06. The laser wavelength is at 1.03μm.

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We then proceed to study the impact of thermal lensing on mode size for three fibers with core diameters of 10μm, 20μm and 30μm. All fibers have a NA of 0.06 and a cladding diameter of 400μm. This study is conducted for a wavelength of 1.06μm. The respective V values are 1.778, 3.557, and 5.335. The 10μm-core fiber is in the single-mode regime and the other two fibers are in the multimode regime. The results are shown in Fig. 1 for both MFD and eMFD (see figure caption for detailed definition). It can be seen that the results from the perturbation method are reasonable for a small normalized thermal lensing parameter, but overestimate the effect of thermal lensing when ξQ₀w₀²>0.1. The results for the three fibers at ξQ₀w₀²>2 are very close. In this regime, wave guidance is almost entirely from the effect of thermal lensing and the original waveguide plays a very small part. At smaller ξQ₀w₀², the single-mode fiber suffers slightly more mode size reduction. The mode is much larger than the core in the single-mode regime and this enhances the impact of thermal lensing on mode size. MFD and eMFD follows each other fairly closely. A convenient place to set the thermal lensing threshold is ξQ₀w₀² = 0.5, where the mode size is reduced by ~10% and the effective mode area by ~20%. The thermal lensing limit is put at ξQ₀w₀² = 0.18 in [7]. This study is able to quantify the change in mode size for this condition for the first time; the mode size is reduced by ~4% and the effective mode area by ~8% at ξQ₀w₀² = 0.18.

Using the numerical study shown in Fig. 1, we can also provide a theoretical mode size change for the experimental work in [9]. Quantum defect heating is used to convert heat load to extracted power. The average heat load over the amplifier is used, since this can be easily found in the paper but not the more relevant peak heat load. This will underestimate the effect of thermal lensing in this case. This is shown in Fig. 5. For three fibers with smallest MFDs, the predicted MFDs are very close to those measured (within few percent). The divergence becomes large for the fibers with large MFD (>80μm). The measured data for those fibers are highly scattered, indicating much larger measurement errors in this regime. This model does not consider effects such as anti-crossings with modes originated in the cladding and photo-darkening, which are known to take place in some of these fibers. It nevertheless provides a reasonable qualitative agreement with the measurements even for fibers with large MFD.

 

Fig. 5 Simulated MFD for the fibers in [9]. Quantum defect heating is used to convert heat load to extracted power. Average heat load is used.

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4. The impact of thermal lensing on TMI

There is a strong interest in understanding the impact of thermal lensing on TMI. TMI can be quantified by the TMI nonlinear coefficient χ defined in [6]. TMI threshold is inverse proportional to χ and also dependent on input higher-order-mode power and amplifier configuration. Since we can easily evaluate the refractive index profile of the fiber under thermal loading, we just need to find the fundamental and higher-order modes to evaluate χ under the influence of thermal lensing. We have assumed that the fundamental mode is dominating in this case and conducted this study for the LP₁₁ mode. Our cylindrical optical vector cannot directly find the LP₁₁ mode, but can find its constituent TE₀₁ and HE₂₁ modes. These modes have the same radial intensity profile as LP₁₁ (see Table14-1 in [18]). We therefore found the radial mode intensity profile for the HE₂₁ mode and used this in Eq. (28) in [6] for our study.

We studied 5 fibers with core diameters of 10μm, 15μm, 20μm, 25μm and 30μm respectively. The fiber NA is 0.06 and cladding diameter is 400μm. This study is conducted at a wavelength 1.06μm. The V values are respectively 1.778, 2.667, 3.557, 4.446, and 5.335. The results are shown in Fig. 6. The 10μm-core fiber is single mode at low heat load, but the LP₁₁ mode is guided when ξQ₀w₀²>0.084. Its TMI nonlinear coupling coefficient χ increases until ξQ₀w₀² = ~1 as the LP₁₁ mode is increasingly guided. When ξQ₀w₀²>1, thermal lensing becomes significant and χ starts to decrease. Thermal lensing pulls all modes to the core center, but this is more significant for the fundamental mode (see Fig. 7). This lowers the overlap between the LP₀₁ and LP₁₁ modes, consequently leading to a reduction in χ. A similar trend can be seen for the 15μm-core fiber. Since the LP₁₁ is already well guided in this fiber with heat load, the initial increase in χ at low heat load is less pronounced. χ starts to decrease when ξQ₀w₀²>0.3. The remaining fibers have similar χ at low heat load; this is due to that fact that χ changes very slowly at large V [6]. For these fibers, χ starts to decrease significantly when ξQ₀w₀²>0.3. Similar studies were conducted for LP₀₂ mode and similar trends were obtained. This reduction of TMI at high thermal load is, however, of limited practical use as the effect of thermal lensing is significant at this point. For most practical fibers, the TMI threshold is also well below this thermal load.

 

Fig. 6 Simulated TMI nonlinear coupling coefficient for fibers with core diameters of 10μm, 15μm, 20μm, 25μm and 30μm respectively. The fiber NA is 0.06 and cladding diameter is 400μm. This study is conducted at a wavelength of 1.06μm. Contour lines indicate constant thermal load.

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Fig. 7 The LP₀₁ and LP₁₁ modes at 1.06μm in a fiber with a NA of 0.06 and a core diameter of 20μm without heat load and with a heat load of 1000w/m, i.e. ξQ₀w₀² = 2.19. The cladding diameter is 400μm.

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5. Conclusions

We have studied thermal lensing effects in optical fibers with both a perturbation method and numerical method and have developed a normalized thermal lensing parameter. A simple thermal lensing threshold condition is also developed. We have further studied the impact of thermal lensing on TMI and found that strong thermal lensing leads to a reduction in TMI.

6 Appendix A. Detailed perturbation analysis of thermal lensing

This derivation is very similar to that in [13]. Readers are encouraged to read [13] for a fuller description. With a perturbation of a temperature profile ΔT(r), the waveguide equation can be re-written in a normalized cylindrical coordination where R=wr and w is the spot size of the optical mode.

(5)$${{\displaystyle {{\displaystyle \nabla }}_t\psi (R)+\left\{{{\displaystyle w}}^2{{\displaystyle k}}_0^2{{\displaystyle n}}_{0w}^2(R)\left[1+2{{\displaystyle k}}_T\Delta T(R)\right]-{{\displaystyle w}}^2{{\displaystyle \beta }}^2-\frac{{{\displaystyle m}}^2}{{{\displaystyle R}}^2}\right\}\psi (R)=0}}_{}$$

where ∇_t = ∂²/∂R² + (1/R)∂/∂R is the transverse operator; ψ(R) is radial field profile of the local mode; k₀ is the vacuum wave number, m is the azimuthal mode number, β is the propagation constant of the local mode, and n_0w is the index profile of the unperturbed fiber in the normalized coordination where R = wr.

For the unperturbed fiber, similarly we have

(6)$${{\displaystyle {{\displaystyle \nabla }}_t{{\displaystyle \psi }}_0(R)+\left\{{{\displaystyle w}}_0^2{{\displaystyle k}}_0^2{{\displaystyle n}}_{0w0}^2(R)-{{\displaystyle w}}_0^2{{\displaystyle \beta }}_0^2-\frac{{{\displaystyle m}}^2}{{{\displaystyle R}}^2}\right\}{{\displaystyle \psi }}_0(R)=0}}_{}$$

where the subscript 0 indicates it belongs to mode of the unperturbed fiber, R = w₀r is normalized against w₀ in this case and n_0w0 is the index profile of the unperturbed fiber in the normalized coordination where R = w₀r.

We can now multiply Eq. 5 by ψ₀ and then subtract Eq. 6 multiplied by ψ. The resulting equation is integrated over the cross section of the fiber, i.e. A∞. We then have

(7)$${{\displaystyle {\displaystyle \underset{A\infty }{\iint }\left[{{\displaystyle w}}_0^2\left({{\displaystyle k}}_0^2{{\displaystyle n}}_{0w0}^2-{{\displaystyle \beta }}_0^2\right)\right]-{{\displaystyle w}}^2\left({{\displaystyle k}}_0^2{{\displaystyle n}}_{0w}^2-{{\displaystyle \beta }}^2\right)\psi {{\displaystyle \psi }}_{0}dA=2{{\displaystyle k}}_T{{\displaystyle w}}^2{{\displaystyle k}}_0^2{\displaystyle \underset{A\infty }{\iint }{{\displaystyle n}}_{0w}^2\Delta TdA+}}{\displaystyle \underset{A\infty }{\iint }{{\displaystyle \psi }}_0{{\displaystyle \nabla }}_t\psi -\psi {{\displaystyle \nabla }}_t{{\displaystyle \psi }}_{0}dA}}}_{}^{}$$

The last term in Eq. (7) can be converted into a line integral at infinity and is equal to zero for a confined optical mode. We then have

(8)$${{\displaystyle {{\displaystyle w}}^2{{\displaystyle \beta }}^2-{{\displaystyle w}}_0^2{{\displaystyle \beta }}_0^2=2{{\displaystyle k}}_T{{\displaystyle w}}^2{{\displaystyle k}}_0^2{{\displaystyle n}}_T^2-{{\displaystyle k}}_0^2\left({{\displaystyle w}}_0^2{{\displaystyle n}}_a^2-{{\displaystyle w}}^2{{\displaystyle n}}_b^2\right)}}_{}^{}$$

This can be written as

(9)$${{\displaystyle {{\displaystyle w}}^2=\frac{{{\displaystyle w}}_0^2\left({{\displaystyle k}}_0^2{{\displaystyle n}}_a^2-{{\displaystyle \beta }}_0^2\right)-2{{\displaystyle k}}_T{{\displaystyle w}}^2{{\displaystyle k}}_0^2{{\displaystyle n}}_T^2}{{{\displaystyle k}}_0^2{{\displaystyle n}}_b^2-{{\displaystyle \beta }}^2}}}_{}^{}$$

where

(10)$${{\displaystyle {{\displaystyle n}}_a^2=\frac{{\displaystyle \underset{A\infty }{\iint }{{\displaystyle n}}_{0w0}^2(R)\psi (R){{\displaystyle \psi }}_0(R)dA}}{{\displaystyle \underset{A\infty }{\iint }\psi {{\displaystyle (R)\psi }}_0(R)dA}}}}_{}^{}{{\displaystyle {{\displaystyle n}}_b^2=\frac{{\displaystyle \underset{A\infty }{\iint }{{\displaystyle n}}_{0w}^2(R)\psi (R){{\displaystyle \psi }}_0(R)dA}}{{\displaystyle \underset{A\infty }{\iint }\psi (R){{\displaystyle \psi }}_0(R)dA}}}}_{}^{}{{\displaystyle {{\displaystyle n}}_T^2=\frac{{\displaystyle \underset{A\infty }{\iint }{{\displaystyle n}}_{0w}^2(R)\Delta T(R)\psi (R){{\displaystyle \psi }}_0(R)dA}}{{\displaystyle \underset{A\infty }{\iint }\psi {{\displaystyle (R)\psi }}_0(R)dA}}}}_{}^{}$$

We then repeat this process with the waveguide equation in cylindrical coordination without normalization. The waveguide equation for the fiber perturbed by the temperature profile in this case is

(11)$${{\displaystyle {{\displaystyle \nabla }}_t\psi (r)+\left\{{{\displaystyle k}}_0^2{{\displaystyle n}}_0^2(r)\left[1+2{{\displaystyle k}}_T\Delta T(r)\right]-{{\displaystyle \beta }}^2-\frac{{{\displaystyle m}}^2}{{{\displaystyle r}}^2}\right\}\psi (R)=0}}_{}$$

The waveguide equation for the unperturbed fiber is

(12)$${{\displaystyle {{\displaystyle \nabla }}_t{{\displaystyle \psi }}_0(r)+\left\{{{\displaystyle k}}_0^2{{\displaystyle n}}_0^2(r)-{{\displaystyle \beta }}_0^2-\frac{{{\displaystyle m}}^2}{{{\displaystyle R}}^2}\right\}{{\displaystyle \psi }}_0(r)=0}}_{}$$

Following a similar procedure to that for Eqs. (5) and 6, we have

(13)$${{\displaystyle {{\displaystyle \beta }}^2-{{\displaystyle \beta }}_0^2=2{{\displaystyle k}}_T{{\displaystyle k}}_0^2{{\displaystyle n}}_{Tr}^2}}_{}^{}$$

where

(14)$${{\displaystyle {{\displaystyle n}}_{Tr}^2=\frac{{\displaystyle \underset{A\infty }{\iint }{{\displaystyle n}}_0^2(r)\Delta T(r)\psi (r){{\displaystyle \psi }}_0(r)dA}}{{\displaystyle \underset{A\infty }{\iint }\psi (r){{\displaystyle \psi }}_0(r)dA}}}}_{}^{}$$

We can evaluate Eqs. 10 and 14 for the parabolic temperature expressed in Eq. 2 and Gaussian mode field profile expressed in

(15)$${{\displaystyle \psi (r)={{\displaystyle {{\displaystyle e}}_{}^{}}}^{-\frac{1}{2}\frac{{{\displaystyle r}}_{}^2}{{{\displaystyle w}}_{}^2}}}}_{}^{}$$

The results are

(16)$${{\displaystyle {{\displaystyle n}}_a^2\approx {{\displaystyle n}}_{co}^2}}_{}^{}{{\displaystyle {{\displaystyle \begin{array}{cc}& \end{array}n}}_b^2\approx {{\displaystyle n}}_{co}^2}}_{}^{}{{\displaystyle {{\displaystyle \begin{array}{cc}& \end{array}n}}_T^2\approx \frac{{{\displaystyle n}}_{co}^2{{\displaystyle Q}}_0}{8\pi \kappa {{\displaystyle \rho }}^2}\left({{\displaystyle \rho }}^2+\frac{{{\displaystyle w}}^2}{2}\right)}}_{}^{}$$

(17)$${{\displaystyle {{\displaystyle n}}_{Tr}^2\approx \frac{{{\displaystyle n}}_{co}^2{{\displaystyle Q}}_0}{4\pi \kappa {{\displaystyle \rho }}^2}\left({{\displaystyle \rho }}^2+\frac{2}{\frac{1}{{{\displaystyle w}}^2}+\frac{1}{{{\displaystyle w}}_0^2}}\right)}}_{}^{}$$

Substituting Eq. 17 into Eq. 13, we have

(18)$${{\displaystyle {{\displaystyle \beta }}_{}^2-{{\displaystyle \beta }}_0^2=\frac{{{\displaystyle k}}_T{{\displaystyle k}}_0^2{{\displaystyle n}}_{co}^2{{\displaystyle Q}}_0}{2\pi \kappa {{\displaystyle \rho }}^2}\left({{\displaystyle \rho }}^2+\frac{2}{\frac{1}{{{\displaystyle w}}^2}+\frac{1}{{{\displaystyle w}}_0^2}}\right)}}_{}^{}$$

Substituting Eqs. 16 and 18 into Eq. 9, we have

(19)$${{\displaystyle \frac{1}{2}{{\displaystyle \xi {{\displaystyle Q}}_0{{\displaystyle w}}_0^2\gamma }}^6+\left({{\displaystyle U}}^2+3\xi {{\displaystyle Q}}_0{{\displaystyle \rho }}^2+\frac{9}{2}\xi {{\displaystyle {{\displaystyle Q}}_{0}w}}_0^2\right){{\displaystyle \gamma }}^4+3{{\displaystyle \xi {{\displaystyle Q}}_0{{\displaystyle \rho }}^2\gamma }}^2-{{\displaystyle U}}^2=0}}^{}$$

where γ=w/w₀, U is the core parameter as normally defined for optical fibers, and

(20)$${{\displaystyle \xi =\frac{{{\displaystyle k}}_T{{\displaystyle k}}_0^2{{\displaystyle n}}_{co}^2}{4\pi \kappa }}}_{}^{}$$

7 Appendix B. Temperature profile with arbitrary cylindrical heating

The cylindrical heat load is described by Q(r) in w/m/μm² in a cylindrical coordination. The cladding radius is b. We will also assume that the fiber is cooled and the cladding surface temperature is constant at T₀, i.e. T(b)= T₀. The heat conduction equation in this case is given by

(21)$${{\displaystyle \frac{{{\displaystyle \partial }}^{2}T(r)}{{{\displaystyle \partial r}}^2}+\frac{1}{r}\frac{\partial T(r)}{\partial r}+\frac{Q(r)}{\kappa }=0}}_{}^{}$$

In areas with no source, the solution is known,

(22)$${{\displaystyle T(r)={{\displaystyle C}}_1+{{\displaystyle C}}_2\ln (r)}}_{}^{}$$

where C₁ and C₂ are constants to be determined by the boundary conditions. In areas with a uniform heat source with a heat load Q, the solution is also known

(23)$${{\displaystyle T(r)=C-\frac{Q}{4\kappa }{{\displaystyle r}}^2}}_{}^{}$$

where C is a constant to be determined by the boundary condition. The heat load extends to r = a. There is no heat load beyond r = a.

We first divide the area between r=0 and r=a into N concentric layers with an equal thickness of a/N. We further divide each layer into two concentric sub-layers, one with a constant source of the heat load for the layer with an outer edge at r_na and one without any source with an outer edge at r_n=n×a/N (see Fig. 8). Starting from the innermost circle, i.e. n=1, the solution for both the areas with source and without source can be found while ensuring continuity and the 1st order derivative continuity at the internal boundary. They are respectively

(24)$${{\displaystyle {{\displaystyle T}}_1(r)={{\displaystyle T}}_c-\frac{{{\displaystyle Q}}_1}{4\kappa }\frac{{{\displaystyle r}}_1^2}{{{\displaystyle r}}_{1a}^2}{{\displaystyle r}}^2}}_{}^{}$$

(25)$${{\displaystyle {{\displaystyle T}}_{1a}(r)={{\displaystyle T}}_c-\frac{{{\displaystyle Q}}_1}{4\kappa }{{\displaystyle r}}_1^2-\frac{{{\displaystyle Q}}_1}{2\kappa }{{\displaystyle r}}_1^2\ln \left(\frac{r}{{{\displaystyle r}}_{1a}}\right)}}_{}^{}$$

where T_c is the temperature at the core center. For the next section n = 2, ensuring continuity at the boundary between the 1st and the 2nd layers and continuity and the 1st order derivative continuity at the internal boundary, we similarly have

 

Fig. 8 Illustration of the scheme for solving the heat conduction equation with an arbitrary heat load profile with cylindrical symmetry.

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(26)$${{\displaystyle {{\displaystyle T}}_2(r)={{\displaystyle T}}_{1a}({{\displaystyle r}}_1)+\frac{{{\displaystyle Q}}_2}{4\kappa }\frac{{{\displaystyle r}}_2^2-{{\displaystyle r}}_1^2}{{{\displaystyle r}}_{2a}^2-{{\displaystyle r}}_1^2}\left({{\displaystyle r}}_1^2-{{\displaystyle r}}^2\right)}}_{}^{}$$

(27)$${{\displaystyle {{\displaystyle T}}_{2a}(r)={{\displaystyle T}}_{1a}({{\displaystyle r}}_1)+\frac{{{\displaystyle Q}}_2}{4\kappa }\left({{\displaystyle r}}_1^2-{{\displaystyle r}}_2^2\right)-\frac{{{\displaystyle Q}}_2}{2\kappa }\frac{{{\displaystyle r}}_2^2-{{\displaystyle r}}_1^2}{{{\displaystyle r}}_{2a}^2-{{\displaystyle r}}_1^2}{{\displaystyle r}}_{2a}^2\ln \left(\frac{r}{{{\displaystyle r}}_{2a}}\right)}}_{}^{}$$

The continuity of the 1st order derivative at the boundary between the first and the second layer requires (note that r_1a can be chosen arbitrarily)

(28)$${{\displaystyle {{\displaystyle r}}_{2a}^2={{\displaystyle r}}_1^2+\frac{{{\displaystyle Q}}_2}{{{\displaystyle Q}}_1}\left({{\displaystyle r}}_2^2-{{\displaystyle r}}_1^2\right)}}_{}^{}$$

For the next layer n=3, we have

(29)$${{\displaystyle {{\displaystyle T}}_3(r)={{\displaystyle T}}_{2a}({{\displaystyle r}}_2)+\frac{{{\displaystyle Q}}_3}{4\kappa }\frac{{{\displaystyle r}}_3^2-{{\displaystyle r}}_2^2}{{{\displaystyle r}}_{3a}^2-{{\displaystyle r}}_2^2}\left({{\displaystyle r}}_2^2-{{\displaystyle r}}^2\right)}}_{}^{}$$

(30)$${{\displaystyle {{\displaystyle T}}_{3a}(r)={{\displaystyle T}}_{2a}({{\displaystyle r}}_2)+\frac{{{\displaystyle Q}}_3}{4\kappa }\left({{\displaystyle r}}_2^2-{{\displaystyle r}}_3^2\right)-\frac{{{\displaystyle Q}}_3}{2\kappa }\frac{{{\displaystyle r}}_3^2-{{\displaystyle r}}_2^2}{{{\displaystyle r}}_{3a}^2-{{\displaystyle r}}_2^2}{{\displaystyle r}}_{3a}^2\ln \left(\frac{r}{{{\displaystyle r}}_{3a}}\right)}}_{}^{}$$

The continuity of the 1st order derivative at the boundary between the second and the third layer requires

(31)$${{\displaystyle {{\displaystyle r}}_{3a}^2={{\displaystyle r}}_2^2+\frac{{{\displaystyle Q}}_3}{{{\displaystyle Q}}_2}\left({{\displaystyle r}}_3^2-{{\displaystyle r}}_2^2\right)\frac{{{\displaystyle r}}_{2a}^2-{{\displaystyle r}}_1^2}{{{\displaystyle r}}_2^2-{{\displaystyle r}}_1^2}\frac{{{\displaystyle r}}_2^2}{{{\displaystyle r}}_{2a}^2}}}_{}^{}$$

For n=N, we have

(32)$${{\displaystyle {{\displaystyle T}}_N(r)={{\displaystyle T}}_{(N-1)a}({{\displaystyle r}}_{N-1})+\frac{{{\displaystyle Q}}_N}{4\kappa }\frac{{{\displaystyle r}}_N^2-{{\displaystyle r}}_{N-1}^2}{{{\displaystyle r}}_{Na}^2-{{\displaystyle r}}_{N-1}^2}\left({{\displaystyle r}}_{N-1}^2-{{\displaystyle r}}^2\right)}}_{}^{}$$

(33)$${{\displaystyle {{\displaystyle T}}_{Na}(r)={{\displaystyle T}}_{(N-1)a}({{\displaystyle r}}_{N-1})+\frac{{{\displaystyle Q}}_N}{4\kappa }\left({{\displaystyle r}}_{N-1}^2-{{\displaystyle r}}_N^2\right)-\frac{{{\displaystyle Q}}_N}{2\kappa }\frac{{{\displaystyle r}}_N^2-{{\displaystyle r}}_{N-1}^2}{{{\displaystyle r}}_{Na}^2-{{\displaystyle r}}_{N-1}^2}{{\displaystyle r}}_{Na}^2\ln \left(\frac{r}{{{\displaystyle r}}_{Na}}\right)}}_{}^{}$$

The continuity of the 1st order derivative at the boundary between the (N-1)^th and the N^th layer requires

(34)$${{\displaystyle {{\displaystyle r}}_{Na}^2={{\displaystyle r}}_{N-1}^2+\frac{{{\displaystyle Q}}_N}{{{\displaystyle Q}}_{N-1}}\left({{\displaystyle r}}_N^2-{{\displaystyle r}}_{N-1}^2\right)\frac{{{\displaystyle r}}_{(N-1)a}^2-{{\displaystyle r}}_{N-2}^2}{{{\displaystyle r}}_{N-1}^2-{{\displaystyle r}}_{N-2}^2}\frac{{{\displaystyle r}}_{N-1}^2}{{{\displaystyle r}}_{(N-1)a}^2}}}_{}^{}$$

For area without source, i.e. r>a

(35)$${{\displaystyle T(r)={{\displaystyle T}}_{(N-1)a}({{\displaystyle r}}_{N-1})+\frac{{{\displaystyle Q}}_N}{4\kappa }\left({{\displaystyle r}}_{N-1}^2-{{\displaystyle r}}_N^2\right)-\frac{{{\displaystyle Q}}_N}{2\kappa }\frac{{{\displaystyle r}}_N^2-{{\displaystyle r}}_{N-1}^2}{{{\displaystyle r}}_{Na}^2-{{\displaystyle r}}_{N-1}^2}{{\displaystyle r}}_{Na}^2\ln \left(\frac{r}{{{\displaystyle r}}_{Na}}\right)}}_{}^{}$$

The boundary condition T(b)=T₀ requires

(36)$${{\displaystyle {{\displaystyle T}}_0={{\displaystyle T}}_{(N-1)a}({{\displaystyle r}}_{N-1})+\frac{{{\displaystyle Q}}_N}{4\kappa }\left({{\displaystyle r}}_{N-1}^2-{{\displaystyle r}}_N^2\right)-\frac{{{\displaystyle Q}}_N}{2\kappa }\frac{{{\displaystyle r}}_N^2-{{\displaystyle r}}_{N-1}^2}{{{\displaystyle r}}_{Na}^2-{{\displaystyle r}}_{N-1}^2}{{\displaystyle r}}_{Na}^2\ln \left(\frac{b}{{{\displaystyle r}}_{Na}}\right)}}_{}^{}$$

Funding

Joint Technology Office High Energy Lasers MRI program (911NF-12-1-0332).

Acknowledgment

This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under contract/grant number W911NF-12-1-0332 through a Joint Technology Office MRI program.

References and links

1. T. Katsouleas, “Accelerator physics: Electrons hang ten on laser wake,” Nature 431(7008), 515–516 (2004). [CrossRef]   [PubMed]  

2. J. J. Macklin, J. D. Kmetec, and C. L. Gordon 3rd, “High-Order Harmonic Generation Using Intense Femtosecond Pulses,” Phys. Rev. Lett. 70(6), 766–769 (1993). [CrossRef]   [PubMed]  

3. J. Hecht, “Half a century of laser weapons,” Opt. Photonics News 20(2), 14–21 (2009). [CrossRef]  

4. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011). [CrossRef]   [PubMed]  

5. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011). [CrossRef]   [PubMed]  

6. L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013). [CrossRef]   [PubMed]  

7. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008). [CrossRef]   [PubMed]  

8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express 19(24), 23965–23980 (2011). [CrossRef]   [PubMed]  

9. F. Jansen, F. Stutzki, H. J. Otto, T. Eidam, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “Thermally induced waveguide changes in active fibers,” Opt. Express 20(4), 3997–4008 (2012). [CrossRef]   [PubMed]  

10. C. Jauregui, H. J. Otto, F. Stutzki, J. Limpert, and A. Tünnermann, “Simplified modelling the mode instability threshold of high power fiber amplifiers in the presence of photodarkening,” Opt. Express 23(16), 20203–20218 (2015). [CrossRef]   [PubMed]  

11. F. Jansen, F. Stutzki, H. J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “High-power thermally guiding index-antiguiding-core fibers,” Opt. Lett. 38(4), 510–512 (2013). [CrossRef]   [PubMed]  

12. L. Kong, J. Cao, S. Guo, Z. Jiang, and Q. Lu, “Thermal-induced transverse-mode evolution in thermally guiding index-antiguided-core fiber,” Appl. Opt. 55(5), 1183–1189 (2016). [CrossRef]   [PubMed]  

13. L. Dong, “Approximate treatment of nonlinear waveguide equation in the regime of nonlinear self-focus,” J. Lightwave Technol. 26(20), 3476–3485 (2008). [CrossRef]  

14. L. Dong, “Formulation of a complex mode solver for arbitrary circular acoustic waveguides,” J. Lightwave Technol. 28, 3162–3175 (2010).

15. L. Dong, “Limits of stimulated Brillouin scattering suppression in optical fibers with transverse acoustic waveguide designs,” J. Lightwave Technol. 28, 3156–3161 (2010).

16. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68(9), 1196–1201 (1978). [CrossRef]  

17. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001). [CrossRef]  

18. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).