1. Introduction

Over the past few decades, high power Yb-doped fiber lasers (YDFL) have attracted intense attentions due to several advantages compared to other solid state lasers (SSL), including high optical-optical conversion efficiency, compact structure and excellent beam quality [1]. Although YDFLs are thought to be of advantage to reduce the thermal-optic distortion because of its lower thermal load per unit length, lower quantum defect, large surface-to-volume ratio and guiding-wave physics, thermal effect is still one of some serious problems in high power YDFL due to steep radial temperature gradient [2, 3]. In particular, it has been proposed that the refractive index change induced by lateral heat dissipation in fibers may cause the mode field deformation, multimode excitation [4] and thus the degradation of beam quality. Such effect has been studied extensively by solving wave equation directly [4] and by beam propagation method (BPM) [5, 6]. Many interesting results have been published, such as the differences between step-index fibers (SIF) and photonic crystal fibers (PCF) [4, 7] and the mode instability phenomenon [5–9]. These effects, so far, were studied mostly on an individual basis. It is expected that different fiber structures with proper thermal load may cause a similar behavior on the mode distortion. As long as the parameters involved can be normalized properly, some general physical properties would be obtained and similar cases would be examined together. In this paper, we propose a general physical model which describes the thermal induced mode distortion in the high power SIF lasers. It is shown that there are two normalized parameters that may fully determine the mode characteristics in heated SIFs. Shrinking of the mode fields and excitation of the high order modes (HOM) by the thermal effect are investigated in the framework of the model. A simplified power amplification model is then set up to describe the output power redistribution on various guiding modes for both single-mode and multi-mode fiber.

At the same time, according to the power scaling estimation [10], the thermally induced mode deformation is one of the most important factors to limit the possible output power of YDFL, especially for the large mode area (LMA) fibers. Different values of output power limit from 36kW to 70kW have been suggested [10, 11], depending on different estimation of the stimulated Raman scattering (SRS) effect and pumping schemes. It must be pointed out that various physical processes in the power scaling analysis are coupled with each other. In particular, the nonlinear effect and the optical damage are coupled with the thermal effect via thermally induced mode-field shrinking. It means that the output power limited by the nonlinear effect and the optical damage in fact depends on the thermal effect. Moreover, the thermal effect produces an independent limitation as well to avoid a multi-mode output. All of these considerations were achieved in literatures under the assumption that the mode field diameter (MFD) decreased by the thermal-optic effect to 0.7-0.8 times of the core diameter [10, 11]. This assumption, however, is not clear in physics and does not depict accurately the coupling among various physical processes. Alternatively, it has a more obvious physical meaning to use the condition of quasi-single-mode output as a constraint for the thermal effect. It may be equivalent as to impose an advanced fraction of the output power to the fundamental mode. This condition, however, corresponds to variable mode field size under different pumping load. A dynamic coupling between the nonlinear-effect/optical-damage and the thermal effect will then be produced. The model proposed here provides a framework to improve the power scaling analysis of YDFL by incorporating this kind of dynamic coupling.

2. Theoretical model

2.1 Normalized parameters governing mode characteristics

Since the ideal axial-symmetric weakly guiding fiber is considered here, the radial function of mode ψ(r) can be described with the scalar wave equation:

In Eq. (1), we have ignored the temperature inhomogeneity along fibers. As long as mode characteristic described by Eq. (1) is studied completely, the longitudinal inhomogeneity can be obtained by the eigen-mode expansion method step by step. Furthermore, in unheated SIFs, the guiding modes in the core are defined clearly by requirement of the real propagation constant. In greatly heated SIFs, however, there are many additional modes that are guided by the thermal induced index gradient instead of the original step-index. In general, the main energy of those modes is not confined in the core. So only a few guiding modes would need to be considered. In theory this can be guaranteed by taking sufficient large cladding thickness, e.g., five to ten times of the core radius.

There are several physical parameters involving in the model. Noticing that ∆T_codn/dT always keeps small value in the fiber lasers, we may ignore its second order terms. Then by introducing variable x = r/r_co, Eq. (1) can be rewritten as

In Eq. (3), the thermally induced index shifting item, namely (T_co-T₀)dn/dT, has been ignored. This is because its major effect is shifting the effective index of the mode and it affects the mode profile lightly.

Therefore, the mode properties of the heated SIF are fully decided by two parameters, namely V and Z. V is nothing but the conventional V-parameter of SIF. The definition of Z-parameter is analogous to V whereas the n_co-n_cl part of V is replaced by the thermally induced refractive index difference in the core, i.e., ∆T_codn/dT. The physical origination of Z-parameter is that the mode fields are deformed by local heat load. In other words, the thermal-optic effect in fibers is not cumulated with the propagation of laser beam. This is essentially different from the bulk SSLs.

The analysis proposed above allows us to discuss the thermally induced mode field properties in the high power SIF lasers in a general framework. As an example, the heat power density can be expressed as q = ηαP_p/πr_co², where η denotes the quantum defect, α is the absorption coefficient of pumping light and P_p is the pumping power. Consequently Z is proportional to r_co(ηαP_p)^1/2. In order to maintain the same mode deformation and mode number, Z must be fixed. It implies that for the same pump configuration the absorbed pump power αP_p should be decreased as the core diameter of fiber increases. Consequently decreasing pump power or extension of fiber length is necessary in LMA fibers in order to avoid serious thermal induced mode distortion. Therefore a tradeoff design between thermal and nonlinear effects should be carefully considered in the high power LMA fiber lasers. The same conclusion can be obtained by the ABCD transformation of the Gaussian beam given in the appendix of [10], with the assumption of the zero numerical aperture (NA) or strong thermal load. According to the normalized model proposed in this paper, we can see that the conclusion holds true for any fiber with the same value of the V-parameter.

2.2 Simplified power amplification model

In order to analyze thermal effect in fiber amplifiers, a power amplification model is required. One type of strict model is based on BPM, where the temperature distribution, mode evolution and laser dynamics are all coupled together. Some important phenomena have been revealed, e.g., the refractive index grating and mode instability caused by heat dissipation and mode beating [5, 6]. Another type of model simply considers the power amplification process [13] which can describe the power redistribution among guiding modes as well when the thermal effect is taken into account. In fact, for a practical amplifier, the input mode field can be expanded into the eigen-modes of the heated fiber amplifier and to be amplified together. It results in the power distribution of different modes at the output port. This simple model may also be useful to distinguish the importance among different ingredients. The assumption of uniform heat power density along active fiber will be used here which is the same as the amplifier power scaling model in [10]. Then the results of the calculation would be applicable in power scaling analysis directly.

As long as the laser beam with power P_in is launched into the active fiber, the input mode field can be expanded as

Within continuous-wave amplifier, all these guiding modes share the same population inversion. Therefore according to the power amplification model, the output power of the k-th mode is

3. Mode field properties in high power SIF lasers

3.1 Thermal effect on guiding modes

The thermal effect on fiber guiding mode will be discussed here with the normalized parameters V and Z. In order to understand Z-parameter more intuitively, we consider a λ = 1064nm laser in the silica SIF as an example. Taking n_co = 1.45, dn/dT = 11.8 × 10⁻⁶/K, κ = 1.38W/m/K, we have the linear relationship Z≈0.0083 × r_co(ηαP_p)^1/2, where r_co is in µm and the heat power per unit fiber length ηαP_p is in W/m. Then Z = 1 corresponds to a massive heat load of 145.3W/m in a fiber with the core diameter d_co of 20µm but an acceptable heat load of 36.3W/m in a d_co = 40µm fiber. Three guiding modes, namely LP₀₁ (fundamental mode), LP₁₁ (the first HOM) and LP₀₂ (the second HOM with ν = 0), are considered here for different values of V-parameter, which governs the unheated SIF to be single mode or multi-mode.

Figure 1 shows the radial amplitude profiles of the three modes with different values of Z-parameter for V = 2.4 (single mode) and V = 3.8 (double modes). It can be seen that the mode fields shrink gradually with the increase of Z. Moreover, Z affects weakly on those modes supported by the original cooling fiber, but strongly inñuences the modes excited by the temperature gradient. The fields of thermally excited modes are mainly located in cladding for Z below 0.4 and would rapidly shrink into core as Z increases to 1~2. A critical value of 0.33 for the confinement factor may be used to judge the excited HOM as the core modes or not [4].

 

Fig. 1 Amplitude profiles of LP₀₁, LP₁₁ and LP₀₂ modes for different Z and V = 2.4 (a, b, c) and V = 3.8 (d, e, f).

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The variation of confinement factor and MFD along with Z for several different V-values are shown in Fig. 2. Here V = 2.4 and V = 3.8 are the critical values for single-mode and double-mode fibers, and V = 6 corresponds to the possible LMA fibers (e.g. d_co = 50µm, NA = 0.04, λ = 1064nm) or some high NA fibers (e.g. d_co = 15μm, NA = 0.13, λ = 1064nm) which have also been proposed to achieve high power output with single mode operation [15]. HOM excitation is clearly seen, though a small confinement factor indicates that their intensities locate in cladding mainly for small Z. The increasing Z leads to the increase and converging to 1.0 of the confinement factor and the gradual decrease of MFD/d_co. It is clear that such trend is controlled by V and Z simultaneously, and can be classified into three zones: The V-zone where the variation of the field parameters are slowly; the V-Z zone; and the Z-zone where the effect of V is negligible and all curves coincide. Particularly for the fundamental mode in the Z-zone, since the refractive index difference induced by thermal load is sufficiently large, the approximation of Gaussian beam transformation method can be performed in the core as shown in the appendix of [10], and the related result can be rewritten in a simple form as

 

Fig. 2 The variation of confinement factor (a, b, c) and MFD/d_co (d, e, f) along with Z for several V values.

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It should be noted that the field parameters remains almost the same in a wide Z-range when the V value is large (e.g. V = 6). It means that the fibers with large V value are more insensitive to the thermal effect.

3.2 Thermally induced mode competition

The thermally induced mode field shrinking and multi-mode excitation lead to the mode competition in the heated amplifier and multi-mode output eventually, even though a sole eigen-mode of cooling delivery fiber is input. The output power redistributes among various modes and can be analyzed by the simplified model proposed in section 2.2.

Here we give an example in which the input laser field is a fundamental mode of a passive fiber whose structure is the same as the active fiber. We further ignore the fiber bending and assume that junctures between different fiber slices is ideally matched. Then only LP_0k modes are excited due to axial symmetry. The number of LP_0k modes used in the calculation is large enough to assure that ∑_k|c_k|²>0.99. The gain G of the amplifier is set as 10.

Figure 3 shows the power fractions of LP₀₁ and LP₀₂ at the output port of the amplifier as functions of Z for several V-values. It can be seen that the fibers with larger V-value is more robust in keeping power in LP₀₁ content. This is because a stronger mode field confinement leads to insensitivity to the thermal effect. Therefore, in certain cases the fibers with high V-value may provide better beam quality due to their weaker LP₀₁ mode deformation and fewer thermally induced HOMs. Being accompanied by the decrease of LP₀₁ content, the power of LP₀₂ mode increases gradually. It will reach a maximum of about 30% and the power transfers to LP₀₃ mode further and so on.

 

Fig. 3 The output power fractions of LP₀₁ and LP₀₂ modes as a function of Z.

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We compare the above results with the one obtained in [5] where a forward pumped amplifier was studied by a much more precise and complicated model based on the BPM method in detailed. In this case, the longitudinal temperature distribution within the forepart of amplifier (about 0.5m) can be roughly regarded as uniform. The content of LP₀₁ and LP₀₂ was found to be around 75% and 25% at z = 0.5m, respectively. The V-value of the fiber was 2.28, and Z≈6 is obtained according to Fig. 17 in [5] (∆T_co≈20K). Meanwhile, our model show that V = 2.3 and Z = 6 leads to the LP₀₁ fraction of 75% and the LP₀₂ fraction of 21%. The similar results indicate that the overlap between the input mode and the thermally induced HOMs in the heated fibers may be an important reason of which to cause the power transferring. It may further enhance the mode instability caused by the longitudinal refractive index grating which is induced by thermal load and mode beating [9].

On the other hand, Z = 2 can be regarded as a critical point where power fraction of the LP₀₁ mode starts to decrease rapidly from almost 100%. For the fiber with a smaller core size, Z = 2 corresponds to a high thermal or pumping load. For example, a 3.71kW/m of pump absorption is needed to produce Z = 2 in a fiber with d_co = 25µm, η = 0.1, α = 1.26dB/m. Therefore for an ideal cylindrically symmetric fiber, the input of pure fundamental mode can guarantee approximate single mode output at a high power situation because the non-symmetric mode cannot be excited. Unfortunately, in real fiber lasers, the fibers are always bent and there is always a slight mismatch at the juncture of fibers. All ingredients cause the LP₁₁ mode excitation from LP₀₁. It can be seen from Fig. 2 in section 3.1 that the confinement factor of LP₁₁ is around 0.7 for Z = 2. Therefore the LP₁₁ mode may be amplified effectively due to its complementary pattern compared to LP₀₁ [1]. Such behavior cannot be simulated by the simplified power amplification model in this paper while the other ingredient such as thermally induced grating may dominate the power transferring process [5, 6]. For fibers with large core size, the condition Z>2 is easy to reach. For example, 1.45kW/m and 3.27kW/m of pump absorption corresponds to Z = 4 and Z = 6 respectively for a LMA fiber with d_co = 80µm, α = 2.2dB/m (η = 0.1 again). It indicates that the LMA fibers face a serious multi-mode problem though they may take advantage of avoiding strong nonlinear effect.

4. Application on power scaling

The results above can be applied to the power scaling analysis of amplifier. It has been proposed that the output power of amplifier is limited by six factors, namely pump brightness, thermal fracture, melting of core, optical damage, nonlinear effect (SRS here) and thermally induced mode deformation [10]. The maximal power is the minimum of the six power limits. The related formulae and parameter definitions can be found in Appendix A.

Our analysis is within the same framework but with two improvements. First, for the calculation of $P_{\max }^{damage}$ and $P_{\max }^{SRS}$, effective mode area A_eff was always set as Γ²πr_co² in the original calculation. However, it is clearly that A_eff is affected by thermal load and the change is notable especially in high power amplifier. Thus these two power limits should be coupled with thermal effect. According to the assumption of the scaling model, the thermal load q can be estimated by the laser output power P as q = η_heat/η_laser∙P/πr_co²L with L as the fiber amplifier length. Consequently A_eff can be calculated with A_eff = f(P), where f can be proved to be a monotonic decreasing function. If the power limit of a constraint condition (say optical damage and SRS here) is given by $P_{\max }^{condition}=g(A_{eff})$ with g the monotonic increasing function, the inequation $g(f(P))\le P_{\max }^{condition}$ is required according to self-consistency and the power limit of the constraint condition coupled with thermal effect is given by the unique solution of the equation

The calculation results are shown in Fig. 4, where Figs. 4(a), 4(b) and 4(c) correspond to V = 2.4, 3.8 and 6.0 together with LP₀₁ power fraction of 90%, Figs. 4(d), 4(e) and 4(f) correspond to LP₀₁ power fraction of 80%. Here we ignore the fiber bending and alignment error at the fiber junctures; hence LP₁₁ mode cannot be excited. It can be seen that the maximal power are always limited by optical damage (yellow region), SRS (orange region) and thermal effect (brown region). The intersection of three regions gives the maximal output power achieved by the minimal core diameter and the shortest fiber length. For V = 2.4 and LP₀₁ power fraction of 90%, the maximal output power allowed is about 67.6kW. When the constraint of LP₀₁ content is relaxed, the thermal effect area shifts to right-hand side and the maximal power increases as shown in Fig. 4(d). For the cases of the multi-mode fibers shown in Figs. 4(b), 4(c), 4(e) and 4(f), the maximal powers are much higher compared to single mode case with the same LP₀₁ content constraint though the required core diameter is larger as well. For example, the maximal power of Fig. 4(f) is around 97kW while the minimal core diameter required exceeds 200μm (which may only be implemented in photonic crystal fibers (V = 6.0)). That is because multi-mode fibers can carry higher thermal load as shown in section 3.2. However, for the lower output power requirement, single mode fibers are able to achieve it with a smaller core. For example, d_co = 50µm is sufficient to achieve 20kW output for a fiber with V = 2.4 while for the fiber with V = 6, d_co = 70µm is required. It can be understood that the weaker field confinement of single mode fibers leads to the larger effective mode area and consequently allows higher output power.

 

Fig. 4 Power scaling analysis with the improved model. The title of each subfigure provides the maximal power, the V-value, and the LP₀₁ content.

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5. Conclusion

In conclusion, by analyzing the scalar wave equation integrated with thermally induced refractive index gradient, two normalized parameters, namely V, Z, are found to be able to determine the mode properties in high power SIF lasers. We show that the Z-parameter governs the waveguide capability of the radial temperature gradient, similar to the V-parameter for unheated SIFs. The thermal effect on fiber guiding modes is investigated with these parameters in general. A simplified power amplification model is then set up to describe the power distribution on different modes at the output port of fiber amplifier. The results are found to be consistent with those obtained by the BPM method. It indicates that the overlap between the input mode and the HOMs of heated fiber may be a considerable factor to cause the output power transferring. The model also implies that the LMA fibers face a serious multi-mode problem though they may take advantage in avoiding strong nonlinear effect. The power scaling of YDFL is investigated with an improved model, where the optical damage and SRS limitations are coupled with thermal effect dynamically and the original thermal lens criterion is replaced with a more practical one based on the power content of fundamental mode. It predicts that the maximal output power is from 67kW to 97kW, depending on power fraction of the fundamental mode and V-parameter of SIFs.

Appendix A. Formulae and parameters of the power scaling model

According to [10] and [11], there are six factors that limit the output power of a fiber amplifier, namely pump brightness, thermal fracture, melting of core, optical damage, nonlinear effect (SRS here) and thermally induced mode deformation. The maximal output power corresponding to each factor can be written as

(13)$$\begin{array}{c}{P}_{\max }^{pump}={\eta }_{laser}{I}_{pump}\cdot \pi r_{cl}{}^2\cdot \pi N{A}^2\\ P_{\max }^{rupture}=\frac{{\eta }_{laser}}{{\eta }_{heat}}\frac{4\pi R_m}{1-r_{co}{}^2/(2r_{cl}{}^2)}L\\ P_{\max }^{temp}=\frac{{\eta }_{laser}}{{\eta }_{heat}}\frac{4\pi \kappa (T_{\max }-T_{cooling})}{1+2\kappa /(r_{cl}h)+2\ln (r_{cl}/r_{co})}L\\ P_{\max }^{damage}=A_{eff}{I}_{damage}\\ P_{\max }^{SRS}=\left(20.3-\ln \beta +\ln \frac{A_{eff}}{g_R{L}_{eff}}\right)\frac{A_{eff}}{g_R{L}_{eff}}G\\ P_{\max }^{lens}=\frac{{\eta }_{laser}}{{\eta }_{heat}}\frac{2\kappa {\lambda }^2}{{\Gamma }^4\pi n_{co}\frac{dn}{dT}}\frac{L}{r_{co}{}^2}\end{array}$$

where L is the fiber length, n_co = 1.45, κ = 1.38W/m/K, dn/dT = 11.8 × 10⁻⁶/K, λ = 1064nm. Definition of other symbols used in Eq. (13) and their values are further listed in Table 1. Here the tandem-pumping scheme is assumed. The maximal allowed power is the minimum of the above six power limits.

Table 1. Definition of some symbols in Eq. (13) and their values.

View Table

Acknowledgments

This work was partly supported by Key Laboratory of Science and Technology on High Energy Laser, under Grant No. LJG2012-07.

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