1. Introduction

Several applications, including soft-tissue medicine, remote sensing, communication, industrial processing and defense, are currently pushing the research of high power sources in the 2 μm - 3 μm range, which is also attractive thank to of the eye-safe nature of the scattered light at these wavelengths [1]. Thulium-doped fiber lasers are of particular interest for these applications, being able to combine high brightness, premium beam quality and compactness, together with the possibility to obtain emission on a relatively wide band, spanning from 1.9 μm to 2.1 μm, when pumped with widely-available diodes emitting at 790 nm.

A huge research effort has been dedicated to the improvement of high power fiber lasers in the last decade, mostly focusing on ∼1 μm-sources, which can exploit the excellent spectroscopic properties of Yb [2, 3]. The development of innovative Double Cladding Photonic Crystal Fibers (DC-PCFs) has been one of the key factors for power scaling of these devices, leading to a significant enlargement of the mode effective area without compromising Single-Mode (SM) guiding, therefore allowing the increase of nonlinear threshold while maintaining high output beam quality. In principle, the same guidelines used for Yb-doped PCFs can be applied to the design of Tm-doped fibers [4, 5], with the advantage that operation at longer wavelengths allows SM propagation with larger effective area, therefore pushing the threshold for nonlinear phenomena towards higher power levels. Nevertheless, the development of PCFs with robust single-mode operation and very large mode-field diameter, preferably exceeding 50 μm, for high peak power pulsed sources is hindered by the much larger quantum defect of Tm ions with respect to Yb, that causes remarkable heating of the doped core at relatively low pump power. The consequent thermal gradient along the fiber cross-section is responsible for an increase of the refractive index with a parabolic shape in the core and a logarithmic decay in the cladding, which severely alters the modal properties [6], allowing propagation of High-Order Modes (HOMs). A mitigation of the quantum defect and an increase of the laser efficiency can be obtained by exploiting cross-relaxation [7], but the high Tm concentration required for these process to occur results in an increased index contrast between the core and the cladding, which is detrimental to SM properties [1].

As a consequence of the strong thermal effects, the adaptation of active PCFs conceived for operation at ∼1μm to Tm-doped devices is not straightforward, and new design approaches are worth exploring. One interesting solution, which has been recently applied to both air silica and all-solid LMA fibers with good results, is to break the C_6ν symmetry of the fiber cladding to improve delocalization of the most detrimental HOMs, while keeping the Fundamental Mode (FM) confined in the doped core [8, 9]. In this paper the design of a Tm-doped DC-PCF with Symmetry-Free cladding (SF-PCF) and a core diameter of ∼80μm is presented, and the fiber guiding properties are thoroughly analyzed by means of a full-vector modal solver based on the finite-element method [10, 11]. Moreover, a thermal model is applied to investigate how thermally-induced refractive index change affects the SM properties [12]. The results are compared with those obtained for a Large-Pitch Fiber (LPF) with similar core size, which represents the current state-of-art of large-mode area Tm-doped fibers [13].

The paper is organized as follows. The next section will describe the characteristics of the SF-PCF and illustrate the simulation procedure. In section 3 the mode discrimination properties of the fiber will be shown, and the effect of different design parameters and heating conditions will be considered. Section 4 will focus on the effective area, showing that the SF-PCF is capable of maintaining a very large mode area on a wide transmission band even at high pump power levels. Conclusion will be drawn in the final section.

2. SF-PCF analysis

The schematic of the considered SF-PCF cross-section is reported in Fig. 1(a). The inner cladding is obtained from a common triangular lattice where several air-holes have been removed to form two hexagonal layers, rotated with respect to each other. Three more air-holes, marked with arrows in the picture, have been kept to further weaken symmetry of the cross-section, similarly to what was done with the all-solid fiber in [9]. The 19 innermost unit cells are replaced with Tm-doped silica elements to obtain the active core. The DC SF-PCF considered in the simulations is shown in Fig. 1(b), where the key parameters are also indicated. All the considered symmetry-free fibers have a underlying lattice pitch Λ_SFF = 14.4 μm, which is also the distance between closest air-holes. The fiber core has a corner-to-corner distance d_cc of about 80 μm, which would be the distance between corner air-holes if the structure were not rotated with respect to the triangular lattice, and the edge a of resulting doped hexagonal region is 36 μm. A 6 μm-thick air-cladding with inner radius of 260 μm surrounds the cross-section, to provide pump guiding. To assess the advantages provided by the SF-PCF over current state-of-art fibers, the simulation results have been compared with those obtained with the LPF shown in Fig. 1(c), which is characterized by hole-to-hole spacing Λ_LPF = 45 μm, core corner-to-corner distance d_cc ≈ 81 μm and doped region with edge a = 31.5 μm [13]. Down-doping of the active region to compensate for the thermally-induced refractive index increase has been taken into account for both fibers.

 

Fig. 1 (a) Schematic of the cross-section of the symmetry-free PCF. Cross-sections of the simulated (b) symmetry-free and (c) large-pitch fibers.

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The degree of confinement of the guided modes has been evaluated considering their overlap integral Γ over the doped area, which has been calculated according to the expression

Thermal effects have been implemented by calculating the temperature distribution along the fiber cross-section T = T(x, y) with a computationally-efficient thermal model, which has been already successfully applied to study the thermo-optic effect in rod-type PCFs with resonant structures [12, 16]. The consequent refractive index change in silica and doped silica Δn_th has been obtained by applying thermo-optic relation Δn_th = β · (T − T₀), being β = 1.16 · 10⁻⁵ the thermo-optic coefficient for silica and T₀ = 25 °C.

Both the SF-PCF and the LPF have been assumed to be 1 m-long rod-type fibers, with outer diameter of 1.7 mm. Convection cooling with forced water flow at the temperature of 14 °C and convective transfer coefficient h = 2000 W/(m²·K) has been applied as boundary condition at the outer fiber edge. Temperature gradient on the fiber cross-section is generated by quantum defect-heating of the active core, which is responsible of a heat load q′

3. Single mode regime

In order to optimize the SF-PCF design to provide the best SM properties in different operating scenarios, the ΔΓ value has been calculated for different choice of the air-hole diameter d between 0.18Λ_SFF and 0.30Λ_SFF, taking into account heat load values spanning from 0 W/m to 340 W/m. λ = 2 μm has been considered as operating wavelength and 80 modes have been calculated for each parameter combination to find the most confined ones. The simulation results are shown in Fig. 2(a)–2(c), for a core down-doping Δn = 0, −10⁻⁴ and −2·10⁻⁴, respectively. The same procedure has been applied to the large-pitch fiber for d/Λ_LPF values between 0.16 and 0.22, and the results are shown in Fig. 2(d)–2(f).

 

Fig. 2 Overlap integral difference ΔΓ between the FM and the most detrimental HOM of the SF-PCF, as a function of the normalized air-hole diameter and heat load, obtained with (a) no core down-doping, (b) with core down-doping Δn = −10⁻⁴, and (c) with Δn = −2·10⁻⁴. Same results obtained for large-pitch fibers are reported in (d) for no core down-doping, (e) for Δn = −10⁻⁴ and (f) for Δn = −2 · 10⁻⁴. White lines are drawn at ΔΓ = 0.3.

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Clearly, none of the considered designs of the SF-PCF or LPF is able to provide SM guidance over the full range of considered heat load values. All fibers show a similar temperature-dependent behavior, which is more evident in down-doped fibers. At low heat load values, for example approximately below 80–100 W/m for the SF-PCFs with Δn = −10⁻⁴ of Fig. 2(b), the confinement of the FM is poor and its overlap integral is close the one of the HOMs, which is below 0.4 in most cases. In some extreme conditions, corresponding to the black areas of Fig. 2(b) and 2(c), the overlap integral of the most confined HOM is even higher than the one of the FM. All the modes propagate mostly outside from the doped core, and poor amplification efficiency is expected. As the heat load increases the FM becomes significantly better confined than the HOMs, and the modal discrimination increases up to the maximum value, which is found at about 150 W/m for the SF-PCFs with Δn = −10⁻⁴. After that point, any further increase of the core temperature decreases the modal discrimination by providing better confinement to the HOMs, up to the point where ΔΓ is again below 0.3 and the fiber operates in multimode regime.

By comparing Fig. 2(a)–2(c) it is possible to notice that SM operation can be shifted to significantly higher power levels by acting on core refractive index. Indeed, the SF-PCF without core down-doping can provide SM propagation for heat load between about 25 W/m and 180 W/m, while with Δn = −10⁻⁴ the SM regime is found between ∼90 W/m and ∼ 290 W/m and from ∼150 W/m up to more than 350 W/m for Δn = −2 · 10⁻⁴. This means that a SF-PCF with suitable core down-doping is capable to maintain SM propagation even with a coupled pump power larger than 300 W. On the other side, the use of a too depressed core is detrimental for the operation at low pump power, and therefore core refractive index must be chosen according to the application. It is worth noting that in a real-world scenario the heat load is not uniform along the fiber, being significantly lower far away from the pumping end, that is at the seed side assuming a counter-propagating pumping scheme. As a consequence, the FM is less confined and its overlap with the doped area is lower, resulting in a decrease of the amplifier efficiency. Considering the results of Fig. 2, it is thereby preferable to choose a fiber design that allows operation close to the upper edge of the SM region at the pumping end, so that FM confinement and SM regime are preserved throughout most of the fiber length.

Air-hole size has a weaker impact on the guiding properties of the SF-PCF with respect to down-doping, resulting in a small shift of the SM region towards lower heat load values for increasing d/Λ_SFF, that can be explained with the increased index contrast of the fibers with largest air-holes.

The main advantage of the SF-PCF approach is the possibility to provide very high modal discrimination at high pump power levels. The maximum value of ΔΓ achieved is above 0.54 for all the three considered values of core down-doping, and it is found at a heat load of about 100 W/m, 150 W/m and 200 W/m for the fibers having Δn = 0, −10⁻⁴ and −2 · 10⁻⁴, respectively. Remarkably, similar mode discrimination has been reported for the all-solid version of the symmetry-free fiber without considering thermal effects [9], and is among the highest values reported so far for any index-guiding LMA fiber design. By choosing a suitable level of core down-doping it is possible to obtain SM guiding at high heat load also with the LPF, but the index depression is detrimental to the modal discrimination, as can be inferred by the comparison of Fig. 2(d) and 2(f). Indeed, with the LPF without core down-doping it is possible to reach a ΔΓ value larger than 0.5 at about 50 W/m, while a maximum overlap difference of only 0.42 is obtained with a core down-doping of −2 · 10⁻⁴.

To give a deeper insight on the effect of fiber heating on mode confinement, Fig. 3(a) reports the overlap integral of the four most relevant modes of the SF-PCF, calculated at λ = 2 μm for the fiber with d/Λ_SFF = 0.24 and Δn = −2 · 10⁻⁴, as a function of the heat load. Notice that when the fiber is relatively cold both the fundamental LP₀₁-like mode, whose field intensity distribution is shown in Fig. 3(c), and the LP₁₁-like one, shown in Fig. 3(d), are poorly confined, with Γ values below 0.2. Under this heating conditions the mode with the highest overlap is the LP₀₃-like, shown in Fig. 3(f), whose overlap integral may even exceed 0.5. This behavior is due to core down-doping, which causes the modes that are naturally localized in the core to be poorly guided due to the lower refractive index of the doped region with respect to the cladding. The increase of the heat load compensates the core down-doping, improving the confinement of the LP₀₁-like mode while decreasing the overlap integral of the LP₀₃-like one. The FM has the highest overlap integral for q′ higher than 120 W/m and its value rapidly increases with the heat load. Between q′ = 145 W/m and q′ = 200 W/m the role of the most detrimental HOM is played by the LP₀₂-like, shown in Fig. 3(e), whose Γ value is anyway below 0.25. As a consequence, for q′ > 150 W/m the SF-PCF is SM. Finally, at very high power levels the thermally-induced refractive index gradient is so strong to force the fiber to behave effectively as graded-index C_6ν-symmetric waveguide. For q′ > 200 W/m the LP₁₁-like HOM becomes the most detrimental one, its overlap integral increasing almost linearly with the heat load beyond this heat load value. Finally, it is worth noting that this SF-PCF is not only capable of providing strong delocalization of the HOMs even at high thermal load, but also to guarantee tight FM confinement, with overlap higher than 0.9 at q′ > 225 W/m.

 

Fig. 3 (a) Overlap integral of the most relevant guided modes of the SF-PCF as a function of the heat load, calculated at λ = 2 μm. (b) ΔΓ between the LP₀₁-like and the LP₁₁-like modes of the SF-PCF (solid lines) and of the LPF (dashed lines). Magnetic field modulus distribution of the (c) LP₀₁-like, (d) LP₁₁-like, (e) LP₀₂-like and (f) LP₀₃-like modes of the SF-PCF.

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In order to demonstrate that the good HOM suppression properties of the SF-PCF are maintained over the whole spectrum of interest for Tm emission, the modal discrimination between the FM and the LP₁₁-like HOM has been calculated between 1800 nm and 2100 nm in two representative cases, and compared with the results provided by a LPF. Simulations have been performed on a SF-PCF with d/Λ_SFF = 0.24 and core down-doping Δn = −10⁻⁴, with q′ = 170 W/m, which roughly corresponds to the condition where the maximum ΔΓ is observed at λ = 2 μm, and on a SF-PCF with the same pitch but Δn = −2 · 10⁻⁴ at q′ = 340 W/m, which is the maximum value of the heat load taken into account. The results have been compared with those obtained for a LPF with d/Λ_LPF = 0.20 with Δn = −2 · 10⁻⁴, operating with q′ = 340 W/m. The overlap difference for both SF-PCF slightly increases with the wavelength, with a variation of about 0.08 over the considered range. The SF-PCF operating at q′ = 170 W/m shows a remarkable overlap difference close to 0.5 over the whole band, with maximum value of 0.54 at 2100 nm. Despite a lower ΔΓ, also the SF-PCF operating at q′ = 340 W/m is SM over the whole range, reaching Δn values between 0.33 and 0.4. These performances are better than those obtained by the LPF for the same heat load, being its overlap difference close to the threshold for SM operation at ΔΓ = 0.3 throughout the considered wavelength span.

4. Effective area

Figure 4(a) shows the effective area calculated at λ = 2 μm for the LP₀₁-like mode of the SF-PCFs with the three considered values of core down-doping and d/Λ_SFF = 0.24, as a function of the heat load. The curves have been drawn only for values of q′ that ensure an overlap larger than 0.7, since lower values of Γ are not suitable for amplification, and are dashed where the fibers operate in multi-mode regime. The intense heating of the fiber has a remarkable effect on A_eff, roughly halving its value from about 5000 μm² to about 2600 μm² at the maximum heat load allowed for SM operation. Notice that down-doping does not affect these limits, but only determines the heat load at which these values of A_eff are obtained. For example, an effective area of 3000 μm is reached by the fiber without core down-doping at q′ = 120 W/m, at q′ = 200 W/m by the fiber with Δn = −10⁻⁴ and at q′ = 280 W/m by the fiber with Δn = −2 · 10⁻⁴. Notice also that the effective area shrinking due to heating is comparable to the values that have been found experimentally for the Tm-doped fiber laser based on the LPF with similar core diameter [13].

 

Fig. 4 (a) Effective area of the SF-PCF with d/Λ_SFF = 0.24 and different values of core down-doping, calculated at λ = 2 μm as a function of the heat load. Dashed lines are used where fibers operate in multi-mode regime. (b) Effective area of the SF-PCF with d/Λ_SFF = 0.24 and Δn = −2 · 10⁻⁴ for a heat load of 340 W/m (red line) and of the SF-PCF with d/Λ_SFF = 0.24 and Δn = −10⁻⁴ for a heat load of 170 W/m (blue line).

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The effective area as a function of the wavelength of the two SF-PCFs already considered in Fig. 3(b), with the same heat load conditions previously used, is reported in Fig. 4(b). In both cases, the effective area change with the wavelength has a similar slope, with an increase over the considered wavelength range of about 350 μm². When the maximum mode discrimination is achieved, that is by the fiber with Δn = −10⁻⁴ at q′ = 170 W/m, A_eff is between 3100 μm² at λ = 1800 nm and 3400 μm² at λ = 2100 nm. On the other hand, by considering the fiber with Δn = −2 · 10⁻⁴ at q′ = 340 W/m, an effective area between 2400 μm² at λ = 1800 nm and 2700 μm² at λ = 2100 nm is found.

5. Conclusion

In this paper the design of a double-cladding photonic crystal fiber with ∼80 μm-core and reduced cladding symmetry has been investigated, aiming to obtain robust SM guiding even at high heat load values, such as those generated during operation of high-power Tm-doped fiber lasers. A full-vector modal solver based on the finite-element method with integrated thermal model to calculate temperature-induced refractive index change has been applied to analyze the effects of the main fiber design parameters on the modal discrimination for different heating conditions. The results have shown that the SF-PCF is capable of providing efficient suppression of the HOMs, with maximum overlap difference between the FM and the most detrimental HOM larger than 0.5, even when relevant heat load is assumed. This modal discrimination is among the highest reported so far for index guiding PCFs, being surpassed only on narrow transmission bands by PCF exploiting resonant features [17]. Moreover, SM propagation with A_eff larger than 2500 μm² has been demonstrated at heat load of 340 W/m.