Effects of ion clustering and excited state absorption on the performance of Ho-doped fiber lasers

Jiachen Wang; Narae Bae; Sang Bae Lee; Kwanil Lee

doi:10.1364/OE.27.014283

1. Introduction

Owing to their inherent advantages regarding operation wavelength, silica based holmium (Ho) doped fiber lasers are attracting great attention. The lasers offer a unique opportunity of efficiently operating at 2~2.2 μm, a spectral region of great interest for a wide range of industrial and scientific applications [1,2]. Alternative active fibers for generating 2 μm radiation are thulium (Tm) doped fibers [3] and Tm-Ho codoped fibers [4]. These fibers cannot displace Ho-doped fibers given the facts that Ho-doped fibers lase in the wavelength region which is beyond the coverage of Tm-doped fibers, and achieve lasing efficiency that is higher than Tm-Ho codoped fibers. Lasers based upon Ho-doped soft-glass fibers (i.e., chalcogenide fibers and fluoride fibers) are capable of operating at even longer wavelength [5,6]. The applications of soft-glass fiber lasers, however, are significantly constrained by the low mechanical, optical, and thermal resistibility of the soft glasses [7]. All these facts make Ho-doped silica fiber laser one of the most promising 2 μm light sources.

A notable concern over Ho-doped fiber lasers is their lower-than-expected slope efficiencies. Previously reported systems are of slope efficiencies that are remarkably less than the theoretical limit indicated by the quantum defect. The problem is observed in both non-resonantly pumped systems (which exploit the 1150 nm-centered absorption band) and resonantly (in-band) pumped systems (which exploit the 1950 nm-centered absorption band). The quantum defect of the non-resonant pumping scheme allows slope efficiency higher than 50%; in most reported experimental demonstrations, however, the achieved slope efficiencies are 30%–37% [8–12]. Only in few cases the slope efficiencies higher than 40% are realized, and even in those cases the achievable values merely correspond to ~80% of the theoretical limit [13–15]. With respect to the resonantly pumped systems, the low quantum defect allows slope efficiency higher than 90%. Most reported systems, however, are of slope efficiencies lower than 65% [16–21]. Several literatures report slope efficiencies higher than 70%, which still merely correspond to ~80% of the theoretical limit [22,23]. Only recently a record high slope efficiency (87%) that is close to the theoretical limit is reported [24].

The sub-optimal slope efficiency significantly hampers the development of Ho-doped fiber lasers, in particular the high power systems for which a high scalability is required. The efficiency degradation is observed in almost all reported experimental works [8–23]. Thus far, the cause of this undesired behavior is still not well understood. Several probable causes are discussed in the previous works. A frequently proposed cause is the high background loss of silica fiber in the 2 μm wavelength region, however it cannot explain the low slope efficiencies observed in the systems comprising very short fibers [17,21]. The loss-associated theory also contradicts the fact that in most cases researchers employ high-quality fibers of which the background loss is effectively suppressed (mainly by minimizing the OH contamination). Other postulated causes are associated with quenching mechanisms, such as non-radiative decay, excited state absorption (ESA), and energy transfer upconversion (ETU). Few of them are validated, either experimentally or theoretically.

In a previous work of us, a rapid upconversion process called pair induced quenching (PIQ) that predominately occurs in clustered ions is identified as the dominant cause of the lower-than-expected slope efficiencies observed in Ho-doped fiber lasers [25]. Our finding is supported by the experimental studies reported by A. S. Kurkov et al. [10,26]. Nevertheless, the scope of the early research is limited to heavily doped fibers or low-quality fibers, in which the presence of ion clustering is apprehensible. Regarding the fibers of low Ho³⁺ concentrations, as well as high-quality compositions, the validity of our conclusion is not tested. Moreover, our previous work is exclusively devoted to the resonantly pumped systems. For such systems we successfully attest that ion clustering is the dominant cause of the lower-than-expected slope efficiencies, and disqualify other probable causes. With respect to non-resonantly pumped Ho-doped fiber lasers, on the other hand, the energy level structure of Ho³⁺ suggests that ESA may also play a role. Therefore, it is strongly desired to carry out a comprehensive study on both ion clustering and ESA for better understanding their effects on the performance of Ho-doped fiber lasers.

In this paper, we investigate the effects of ion clustering and ESA occurring in Ho-doped fiber lasers experimentally and theoretically. The experimental work is based upon a resonantly pumped system consisting of commercially available fiber. The fiber is fabricated with state-of-the-art technology, and is of a low Ho³⁺ concentration. The numerical models developed in our previous work [25] are employed to simulate the experimental systems. The results of the experiments and the simulation works verify the presence of ion clustering and its crucial role in the deterioration of slope efficiencies, even as the fiber is of a low doping concentration and advanced composition design. We also analyze the effect of ESA in non-resonantly pumped Ho-doped fiber lasers. The ESA spectrum of Ho³⁺ is acquired through Judd-Ofelt approach, and is evaluated in experiments. Both the calculated and experimental results reveal that the effect of ESA (including the phonon-assisted one) in Ho-doped fiber lasers is negligible. The lower-than-expected slope efficiencies observed in the non-resonantly pumped systems is overwhelmingly induced by ion clustering instead of ESA, given that the mechanism of the clustering-associated efficiency degradation is independent of pumping schemes. In summary, we found that ion clustering is the dominant cause of the sub-optimal performance of Ho-doped fiber lasers, regardless of fiber types or pumping schemes. We also hold the view that ion clustering is an inherent drawback of Ho-doped fibers and is difficult to eliminate. To the best of our knowledge, this is the first research effort to systematically analyze the effects of ion clustering and ESA in Ho-doped fibers. The outcomes of this work should be beneficial for developing high power Ho-doped fiber lasers.

2. Mechanisms

A simplified energy level diagram of Ho³⁺ in silica host is presented in Fig. 1. The ⁵I₈, ⁵I₇, ⁵I₆, ⁵I₅ and ⁵I₄ energy levels (manifolds) are respectively labeled as 0, 1, 2, 3, 4. GSA represent the ground state absorption of pump, among which GSA1 represents the resonant pumping (i.e., ⁵I₈→⁵I₇), GSA2 represents the non-resonant pumping (i.e., ⁵I₈→⁵I₆). The putative ESA process occurs in the form of ⁵I₇→⁵I₄, by which it shares the pump of GSA2. With respect to GSA1 (resonant pumping), ESA is inhibited by the apparent mismatch between energy gaps. The ⁵I₇→⁵I₄ ESA consumes pump without contribution to lasing (the photon energy absorbed by the ESA will be mostly released through non-radiative processes), and thus, is detrimental to the performance of laser.

Fig. 1 Simplified energy level diagram of Ho³⁺ which is of the five lowest energy levels. GSA: ground state absorption. ESA: excited state absorption. RD: radiative decay. NRD: non-radiative decay. ETU: energy transfer upconversion.

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An important mechanism affecting the operation of Ho-doped fiber lasers is ETU [22]. ETU is a non-radiative energy transfer between a pair of excited ions (in this case, the Ho³⁺ in ⁵I₇) induced by electric multi-polar interaction. As a consequence of the interaction one of the ions is promoted to higher energy level whilst the other is demoted to lower energy level. There are two recognized ETU processes involving the ⁵I₇ manifold of Ho³⁺, namely the ETU1 (expressed as ⁵I₇, ⁵I₇→⁵I₆, ⁵I₈) and the ETU2 (expressed as ⁵I₇, ⁵I₇→⁵I₅, ⁵I₈). The two ETU processes can be addressed collectively due to the short lifetime of ⁵I₅ [27]. In fibers that are entirely free of ion clustering, the Ho³⁺ ions are uniformly distributed and are involved in ETU with the same rate. In practice, however, the ions may not be fully evenly distributed, with some ions reside in close proximity to form clusters. The ETU occurring in ion clusters (i.e., PIQ) is much more rapid than that in non-clustered ions. Such clustering-associated inhomogeneous upconversion is deleterious to the performance of laser, as elucidated by us in [25]. It should be noted that in fibers of very high Ho³⁺ concentrations the clustering effect may induce self-pulsing [28], a dynamic process that is difficult to describe. In this work, self-pulsing is ignored since we concentrate on fibers of low Ho³⁺ concentrations.

Another important factor that affects the performance of Ho-doped fiber lasers is the background loss of fibers. Silica fibers suffer from heavy loss in the 2 μm wavelength region due to the intrinsic absorption of silica [24] and the absorption associated with OH combination mode [29]. Although fiber loss is indisputably harmful to the slope efficiency of laser, its effect is limited. The loss in the 2 μm range is predominantly determined by the OH impurities in fiber, which are easy to reduce via improvement of fabrication technique. At present most available Ho-doped fibers are of well-suppressed OH contamination [30]. As a result, the effect of fiber loss only becomes significant in systems using very long fibers.

3. Effect of ion clustering in Ho-doped fiber of low Ho³⁺ concentration

In this section, we investigate the effect of ion clustering in the Ho-doped fiber of low Ho³⁺ concentration and high-quality composition. A resonantly pumped experimental system is established, with the configuration demonstrated in Fig. 2(a).

Fig. 2 (a) Configuration of established Ho-doped fiber laser. TDFL: Tm-doped fiber laser. HDF: Ho-doped fiber. HR FBG: high reflectivity FBG. The length of Ho-doped fiber is variable. (b) Output spectrum of the Ho-doped fiber laser under the pump power of ~20 W. Fiber length: 1.9 m.

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The system is established with commercially available fiber (Nufern SM-HDF-10/130), and is pumped by an in-house built Tm-doped fiber laser which delivers up to 30 W output power at 1950 nm. The output beam of the Tm-doped fiber laser is sent through a cladding mode stripper for eliminating the 793 nm residual pump, and then coupled into the core of the following fibers. A fiber Bragg grating (FBG) with high reflectivity (>99.5%) at 2100 nm is used as the high reflection reflector. Another resonator reflector is performed by the perpendicularly cleaved fiber facet (reflectance: ~4%). The SM-HDF-10/130 fiber is the most recent product of the primary supplier of rare-earth doped fibers, Nufern Inc. It represents the state-of-the-art fabrication technology of Ho-doped fibers. The fiber is of low doping concentration (0.5 wt. %) and appropriate codopant (Al³⁺) to suppress ion clustering. The fiber is also of a very low level OH contamination to minimize the background loss. The output spectrum of the laser is presented in Fig. 2(b). Parasite lasing is prevented, and amplified spontaneous emission (ASE) is suppressed well. Such spectral characteristics make the laser particularly suitable for being modeled.

In order to analyze the effect of ion clustering theoretically, we resort to the method developed in our previous work. In [25], we introduce two models which are devoted to the systems free of ion clustering (denoted by HUC) and the systems affected by ion clustering (denoted by PIQ) respectively. In case of HUC, the rate equations are:

\frac{d N_{0}}{d t} = W_{30} N_{3} + W_{20} N_{2} + W_{10} N_{1} + W_{11} N_{1}^{2} + \frac{(σ_{10} N_{1} - σ_{01}^{s} N_{0}) Γ_{s} λ_{s} P_{s} - σ_{01}^{p} N_{0} Γ_{p} λ_{p} P_{p}}{h c A}

\frac{d N_{1}}{d t} = W_{31} N_{3} + W_{21} N_{2} - W_{10} N_{1} - 2 W_{11} N_{1}^{2} + \frac{σ_{01}^{p} N_{0} Γ_{p} λ_{p} P_{p} - (σ_{10} N_{1} - σ_{01}^{s} N_{0}) Γ_{s} λ_{s} P_{s}}{h c A}

\frac{d N_{2}}{d t} = W_{32} N_{3} - W_{21} N_{2} - W_{20} N_{2}

\frac{d N_{3}}{d t} = - W_{32} N_{3} - W_{31} N_{3} - W_{30} N_{3} + W_{11} N_{1}^{2}

N_{Ho} = N_{0} + N {}_{1}+ N_{2} + N_{3}

where N_Ho represents the total population density of Ho³⁺ ions, N₀, N₁, N₂, and N₃ represent the population densities of the ⁵I₈, ⁵I₇, ⁵I₆, and ⁵I₅ energy level respectively (as labeled in Fig. 1), W_ij represents the spontaneous decay rate between energy level i and j with the exception of W₁₁ which represents the homogenous upconversion rate, Γ_p and Γ_s represent the power filling factors of the pump and signal respectively, λ_p and λ_s represent the wavelength of the pump and signal respectively, h represents the Planck constant, c represents the speed of light, A represents the area of fiber core, σp 01 represents the absorption cross section at the pump wavelength, σs 01 represents the absorption cross section at the signal wavelength, σ₁₀ represents the emission cross section at the signal wavelength, P_p and P_s represent the power of the pump and signal respectively which can be expressed as the sum of the forward ( + ) and backward (–) propagated beam. Here, the population density of ⁵I₄ is omitted, in view of that the transition process which can populate ⁵I₄ is nonexistent in resonantly pumped systems.

The propagation equations are listed as follows:

\frac{d P_{p}^{\pm}}{d z} = \mp {Γ_{p} σ_{01}^{p} N_{0} + ξ_{p}} P_{p}^{\pm}

\frac{d P_{s}^{\pm}}{d z} = \pm {Γ_{s} σ_{10} N_{1} - Γ_{s} σ_{01}^{s} N_{0} - ξ_{s}} P_{s}^{\pm}

where ξ_p and ξ_s represent the loss coefficients at the pump and signal wavelength respectively, of which the values are defined as the sum of the scattering loss, the intrinsic absorption of silica, and the absorption associated with OH impurities. Substituting N₀ and N₁ in Eq. (6) and Eq. (7) with the steady-state solutions of the rate equations, and subjecting Eq. (6) and Eq. (7) to certain boundary conditions, the operation of a clustering-free system can be simulated.

In case of PIQ, the Ho³⁺ ions can be divided into two species: the non-clustered ions and the clustered ions. The rate equation of the non-clustered ions is expressed as

\frac{d N_{0}^{i}}{d t} = W_{10} N_{1}^{i} + W_{11} {(N_{1}^{i})}^{2} + \frac{(σ_{10} N_{1}^{i} - σ_{01}^{s} N_{0}^{i}) Γ_{s} λ_{s} P_{s} - σ_{01}^{p} N_{0}^{i} Γ_{p} λ_{p} P_{p}}{h c A}

where Ni 0 and Ni 1 represent the population densities of the non-clustered Ho³⁺ ions in ⁵I₈ and ⁵I₇ respectively. Here, the ions promoted to ⁵I₆ and ⁵I₅ are neglected given their short lifetimes. Regarding the clustered ions, if we consider the simplest case in which each cluster is composed of only two ions, the Ho³⁺ clusters fall into three states: a). both ions in ⁵I₈, b). one ion in ⁵I₈ whilst one ion in ⁵I₇, and c). both ions in ⁵I₇. The state c is omitted in the modeling work due to the fact that PIQ will instantly demote one ion to ⁵I₈ [25]. Hence, the population densities of the clusters in state a (denoted by $N_{a}^{c}$ ) and state b (denoted by $N_{b}^{c}$ ) can be introduced to the total population density of ions by $N_{a}^{c}$ + $N_{b}^{c}$ = kN_Ho. Here k is the relative number of clusters, by which the population densities of the clustered and non-clustered Ho³⁺ ions can be expressed as 2kN_Ho and (1–2k)N_Ho, respectively. The rate equation of the Ho³⁺ clusters is

\frac{d N_{a}^{c}}{d t} = W_{10} N_{b}^{c} + \frac{(σ_{10} N_{b}^{c} - 2 σ_{01}^{s} N_{a}^{c}) Γ_{s} λ_{s} P_{s} - 2 σ_{01}^{p} N_{a}^{c} Γ_{p} λ_{p} P_{p}}{h c A}

The propagation equations of the PIQ model are

\frac{d P_{p}^{\pm}}{d z} = \mp {Γ_{p} σ_{01}^{p} [N_{0}^{i} + 2 N_{a}^{c} + N_{b}^{c}] + ξ_{p}} P_{p}^{\pm}

\frac{d P_{s}^{\pm}}{d z} = \pm {Γ_{s} σ_{10} [N_{1}^{i} + N_{b}^{c}] - Γ_{s} σ_{01}^{s} [N_{0}^{i} + 2 N_{a}^{c} + N_{b}^{c}] - ξ_{s}} P_{s}^{\pm}

Substituting the population densities in Eq. (10) and Eq. (11) with the steady-state solutions of Eq. (8) and Eq. (9), the operation of a Ho-doped fiber laser affected by ion clustering can be simulated.

We employ the models described above to simulate the experimental system presented in Fig. 2. The simulations are performed with the following parameters: N_Ho = 4 × 10²⁵ m⁻³, σp 01 = 2.93 × 10⁻²⁵ m², σs 01 = 0.194 × 10⁻²⁵ m², σ₁₀ = 1.18 × 10⁻²⁵ m², W₁₀ = 1729.2 s⁻¹, W₂₀ = 72.26 s⁻¹, W₂₁ = 7.1421 × 10⁵ s⁻¹, W₃₀ = 27.97 s⁻¹, W₃₁ = 29.85 s⁻¹, W₃₂ = 4.5454 × 10⁷ s⁻¹, W₁₁ = 4 × 10⁻²³ m³ s⁻¹, ξ_p = 0.695 × 10⁻² m⁻¹, ξ_s = 2.245 × 10⁻² m⁻¹, Γ_p = 0.8112, and Γ_s = 0.7772. Here the value of N_Ho is calculated from the doping concentration (0.5 wt. %), the values of the transition cross sections and spontaneous decay rates are determined with the approach reported in [25], the values of the loss coefficients and power filling factors are calculated from product specifications (provided by Nufern Inc.) through the method illustrated in [25].

Several criteria can be used to evaluate the reliability of the models, such as i) transmission characteristics, and ii) laser output characteristics. We firstly investigate the transmission characteristics. A short piece of SM-HDF-10/130 fiber is spliced to the output port of the 1950 nm pump laser (i.e., the pigtail of the cladding mode stripper). The free end of the tested fiber is angularly cleaved for eliminating Fresnel reflection and preventing lasing. The transmittance of the 1950 nm radiation through the tested fiber is measured under different powers. The measurements are performed for two fiber lengths (0.23 m and 0.55 m). The HUC and the PIQ models are run for numerical simulations. To simulate such a non-laser system the terms representing stimulated emission are eliminated from the rate equations. The boundary conditions are set properly for simulating the non-resonator configuration.

The results are demonstrated in Fig. 3. The fluctuation of the experimental curves reflects the instability of the pump. The experimental results exhibit strong non-saturable absorption. The simulation results acquired with the HUC model manifest saturable absorption, and thus, show substantial discrepancy with the experimental results. On the other hand, the simulation results obtained with the PIQ model can be fitted to the experimental curves by adjusting the value of k. The best fit is achieved with k = 0.05, indicating that ~10% of the Ho³⁺ ions in the tested fibers are grouped as clusters. The non-saturable absorption exhibited by the experimental systems can be attributed to the fact that PIQ forces a large number of clustered ions to stay in ⁵I₈, and consequently, relieves ground state depletion (GSD). It should be noted that the divergence between the HUC curves and the experimental curves only becomes considerable as the pump power is beyond a certain value [31]. Under low pump power, the HUC model may well simulate the transmission characteristics of Ho-doped fibers, as reported in [21].

Fig. 3 Measured and simulated transmission values as a function of the launched pump power for (a) 0.23 m fiber and (b) 0.55 m fiber. Exp.: experimental results.

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The laser output characteristics are also studied. Figure 4(a) presents the experimental and the simulated results of a laser system consisting of 1.9 m fiber. In the simulation using the PIQ model, the value of k is set to be 0.05 given that the value leads to the best fit between the simulated and the experimental results in the investigation of transmission characteristics. The samples tested in this work are all from a single piece of fiber, and therefore, should be of a similar fraction of clustered ions. With k = 0.05 the PIQ-simulated result indeed agrees well with the experimental result, a fact that further verifies the presence of ion clustering in the SM-HDF-10/130 fiber. On the other hand, the HUC-simulated result shows significant disagreement with the experimental result. The experimental and the simulated slope efficiencies of systems comprising various lengths of fibers are presented in Fig. 4(b). With k = 0.05 the PIQ-simulated slope efficiencies are almost identical to the experimental values, whereas the HUC-simulated efficiencies are substantially higher. Without a filter to separate the signal from the unabsorbed pump, here the slope efficiencies of systems using short fibers cannot be determined in experiments; however an optimal fiber length is expected according to our previous research [25]. A noteworthy fact is that the HUC-simulated slope efficiencies are approximate to the theoretical limit (~93%). In view of that the HUC model incorporates loss coefficients (i.e., ξ_p and ξ_s) of which the values are calculated from the product specifications, it stands to reason that the contribution of fiber loss to the slope efficiency degradation is negligible in systems using high-quality fibers.

Fig. 4 (a) Experimental and simulated output power as a function of the absorbed pump power. Fiber length: 1.9 m. (b) Experimental and simulated slope efficiencies as a function of the fiber length. Exp: experimental results.

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In this section the experimental systems comprising Ho-doped fiber of low doping concentration and high quality composition are simulated using the numerical models. With appropriate value of k, the simulation results of the PIQ model which features the incorporation of clustering mechanism exhibit excellent agreement with the experimental results. On the other hand, the simulation results of the HUC model which is devoted to the clustering-free systems show substantial disagreement with the experimental results. The investigated fiber (Nufern SM-HDF-10/130) is fabricated with state-of-the-art technology and is of a low dopant concentration. The experimental and the simulated results reveal that such fiber still suffers from ion clustering. Previous studies have identified ion clustering as the dominant cause of the slope efficiency degradation observed in heavily doped fibers or low-quality fibers fabricated with non-optimized technology [25,26]. The research work presented in this section testifies that the same mechanism also acts in high-quality fiber of low Ho³⁺ concentration. It is reasonable to speculate that Ho³⁺ ions in silica host are of a strong tendency to form clusters, like the well-known case of erbium (Er³⁺) ions. Therefore, ion clustering is an intrinsic flaw of Ho-doped fibers and is difficult to eliminate. One way to mitigate ion clustering in Ho-doped fibers is to further reduce the concentration of Ho³⁺. Recently, A. V. Kir’yanov et al. report a record high slope efficiency achieved from a non-resonantly pumped Ho-doped fiber laser [32]. Employing a fiber with ~1.15 × 10²⁵ m⁻³ population density of Ho³⁺ (corresponding to ~0.14 wt. % concentration), the reported 1.13 μm-pumped system realizes a slope efficiency of 48%, almost 88% of the theoretical limit. The strategy of reducing dopant concentrations of fibers, however, cannot be applied to high power systems in which cladding-pumped fibers are required. The issue relevant to cladding-pumped fibers will be discussed in the later part of this paper.

4. Effect of ESA in non-resonantly pumped Ho-doped fiber lasers

The essence of the clustering-associated slope efficiency degradation is that the rapid ETU occurring within ion clusters (i.e., PIQ) render a large number of Ho³⁺ ions unusable for laser operation. The unusable ions still absorb pump, however they are immediately demoted to ground state by PIQ without contribution to lasing. As a consequence, the slope efficiency is reduced. Given that ETU is a process independent of pumping schemes, the detrimental effect of ion clustering acts on both resonantly and non-resonantly pumped systems. A. S. Kurkov et al. report the concentration-associated slope efficiency degradation observed in non-resonantly pumped Ho-doped fiber laser, and ascribe it to ion clustering [26]. On the other hand, the energy level structure of Ho³⁺ illustrates that the energy gaps involved in the ⁵I₈→⁵I₆ GSA and the ⁵I₇→⁵I₄ ESA are near-resonant. This fact implies that ESA may also contribute to the slope efficiency degradation in non-resonantly pumped systems. Without a comprehensive analysis of ESA, the cause of the lower-than-expected slope efficiencies observed in non-resonantly pumped Ho-doped fiber lasers cannot be ascertained.

Determination of the ESA cross sections via experimental approach is usually difficult or even impossible [33]. In this section, we calculate the cross sections of the ⁵I₇→⁵I₄ ESA in a direct way. The approach used by us is based upon Judd-Ofelt theory. The ESA spectrum can be expressed by [33]

σ_{14} (\tilde{ν}) = \frac{e^{2}}{4 ε_{0} m_{e} c^{2}} f_{14} G (\tilde{ν})

where σ₁₄ represents the cross section of the ⁵I₇→⁵I₄ transition at wavenumber

\tilde{ν}

, e represents the elementary charge, ε₀ represents the vacuum permittivity, m_e represents the rest mass of an electron, c represents the speed of light. G represents the line shape of the inhomogeneously broadened absorption peak of Ho³⁺ in glass matrix, which can be expressed by the Gaussian function

G (\tilde{ν}) = \sqrt{\frac{\ln 2}{π B^{2}}} \exp [- \frac{\ln 2}{B^{2}} {(\tilde{ν} - {\tilde{ν}}_{0})}^{2}]

where B is the half width at half-maximum (HWHM),

{\tilde{ν}}_{0}

is the peak center (all in cm⁻¹).

In Eq. (12), f₁₄ represents the electric dipole oscillator strength of the ⁵I₇→⁵I₄ transition. f₁₄ can be expressed as

f_{14} = \frac{8 π^{2} m_{e} c ν_{m}}{3 h (2 J + 1)} \frac{{(n^{2} + 2)}^{2}}{9 n} S_{14}

in which ν_m is the mean energy of the ⁵I₇→⁵I₄ transition (in cm⁻¹), h is the Planck constant, J is the total angular momentum of the initial state in the transition, n is the refraction index of silica glass. S₁₄ is the electric dipole line strength of the ⁵I₇→⁵I₄ transition that can be calculated by

S_{14} = \sum_{k = 2, 4, 6} Ω_{k} {〈 f^{n} [S L] J ‖ U^{(k)} ‖ f^{n} [S' L'] J' 〉}^{2}

in which Ω_k are the Judd-Ofelt parameters, U^(k) are the irreducible tensor forms of the dipole operator. It should be noted that the magnetic dipole process is not taken into account in our calculations. This is owing to the fact that in the ⁵I₇→⁵I₄ transition the magnetic dipole process is forbidden by the selection rules [34].

The reduced matrix elements in Eq. (15) are commonly assumed to be host independent [35]. In this work the values found in [36] (which are calculated for Ho³⁺ in LaF₃) are adopted. The values of the Judd-Ofelt parameters of Ho³⁺ in silica host are found in [37]. The transition energy (ν_m) and the peak center of the absorption band ( ${\tilde{ν}}_{0}$ ) are strongly dependent on host materials [38]. Unfortunately, there are no available values concerning Ho³⁺ in silica fibers. A set of data associated with Ho³⁺ in silicate glass can be found in [38], however they are of remarkable discrepancy with the experimentally measured absorption spectrum of Ho-doped fibers [39]. In this work we estimated the energies of ⁵I₅, ⁵I₆, and ⁵I₇ directly from the absorption spectrum presented in [39]. The energy of ⁵I₄, however, cannot be acquired in such way. There is no observable spectral peak for the ⁵I₈→⁵I₄ transition, due to the fact that the cross sections of this transition are too small [33]. As an approximation we adopt the energy value of ⁵I₄ found in [33], which is calculated for Ho³⁺ in ZBLAN matrix. Such a management is reasonable, given that the measured locations of other absorption peaks of Ho³⁺ in ZBLAN host [33,40] are almost identical to those of Ho³⁺ in silica fibers [39]. The HWHM linewidths (B) are also from [33].

We employ the approach introduced above to calculate the cross section spectrum of the ⁵I₇→⁵I₄ ESA. The cross section spectra of the ⁵I₈→⁵I₆ and ⁵I₈→⁵I₅ GSA are also calculated with the same formulas. The spectrum of the ⁵I₈→⁵I₇ GSA (that is, the resonant pumping) is not calculated here, owing to the fact that in the ⁵I₈→⁵I₇ transition the magnetic dipole process plays an important role [34]. In the calculation, we use the following parameters: n = 1.45, Ω₂ = 3.6 × 10⁻²⁰ cm², Ω₄ = 2.3 × 10⁻²⁰ cm², Ω₆ = 0.65 × 10⁻²⁰ cm²; the energies of ⁵I₄, ⁵I₅, ⁵I₆, ⁵I₇, and ⁵I₈ are respectively 13285, 11350, 8696, 5128, and 0 (all in unit of cm⁻¹); the values of J, B and the reduced matrix elements are presented in Table 1.

Table 1. Reduced Matrix Elements and Transition Linewidths between Energy Levels of Ho³⁺

View Table

The calculation results are presented in Fig. 5. The calculated lineshapes of the ⁵I₈→⁵I₆ and the ⁵I₈→⁵I₅ GSA are of good agreement with the experimentally measured spectrum, except that the calculation results cannot reproduce the sub-peak (which are most probably caused by the Stark splitting of the energy levels) of the absorption peaks in measured spectrum [39]. For evaluating the validity of the calculation results, we compare their values to the experimentally determined cross sections (denoted by “measured values” here) which is acquired from the measured absorption rates of SM-HDF-10/130 fiber. The maximum cross section calculated with Judd-Ofelt approach for the ⁵I₈→⁵I₅ GSA (~0.22 × 10⁻²⁵ m²) exhibits excellent agreement with the measured value (~0.23 × 10⁻²⁵ m²). On the other hand, the maximum calculated cross section for the ⁵I₈→⁵I₆ GSA (~0.77 × 10⁻²⁵ m²) is smaller than the measured value (~1.1 × 10⁻²⁵ m²). The discrepancy can be ascribed to various reasons, such as the uncertainty of the parameters (e.g., linewidth, reduced matrix elements, etc.) used in the calculations. A particularly noteworthy reason is the effect of Al³⁺ codopant. It is known that in Er-doped fibers the concentrations of Al³⁺ codopant influence the absorption cross sections of Er³⁺ ions by means of affecting Judd-Ofelt parameters (Ω_k) [41]. The SM-HDF-10/130 fiber is also codoped with Al³⁺ for suppressing ion clustering. Hence, it is probable that the effect of Al³⁺ induces the larger absorption cross sections for the ⁵I₈→⁵I₆ GSA in the tested fiber.

Fig. 5 Spectrum of calculated ESA (⁵I₇→⁵I₄, in red solid line) and GSA (⁵I₈→⁵I₅ and ⁵I₈→⁵I₆, in blue dashed line) cross sections.

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In Fig. 5, the non-resonant pumping scheme corresponds to the 1150 nm-centered absorption peak (i.e., the ⁵I₈→⁵I₆ GSA). Previously reported non-resonantly pumped systems are of pump wavelength range from 1100 nm to 1160 nm [8–12,14]. Pump wavelengths longer than 1160 nm are not adopted by researchers, simply due to the fact that i) the strongest absorption occurs at 1150 nm, and ii) there is no efficient light source that can emit wavelength longer than 1160 nm. On the other hand, the center of the ⁵I₇→⁵I₄ ESA is at ~1226 nm. The overlap between the absorption peaks of the ⁵I₈→⁵I₆ GSA and the ⁵I₇→⁵I₄ ESA is very limited. In particular, the calculated cross sections of the ⁵I₇→⁵I₄ ESA in the wavelength region shorter than 1160 nm are tiny, indicating a very weak absorption. Hence, the effect of ESA is negligible in the non-resonantly pumped systems relying on ordinary pump wavelengths (i.e., 1100 nm–1160nm).

As discussed above, the validity of the cross sections calculated by Judd-Ofelt theory is limited. Therefore, we also design an experimental method to investigate the intensity of the ⁵I₇→⁵I₄ ESA. The setup of the experiment is presented in Fig. 6. A YSL SC-5 supercontinuum laser is used as the probe. The in-house built Tm-doped fiber laser operating at 1950 nm is used as the pump. Both the probe and the pump beams are delivered into the tested fiber with a wavelength-insensitive 50:50 coupler. The tested fiber is 1.5 m SM-HDF-10/130 Ho-doped fiber. The transmission spectrum of the tested fiber is monitored by a Yokogawa AQ6370C OSA.

Fig. 6 Experimental setup used for the measurement of the ESA spectrum. TDFL: Tm-doped fiber laser; SC: supercontinuum laser; HDF: 1.5 m SM-HDF-10/130 Ho-doped fiber.

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Figure 7 demonstrates the evolution of the transmission spectrum through the increase of the pump power. The absorption peaks (manifested as depressions in the transmission spectrum) of ESA processes can be observed even as the pump is off, owing to the high output power of the supercontinuum laser. As the pump power is increased, the absorption peaks of the GSA processes become smaller, due to the decreased population of ⁵I₈. On the other hand, the absorption peaks associated with the ESA processes become larger due to the increase of the population of ⁵I₇. After the pump power is beyond 1 W, the spectrum is stabilized, and will not change through further increase of the pump power. This is attributed to the fact that with >1 W pump the population of the ⁵I₇ energy level in the 1.5 m fiber is saturated (under higher power the pump can be still absorbed by the clustered ions residing in ⁵I₈ as discussed in Section 3, however the ions excited in this way cannot stay in ⁵I₇ due to PIQ). Under maximum pump power a number of ESA processes are observed, such as ⁵I₇→⁵F₅ and ⁵I₇→⁵F₄/ ⁵S₂. No absorption peak corresponding to the ⁵I₇→⁵I₄ ESA is detected. The location of the calculated center of the ⁵I₇→⁵I₄ ESA is presented as the vertical dashed line in Fig. 7.

Fig. 7 Measured absorption spectrum of a 1.5 m SM-HDF-10/130 Ho-doped fiber under varying launched pump powers from 0 to 2.3 W.

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The experimental result presented in Fig. 7 indicates that the intensity of the ⁵I₇→⁵I₄ ESA is very small, a fact that accords with the calculation result obtained with Judd-Ofelt approach. In the experiment, almost all excited ions are in the ⁵I₇ energy level since the fiber is resonantly pumped at 1950 nm. This fact is a merit for the ESA processes starting from ⁵I₇, including the target of the investigation, ⁵I₇→⁵I₄. The most reasonable explanation for the absence of the ⁵I₇→⁵I₄ ESA in the transmission spectrum is that its cross sections are tiny, and thus, the absorption peak of it is entirely annexed by the pedestal of the ⁵I₈→⁵I₆ GSA.

Both the theoretical and the experimental research efforts described above bring to a conclusion that the effect of the ⁵I₇→⁵I₄ ESA is negligible in Ho-doped fiber lasers, mainly due to its small cross sections, but also due to its insignificant overlap with the ordinary pump band of non-resonantly pumped systems (i.e., 1100 nm−1160 nm). On the other hand, phonon-assisted ESA may induce a larger overlap and should be taken into account. Here, we use the method developed by L. V. G. Tarelho et al. in [42] to calculate the phonon sideband of the ⁵I₇→⁵I₄ ESA. The spectral location of the phonon sideband can be obtained by shifting the wavelength from λ to λ $_{q}^{-}$ according to the relation λ $_{q}^{-}$ = (1/λ + qħω)⁻¹, where q is the number of the phonons involved in the process, ħω is the average energy of phonon. The amplitude of the phonon sideband can be obtained by scaling the zero-phonon band of the ⁵I₇→⁵I₄ ESA with a factor P $_{q}^{-}$ which is given as P $_{q}^{-}$ = exp[−2n_AS₀](S₀^q/q!)n_A^q, in which n_A = [exp(ħω/k_BT)−1]⁻¹ is the average occupancy of phonon mode at T = 300 K, S₀ = 0.31 is the Huang-Rhys factor [42], and ħω = 505 cm⁻¹ is the effective phonon energy of silica [43] which is different from the maximum phonon energy (1100 cm⁻¹) used in most research works. In the case investigated here, one-phonon sideband is enough to realize a large overlap with the 1100 nm−1160 nm pump band. The value of P $_{1}^{-}$ is calculated as ~0.028. The one-phonon sideband of the ⁵I₇→⁵I₄ ESA is presented in Fig. 8.

Fig. 8 Spectrum of calculated ESA (⁵I₇→⁵I₄, in red solid line), one-phonon sideband of ESA (⁵I₇→⁵I₄, in red dotted line, of which the amplitude is enlarged by 10 times), and GSA (⁵I₈→⁵I₆, in blue dashed line) cross sections.

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As demonstrated in Fig. 8, the one-phonon sideband of the ⁵I₇→⁵I₄ ESA can achieve a large overlap with the 1100 nm−1160 nm pump band. Nonetheless, the values of the cross sections of the sideband are too small. The maximum cross section of the sideband is calculated as 4.84 × 10⁻²⁸ m² at ~1154.4 nm. With such a small cross section the one-phonon sideband of the ⁵I₇→⁵I₄ ESA cannot affect the performance of Ho-doped fiber lasers. Regarding the multi-phonon sidebands (q≥2), the cross sections are even smaller given the decreased value of P $_{q}^{-}$ , and thus, should be ignored in the investigation of the ⁵I₇→⁵I₄ ESA.

In this section, we comprehensively study the role of ESA in non-resonantly pumped Ho-doped fiber lasers. It is found that the effect of ESA on the performance of Ho-doped fiber lasers is negligible. On the other hand, ion clustering still behaves as a detrimental factor in non-resonantly pumped systems, since ETU is a pumping-independent process. In summary, ion clustering is the dominant cause of the sub-optimal slope efficiencies in both resonantly pumped and non-resonantly pumped Ho-doped fiber lasers.

5. Discussion

Hitherto the lower-than-expected slope efficiencies observed in Ho-doped fiber lasers is not well understood. In this work, the problem is well explained by the effect of ion clustering, regardless of pumping schemes. The investigated fiber (Nufern SM-HDF-10/130) is found to be of ~10% clustered ions, even as the fiber is of a low Ho³⁺ concentration (0.5 wt. %) and appropriate codopant (Al³⁺) for suppressing clustering. In light of this, we argue that the Ho³⁺ ions doped in silica host inherently tend to form clusters.

A number of literatures report the concentration dependence of the slope efficiency degradation in Ho-doped fiber lasers [17,26]. The efficiency performance of the lasers comprising heavily doped fibers is usually worse than that of the lasers consisting of low-concentration fibers. This can be attributed to the fact that the homogeneousness of Ho³⁺ ions is more difficult to control in heavily doped fibers. In other words, fibers of higher doping concentrations tend to be of greater fractions of clustered ions. Such concentration effect is a barrier in the development of high power Ho-doped fiber lasers. It is well known that the high power application requires cladding-pumped, multi-clad fibers. In case of Ho-doped fiber, a triple clad structure is desired due to the strong absorption of 2 μm radiation by the polymer coating. To achieve a moderate absorption rate of pump, cladding pumped fibers are ordinarily of much higher doping concentrations than core pumped ones. The high doping concentrations, however, are not acceptable for fibers suffering from ion clustering, due to the penalty on slope efficiencies. A well known instance is Er-doped fiber, in which Er³⁺ ions are of a strong tendency to form clusters. The scalability of Er-doped fiber lasers are significantly limited owing to the absence of heavily doped, double clad Er-doped fibers.

At present, the cladding-pumped Ho-doped fibers are of low doping concentrations that are comparable to the concentrations of the core-pumped ones [18,19]. In the previous work we attest that such low-concentration fibers cannot realize high performance in laser systems, due to the combined effect of high background loss and low pump absorption [25]. A feasible solution to this problem is to increase the core-to-cladding area ratio of fibers, by which high pump absorption can be achieved with short fibers even as the fibers are of low doping concentrations [25,44]. A more promising solution is to employ the technique of nanoparticle doping, by which the Ho³⁺ ions are encaged in a nanoparticle host which can improve the solubility of Ho³⁺. Recently, fiber comprising Ho-doped LaF₃ and Ho-doped Lu₂O₃ nanoparticles has been reported to realize slope efficiency as high as 85% in a MOPA configuration [45].

6. Conclusion

For the first time the effects of ion clustering and ESA on the performance of Ho-doped fiber lasers are comprehensively investigated. We attest that ion clustering is the dominant cause of the lower-than-expected slope efficiencies observed in Ho-doped fiber lasers, regardless of pumping schemes. On the other hand, the effect of ESA is found to be negligible. We hold the view that ion clustering is an inherent defect of Ho-doped fibers and is difficult to eliminate, given that our works are based upon commercially available fiber which is of low Ho³⁺ concentration and high-quality composition. Our works are informative for the future research on Ho-doped fiber lasers, particularly the high power systems in which high-concentration fibers are required.

Funding

R&D Program, Korea Institute of Science and Technology (KIST) (2E29300).

Acknowledgments

The authors acknowledge support by Dr. Adrian Carter, chief technical officer of Nufern, Inc.

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[SL]J	[S’L’]J’	\|\| U⁽²⁾ \|\|	\|\| U⁽⁴⁾ \|\|	\|\| U⁽⁶⁾ \|\|	B [cm⁻¹]
⁵I₇	⁵I₄	0	–0.05786	0.39603	200
⁵I₈	⁵I₅	0	–0.09943	0.30595	150
⁵I₈	⁵I₆	0.09103	–0.19576	0.83173	220

Effects of ion clustering and excited state absorption on the performance of Ho-doped fiber lasers

Abstract

1. Introduction

2. Mechanisms

3. Effect of ion clustering in Ho-doped fiber of low Ho³⁺ concentration

4. Effect of ESA in non-resonantly pumped Ho-doped fiber lasers

5. Discussion

6. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (8)

Tables (1)

Equations (15)

Optics Express

Abstract

1. Introduction

2. Mechanisms

3. Effect of ion clustering in Ho-doped fiber of low Ho3+ concentration

4. Effect of ESA in non-resonantly pumped Ho-doped fiber lasers

5. Discussion

6. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (8)

Tables (1)

Equations (15)

Optics Express

3. Effect of ion clustering in Ho-doped fiber of low Ho³⁺ concentration