Tm/Ho-doped fiber laser systems using coaxial fiber

Krysta A. Boccuzzi; G. Alex Newburgh; John R. Marciante

doi:10.1364/OE.27.027396

1. Introduction

Holmium fiber lasers are an efficient source for producing radiation at 2100 nm and beyond, but the lack of high-power pumps at 1950 nm prohibits the simple power scaling that is available to Yb- and Tm-doped fiber lasers. Holmium is therefore typically exploited via application of tandem laser systems: 790 nm diodes pump a Tm:fiber laser emitting at 1950 nm, which then pumps the Ho:fiber laser [1,2]. Another common method of producing 2100 nm radiation is through a Tm/Ho co-doped fiber laser, where the holmium and thulium are doped into the same core. Typically, co-doped fiber lasers are less efficient than the tandem configuration, as their performance is limited by energy transfer upconversion [3].

A recent demonstration [4] and related patents [5,6] introduced the concept of a coaxial fiber, with a holmium-doped core surrounded by a thulium-doped ring, in order to reduce the complexity of these systems into a single diode-pumped fiber. A cross section of this coaxial fiber concept is shown in Fig. 1. In the initial demonstration [4], 790 nm radiation was used to pump the Tm ring, and the resulting signal pumped the Ho core to produce 2100 nm radiation. Although the slope efficiency of this initial demonstration was 10%, the question of the maximum achievable efficiency still remains.

Fig. 1 Cross section of the coaxial fiber design.

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The coaxial fiber may be more efficient than other methods since the holmium is pumped directly by the intracavity power of the Tm signal. The output coupler can be designed to minimize the cavity losses and maximize the intracavity power at the Tm emission wavelength, allowing for better net conversion efficiency from 790 nm pump to 2100 nm emission. On the other hand, high net conversion efficiency means depletion of the Tm laser signal since the Tm emission is absorbed by the Ho core. This absorption represents intracavity loss for the Tm laser, which may then operate at reduced efficiency compared to an isolated Tm laser.

In order to understand the efficiency limits of this fiber type, the performance of the Tm/Ho coaxial fiber is optimized through simulations and compared to the performance of both the conventional tandem and co-doped configurations. The model is used to understand why the first experimental demonstration was unsuccessful. In Section 2, the physical model used to simulate the various laser systems is presented. The simulation results are presented in Section 3, while a discussion of the results and concluding remarks are presented in Section 4.

2. Physical Mode

Four different fiber lasers were modeled in this study. The first, using a silica fiber with a Tm-doped core, was modeled using the method described by Jackson and King [7]. For this method, the rate equations for the four lowest energy levels of Tm³⁺ are shown in Fig. 2(a) and described by Eqs. (1)-(4):

N_{0} = N_{T m} - N_{1} - N_{2} - N_{3}

\begin{array}{l} \frac{d N_{1}}{d t} = - A_{10}^{'} N_{1} + A_{21}^{'} N_{2} + A_{31} N_{3} + 2 (k_{31} N_{3} N_{0} - k_{13} N_{1}^{2}) + 2 (k_{21} N_{2} N_{0} - k_{10} N_{1}^{2}) \\ - \frac{λ_{T m}}{h c A_{e f f}^{p}} [P_{f}^{T m} (z) + P_{r}^{T m} (z)] [σ_{e}^{T m} (λ_{T m}) N_{1} - σ_{a}^{T m} (λ_{T m}) N_{0}] \end{array}

\frac{d N_{2}}{d t} = A_{32}^{'} N_{3} - (A_{21}^{'} + A_{20}) N_{2} - (k_{21} N_{2} N_{0} - k_{10} N_{1}^{2})

\frac{d N_{3}}{d t} = \frac{λ_{p}}{h c A_{e f f}^{T m}} σ_{a}^{T m} (λ_{p}) [P_{f}^{p} (z) + P_{r}^{p} (z)] N_{0} - (A_{3}^{'} + A_{32}^{'}) N_{3} - (k_{31} N_{3} N_{0} - k_{13} N_{1}^{2})

where

A_{i j}^{'} = A_{i j} + Γ_{i}

and

A_{3}^{'} = A_{30} + A_{31}

.

Fig. 2 Energy level diagrams for (a) thulium and (b) holmium ions, demonstrating the cross relaxation (CR) and upconversion (UC) processes. For simplicity, not all process captured in Eqs. (1)-(14) are shown in the diagram.

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The bi-directional propagation of the power in the Tm:fiber can then be described by:

\frac{d P_{f, r}^{p} (z)}{d z} = \mp P_{f, r}^{p} (z) [σ_{a}^{T m} (λ_{p}) N_{0} Γ_{p}^{T m} + α_{p}]

\begin{array}{l} \frac{d P_{f, r}^{T m} (z)}{d z} = \pm P_{f, r}^{T m} (z) {[σ_{e}^{T m} (λ_{T m}) N_{1} - σ_{a}^{T m} (λ_{T m}) N_{0}] Γ_{T m}^{T m} - α_{T m}} \\ \pm σ_{e}^{T m} (λ_{T m}) N_{1} Γ_{T m}^{T m} h ν_{T m} Δ ν_{T m} \end{array}

Equations (5) and (6) describe the pump and Tm signal powers, respectively, and differ from [7] due to the addition of spontaneous emission, which allows the Tm signal to start from noise. All the variables used in Eqs. (1) – (6) are defined in Table 1.

Table 1. Variable definitions

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The second fiber laser, using a silica fiber with a Ho-doped core, was modeled using the rate equations for the four lowest energy levels of Ho³⁺ [8], which are shown in Fig. 2(b) and described by Eqs. (7)-(9):

N_{4} = N_{H o} - N_{5} - N_{7}

\begin{array}{l} \frac{d N_{5}}{d t} = - A_{54}^{'} N_{5} + A_{75}^{'} N_{7} + \frac{λ_{T m}}{h c A_{e f f}^{T m}} σ_{a}^{H o} (λ_{T m}) [P_{f}^{T m} (z) + P_{r}^{T m} (z)] N_{4} \\ - 2 k_{57} N_{5}^{2} - \frac{λ_{H o}}{h c A_{e f f}^{H o}} [P_{f}^{H o} (z) + P_{r}^{H o} (z)] [σ_{e}^{H o} (λ_{H o}) N_{5} - σ_{a}^{H o} (λ_{H o}) N_{4}] \end{array}

\frac{d N_{7}}{d t} = - (A_{75}^{'} + A_{74}) N_{7} + k_{57} N_{5}^{2}

Since the third energy level in Ho,

N_{6}

, has a very short lifetime, it is assumed that any population in

N_{6}

immediately decays into the

N_{5}

level so the population in

N_{6}

is zero at any given time.

The bi-directional propagation of the power in the Ho:fiber can be described by:

\frac{d P_{f, r}^{p} (z)}{d z} = \mp P_{f, r}^{p} (z) [σ_{a}^{H o} (λ_{p}) N_{4} Γ_{p}^{H o} + α_{p}]

\begin{array}{l} \frac{d P_{f, r}^{H o} (z)}{d z} = \pm P_{f, r}^{H o} (z) {[σ_{e}^{H o} (λ_{H o}) N_{5} - σ_{a}^{H o} (λ_{H o}) N_{4}] Γ_{H o}^{H o} - α_{H o}} \\ \pm σ_{e}^{H o} (λ_{H o}) N_{5} Γ_{H o}^{H o} h ν_{H o} Δ ν_{H o} \end{array}

Equation (10) describes the pump power, which decays due to absorption by the Ho ions and passive loss. Equation (11) describes the gain in Ho power due to stimulated and spontaneous emission from the first excited state, and loss due to reabsorption of the signal light by Ho and passive loss. All the variables used in Eqs. (7)– (11) are defined in Table 2.

Table 2. Variable definitions

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The third fiber modeled was a coaxial Tm/Ho-doped fiber. In the model developed for this fiber geometry, the thulium laser equations in particular must be modified. Specifically, the photons emitted in thulium are also absorbed by the holmium ions. Therefore, at each point in the fiber, the power in the Tm signal is simultaneously created by the absorption of the pump power, and lost to absorption by the holmium. In the coaxial fiber, the rate equations for Tm (Eqs. (1)-(4)) and Ho (Eqs. (7)-(9)) were used. The power propagation in the coaxial fiber is described by Eqs. (12)-(14):

\frac{d P_{f, r}^{p} (z)}{d z} = \mp P_{f, r}^{p} (z) [σ_{a}^{T m} (λ_{p}) N_{0} Γ_{p}^{T m} + α_{p}]

\begin{array}{l} \frac{d P_{f, r}^{T m} (z)}{d z} = \pm P_{f, r}^{T m} (z) {[σ_{e}^{T m} (λ_{T m}) N_{1} - σ_{a}^{T m} (λ_{T m}) N_{0}] Γ_{T m}^{T m} - σ_{a}^{H o} (λ_{T m}) N_{4} Γ_{T m}^{H o} - α_{T m}} \\ \pm σ_{e}^{T m} (λ_{T m}) N_{1} Γ_{T m}^{T m} h ν_{T m} Δ ν_{T m} \end{array}

\begin{array}{l} \frac{d P_{f, r}^{H o} (z)}{d z} = \pm P_{f, r}^{H o} (z) {[σ_{e}^{H o} (λ_{H o}) Ν_{5} - σ_{a}^{H o} (λ_{H o}) N_{4}] Γ_{H o}^{H o} - α_{H o}} \\ \pm σ_{e}^{H o} (λ_{H o}) N_{5} Γ_{H o}^{H o} h ν_{H o} Δ ν_{H o} \end{array}

Finally, the Tm/Ho co-doped fiber laser was modeled using the same method as the coaxial fiber laser, but in this case the Ho core and Tm ring were assumed to have to same diameter, and overlapped perfectly. This model does not incorporate the energy transfer mechanisms [3] that limit efficiency in Tm/Ho co-doped fibers, but instead is used to predict the best case scenario for the co-doped fiber laser.

For each simulation, the radiation was subject to certain boundary conditions. The pump field boundary conditions are described in Eqs. (15) and (16) and the Tm/Ho radiation field boundary conditions are described in Eqs. (17) and (18):

P_{r}^{p} (L) = R_{2} (λ_{p}) P_{f}^{p} (L)

P_{f}^{p} (0) = R_{1} (λ_{p}) P_{r}^{p} (0) + [1 - R_{1} (λ_{p})] P_{l a u n c h}

P_{r}^{T m / H o} (L) = R_{2} (λ_{T m / H o}) P_{f}^{T m / H o} (L)

P_{f}^{T m / H o} (0) = R_{1} (λ_{T m / H o}) P_{r}^{T m / H o} (0)

where

R_{1}

is the reflectance of the mirror on the input side of the laser,

R_{2}

is the reflectance of the output coupler, and

P_{l a u n c h}

is the launched pump power.

The parameters used in the simulation are listed in Table 3. The input mirror was assumed to reflect 1% of the pump light and 99% of both the Tm and Ho signal wavelengths. The output coupler was assumed to reflect 99% of the pump light and 99% of the Tm signal wavelength for the coaxial and co-doped fibers. The cross sections were measured from samples of Tm and Ho doped silica. In all cases, the Tm ring had a large diameter and was therefore highly multimode. As a result, the intensity distribution of the Tm emission was assumed to be approximately uniform, and the overlap integrals for the Tm ions were calculated using the ratio of areas of the Tm ion distribution to the pump area or the Ho ion distribution. In contrast, the Ho core was designed to be single mode. Therefore, the overlap of the Ho signal with the Ho ion distribution was calculated using the mode field diameter approximation [9]. The power propagation equations were solved using the Runge-Kutta method to propagate back and forth along the fiber until a self-consistent solution was found. The steady state solutions of the rate equations were found iteratively with the power propagation equations until a stable solution was found. Stability was defined as a change of less than 1% in output power between subsequent iterations.

Table 3. Simulation parameters

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The performance of three types of fiber lasers emitting at 2100 nm was compared using the power conversion efficiency, defined by $η = P^{T m / H o} / P_{l a u n c h} .$ For each configuration, output coupler reflectivity, and doping density, the fiber was also optimized for length, which nominally varied between 100 mm and 1 m. The length was optimized by simulating fibers of different lengths until the maximum output power was found. Each fiber laser was pumped with 400 W, representing both available power for laboratory experiments and a sufficiently high power to represent the laser operation well above threshold. Despite the high pump power and short length, there was no evidence of ground state bleaching, which would manifest as reduced slope efficiency at high pump powers.

The pump wavelength was set at 805 nm, rather than the typical 790 nm, to match the test configuration that will be used to study the coaxial fibers subsequent laboratory experiments. In the simulations, each fiber laser was optimized in terms of output coupler reflectivity and dopant density. The ranges of these parameters were set to the limits of a physically realizable system. The output coupler reflectivity was allowed to vary from 4% to 99%. Although using a reflectivity less than the 4% Fresnel reflection may yield higher efficiencies, creating such a system is not feasible for the coaxial fibers. Typically, lower reflectivities can be achieved using fiber Bragg gratings (FBGs) and an angle cleaved fiber end. However, an FBG will not work with the coaxial fiber due to the highly multimode nature of the Tm ring. The maximum dopant density considered for Tm and Ho was 7 wt%, to cover the full range of available concentrations using multicomponent silicate glasses.

It should be noted that since the cross sections are wavelength dependent and the power can be spectrally resolved, the model does indeed include gain competition between the Tm and Ho ions. In the cases modeled, however, no parasitic behavior is possible, as will be discussed in Section 3.

3. Simulation results

Throughout this work, the efficiency is characterized by the power conversion efficiency (PCE), defined as the ratio of the signal output power to the total injected pump power. Note that this differs from common laser characterization via slope efficiency since it includes all losses due to threshold and any unabsorbed pump. Therefore, while it is a more conservative metric than the conventional slope efficiency, it is in fact more practical from the perspective of overall system design.

First, a fiber laser with a Tm-doped core was used to pump a fiber laser with a Ho-doped core. The lasers were simulated separately, and each was optimized by varying the reflectivity of the output coupler at the emission wavelength and the dopant ion concentration. The parameters of each fiber were chosen to match the optimal configuration of the coaxial fiber. The Tm-doped fiber was pumped with 400 W of power at 805 nm into a 200-µm cladding, and emitted at 1950 nm from a core with a 60-µm diameter. The optimization surface for this fiber laser is shown in Fig. 3, with a maximum output power of 230 W. The resultant PCE is 58%, similar to experimentally reported values [14]. The Ho-doped fiber was pumped with 400 W of power at 1950 nm into a 60-µm cladding, and emitted at 2100 nm from a 20-µm core. The optimization surface for this fiber laser is shown in Fig. 4, with a maximum output power of 298 W. The PCE is 75%, which closely matches what has been reported for previous experiments [1]. It should be noted that only cladding pumped Ho:fiber lasers were modeled for this part of the simulation. It is possible to obtain higher efficiencies by core pumping the fiber [15]. For completeness and to verify the accuracy of the simulations, the core pumped case was tested and compared to the experiment described in [15], and the results matched. However, core pumping limits the ultimate achievable pump power (since a large number of diffraction-limited beams that would be needed to pump the core can be geometrically combined to pump the cladding) and therefore output power for these lasers. In the current simulations, where high output powers were desired, designs were thus intentionally limited to cladding-pumped fiber lasers.

Fig. 3 Laser output power in W as a function of output coupler reflectivity and dopant density for a Tm-doped fiber laser.

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Fig. 4 Laser output power in W as a function of output coupler reflectivity and dopant density for a Ho-doped fiber laser.

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The efficiency for 2100 nm emission was calculated for the tandem Tm- and Ho-doped fiber lasers. For both fiber lasers, the best performance occurred when the output coupler reflectivity was minimized (e.g. 4% cleaved facet reflectivity) at the laser wavelength. Therefore, the overall efficiency with these output couplers was calculated for each combination of dopant densities. The resulting optimization surface is shown in Fig. 5. When the optimal configuration for each fiber is applied in tandem, the overall PCE of the system, from 805 nm diode pumping to 2100 nm Ho emission, is a maximum of 43%.

Fig. 5 Laser output power conversion efficiency in % as a function of holmium and thulium dopant densities for the tandem fiber laser.

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The performance of the coaxial fiber, depicted in Fig. 6, was optimized through simulations. The optimal coaxial fiber has a Ho-doped core with a diameter of 20 µm, which is surrounded by a Tm-doped ring with an outer diameter of 60 µm, and a pump cladding diameter of 200 µm. The Ho core size was chosen to be large enough to be easily manufacturable while maintaining a single-mode output. The laser was modeled for several different Tm ring diameters and output coupler reflectivities at the Ho emission wavelength, with 60 µm diameter and 4% reflectivity yielding the highest efficiencies in all cases. It should be noted that changing the diameter by 5-10 micron resulted in only small decreases in efficiency, which will manifest in large tolerances when manufacturing the fiber.

Fig. 6 Laser output power conversion efficiency in % as a function of holmium and thulium dopant densities for the coaxial fiber laser with an output coupler with 4% reflectivity at 2100 nm.

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Figure 6 shows the efficiency of the coaxial fiber, as a function of holmium and thulium dopant densities, for an output coupler with 4% reflectivity at 2100 nm. The maximum achievable output power for this laser was calculated to be 207 W, for a power conversion efficiency of 54%. Note that this is much higher than the 43% obtained for the tandem Tm-Ho laser system, validating the coaxial fiber as a viable architecture for high-efficiency holmium lasers. This is also much higher than the 10% efficiency obtained in the experimental demonstration. The simulation results indicate that, while the experimental fiber design was not optimized, this in itself not the sole cause of the low measured efficiency. Other issues and explanations are explored in Section 4.

Finally, the coaxial fiber laser is compared to a Tm/Ho co-doped fiber laser, whose performance is shown in Fig. 7. The co-doped fiber has a core diameter of 20 µm and a pump cladding diameter of 200 µm. Again, several simulation sets were performed, with varying output coupler reflectivities at the Ho emission wavelength, indicating that 4% was the optimal output coupler reflectivity. Figure 7 shows the output power of this laser, as a function of holmium and thulium dopant densities with the optimal output coupler. The maximum achievable output power for this laser was found to be 135 W, for a power conversion efficiency of 34%.

Fig. 7 Laser output power conversion efficiency in % as a function of holmium and thulium dopant densities for the co-doped fiber laser with an output coupler with 4% reflectivity at 2100 nm.

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One issue that may arise in any laser system is that of spectral gain competition. In the case of the coaxial fiber laser, there are two dopants to consider, much like Tm:Ho and Er:Yb co-doped fiber lasers. Several areas of concern may arise for Tm:Ho systems in general, and are investigated for the coaxial fiber laser specifically.

For example, the 99% reflector required to produce efficient 1950 nm Tm emission may lead to Ho emission at this wavelength. In the cases modeled, however, no such parasitic behavior is possible since the Ho emission cross-section is extremely small at 1950 nm. In the worst-case scenario (4% reflection at 2100 nm), lasing at 1950 nm would require 4x higher inversion than lasing at 2100 nm and thus cannot possibly occur.

Similarly, one may consider parasitic reflections in the 2020-2050 nm band. Assuming 1% reflections, Ho lasing in this region would require 1.5x higher inversion than lasing at 2100 nm with a 4% reflection (worst case, highest inversion for 2100 nm lasing). In contrast, Tm requires a relatively small inversion (~5%) to have gain in this region. However, an output coupler with 99% reflection at 1950 nm and 1% reflection in the 1960-2050 nm region causes the gain to be negligible at 1960-2050 nm relative to the gain at 1950 nm.

In all cases modeled, the coaxial fiber laser systems were shown to be extremely robust against these forms of parasitic lasing due to spectral gain competition.

The performance of the coaxial fiber can also be evaluated in an amplifier configuration. In order to pump the Ho, the Tm signal must still be oscillated within the cavity. The coaxial fiber amplifier was pumped with 2 kW at 805 nm, and a 50 W seed was used at 2100 nm. In order to directly compare against the oscillator results, the fiber used for the amplifier simulations matched the fiber designs used in the laser simulations. The intracavity signal power as a function of normalized axial position in the fiber for each of the three cases is plotted in Fig. 8. The resulting coaxial fiber amplifier was able to output nearly 800 W of signal power. When the tandem and co-doped fiber lasers were similarly redesigned as amplifiers, their maximum output powers were 620 W and 275W, respectively. These results shows that coaxial fiber is not only effective as an amplifier, but it also outperforms the tandem and co-doped configurations.

Fig. 8 Intracavity Ho signal power as in W as a function of axial position along the fiber length for amplifiers made with the coaxial (green) and co-doped (purple) fibers, and a Ho amplifier in the tandem configuration.

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4. Discussion and conclusions

From the power conversion efficiencies calculated in Figs. 5-7, it is clear that the coaxial fiber laser is the most efficient of the three laser types. For a pump wavelength of 805 nm, the power conversion efficiency of the tandem laser ranges from 30% to 43%, while the co-doped laser reaches a maximum of 34%. For both of these cases, the coaxial fiber laser offers a significant advantage, with the power conversion efficiency ranging from 46% to 54% for the same set of conditions. Therefore, for the set of conditions represented in these simulations, even the least efficient coaxial fiber laser is more efficient than the most efficient tandem or co-doped fiber laser.

If the pump wavelength is changed to 790 nm, the maximum efficiencies increase for all three laser types, and the coaxial fiber laser is still the most efficient laser in almost all cases. If the tandem fiber laser utilizes a core-pumped Ho:fiber laser, it slightly outperforms the coaxial fiber laser, with a 58% power conversion efficiency. However, the difference is small, and there are other advantages to the coaxial fiber laser. While all three fiber types show efficiency improvement when the pump wavelength is reduced, there is only a small improvement in efficiency for the coaxial fiber laser. Since there is little variation in coaxial fiber laser efficiency between the two pump wavelengths, the coaxial fiber laser has a much larger effective pump bandwidth than the other two laser types, which is critical for low-cost power scaling. A summary of the simulation results, with pump wavelengths at 790 nm and 805 nm, can be found in Table 4 along with a comparison to experimental results from the literature. Table 4 validates the results of our simulations, as the Tm:fiber and co-doped fibers pumped at 790 nm closely match the referenced experiments.

Table 4. Efficiency type comparison

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Though the coaxial fiber is the most efficient of the three fiber types, it also has the highest threshold, near 20 W. In comparison, the tandem fiber laser has a threshold of 15 W and the co-doped fiber laser has a threshold of 12 W. As a result, the tandem fiber laser may end up with higher power conversion efficiency than the coaxial fiber laser for low power applications, e.g. less than 60 W.

The coaxial fiber laser is most efficient high-power architecture because the Ho core is pumped by the intracavity power of the Tm signal, which in simplest terms means that more power is available for pumping the Ho. This effect is greater than the losses in the Tm signal due to absorption by the Ho, which is a minor effect since the overlap between the Tm and Ho ion distributions is small. The large area of the Tm ring provides enough space for the Tm signal to grow sufficiently large. This is not the case for the co-doped fiber, where the large overlap between the Tm and Ho ion distributions causes an increase in the intracavity losses of the Tm signal. In this co-doped case, the absorption loss overpowers the benefits of intracavity pumping, and the efficiency decreases. This is especially problematic when the ratio of Tm to Ho dopant density is small; if the Ho dopant density is too large in comparison to the Tm dopant density, the losses are too great and the laser simply does not work.

There is a significant difference between the efficiency of 10% achieved in the initial experimental demonstration [4] and the 52% achieved in these simulations. There are several reasons for this. First, the fiber design in the experiment was far from optimal. When the physical fiber geometry and dopant densities are inserted into the simulation, the maximum achievable PCE drops to 43%. In addition, the spectral data taken in [4] indicate that the Tm ions are lasing at wavelengths longer than 1950 nm, in the range 1960-2000 nm. Study of the cross sections reveals that not only does Tm have a small emission cross section at these wavelengths, but Ho also has a small absorption cross section at the longer wavelengths. If the simulation is adapted for the Tm ions to lase at a single wavelength in this range, the overall efficiency of the system drops even further, to 8-20%. These results closely match the published experiment, and emphasizes the importance of choosing a good output coupler for obtaining the maximum efficiency for the system. The ideal output coupler would force the Tm ions to only lase near 1950 nm, while also allowing the Ho ions to lase near 2100 nm. This can be achieved, for example, using a fiber Bragg grating in the Ho core and coating the fiber end face with a coating that is highly reflective at 1950 nm and highly transmissive above 1960 nm. An example output coupler features a narrow bandpass near 1950 nm, which forces the Tm ions to lase at 1950 nm for efficient Ho operation. For the restriction band, between 1960 nm and the desired Ho emission wavelength, 1% reflection is more than sufficient to suppress undesired emission. This was confirmed by adapting the model to allow it to self-select the wavelengths at which the Tm and Ho ions lased. When an output coupler meeting these specifications was used, the Tm ions lased at 1950 nm and the efficiency matched the maximum efficiency shown in Fig. 6.

Generally speaking, Ho laser efficiency can be limited at high dopant densities due to ion clustering [17]. However, these effects can be mitigated using glass network modifiers such as aluminum, sodium, or potassium ions [18]. Moreover, Figs. 4-6 show that there is limited benefit from increasing Ho dopant density, as the increase in efficiency due to increasing the number of active ions is countered by the increase in the upconversion rate. As such, using lower Ho doping density can be used to circumvent clustering with little impact on power conversion efficiency in the case of the coaxial fiber laser. Future manufacturers of this fiber should choose the highest Ho dopant density possible for which they can sufficiently reduce clustering using glass network modifiers. Currently, most Ho doped fibers are manufactured with concentrations of 1 wt% or less to avoid the clustering effects [1,4,16,17], but the potential to obtain higher concentrations exists [18]. Tm doped fibers, conversely, often have higher concentrations ranging from 3 to 6 wt% [7,14]. While the efficiency stays relatively constant with increasing Ho concentration, increasing the Tm concentration can have a much greater effect, as can be seen in Figs. 3, 5, and 6. In the case of Tm, fibers with higher concentration benefit from an increased number of active ions as well as an increased cross relaxation rate.

In conclusion, coaxial Ho/Tm fiber lasers were explored via numerical simulations. A complete simulation environment was developed to model Tm/Ho-doped fiber laser systems. These simulations predict a maximum 52% power conversion efficiency for a coaxial fiber laser, which is more efficient than the tandem fiber laser by up to 20% relative and 10% absolute, and more efficient than the co-doped fiber laser by up to 50% relative and 20% absolute. The coaxial fiber laser has a higher power conversion efficiency than both traditional systems due to the benefits from the small overlap between the Tm signal and Ho core, which allows the Tm to lase efficiently, and because the full intracavity Tm signal power is dedicated to pumping the Ho core. The coaxial fiber is therefore promising as a compact and highly efficient method for producing 2100 nm fiber lasers.

References

1. S. D. Jackson, “Midinfrared Holmium Fiber Lasers,” IEEE J. Quantum Electron. 42(2), 187–191 (2006). [CrossRef]

2. S. D. Jackson, “Towards high-power mid-infrared emission from a fibre laser,” Nat. Photonics 6(7), 423–431 (2012). [CrossRef]

3. S. D. Jackson and T. A. King, “CW Operation of a 1.064-μm Pumped Tm-Ho-Doped Silica Fiber Laser,” IEEE J. Quantum Electron. 34(9), 1578–1587 (1998). [CrossRef]

4. D. G. Lancaster and S. D. Jackson, “In-fiber resonantly pumped Q-switched holmium fiber laser,” Opt. Lett. 34(21), 3412–3414 (2009). [CrossRef] [PubMed]

5. D. Hanna, “Optical fibre with doped core and doped inner cladding, for use in an optical fibre laser,” US patent 5291501 (1994).

6. G. A. Newburgh, “Composite laser gain medium,” US patent 9118164 B1 (2015).

7. S. D. Jackson and T. A. King, “Theoretical Modeling of Tm-Doped Silica Fiber Lasers,” J. Lightwave Technol. 17(5), 948–956 (1999). [CrossRef]

8. N. P. Barnes, B. M. Walsh, and E. D. Filer, “Ho:Ho upconversion: applications to Ho lasers,” J. Opt. Soc. Am. B 20(6), 1212–1219 (2003). [CrossRef]

9. D. Marcuse, “Gaussian Approximation of the fundamental modes of graded-index fibers,” J. Opt. Soc. Am. 68(1), 103–109 (1978). [CrossRef]

10. B. Peng and T. Izumitani, “Optical properties, fluorescence mechanisms and energy transfer in Tm³⁺, Ho³⁺ and Tm³⁺-Ho³⁺ doped near-infrared laser glasses, sensitized by Yb³⁺,” Opt. Mater. 4(6), 797–810 (1995). [CrossRef]

11. R. Reisfeld and M. Eyal, “Possible ways of relaxations for excited states of rare earth ions in amorphous media,” J. Phys. (Paris) 46, 349–375 (1985).

12. J. B. Gruber, M. E. Hills, R. M. Macfarlane, C. A. Morrison, G. A. Turner, G. J. Quarles, G. J. Kintz, and L. Esterowitz, “Spectra and energy levels of Tm³⁺:Y₃Al₅O_12.,” Phys. Rev. B Condens. Matter 40(14), 9464–9478 (1989). [CrossRef] [PubMed]

13. C. Huang, Y. Tang, S. Wang, R. Zhang, J. Zheng, and J. Xu, “Theoretical Modeling of Ho-Doped Fiber Lasers Pumped by Laser-Diodes Around 1.125 µm,” J. Lightwave Technol. 30(20), 3235–3240 (2012). [CrossRef]

14. P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-Doped Fiber Lasers: Fundamentals and Power Scaling,” IEEE J. Sel. Top. Quantum Electron. 15(1), 85–92 (2009). [CrossRef]

15. C. C. Baker, E. J. Friebele, A. A. Burdett, D. L. Rhonehouse, J. Fontana, W. Kim, S. R. Bowman, L. B. Shaw, J. Sanghera, J. Zhang, R. Pattnaik, M. Dubinskii, J. Ballato, C. Kucera, A. Vargas, A. Hemming, N. Simakov, and J. Haub, “Nanoparticle doping for high power fiber lasers at eye-safer wavelengths,” Opt. Express 25(12), 13903–13915 (2017). [CrossRef] [PubMed]

16. S. D. Jackson, A. Sabella, A. Hemming, S. Bennetts, and D. G. Lancaster, “High-power 83 W holmium-doped silica fiber laser operating with high beam quality,” Opt. Lett. 32(3), 241–243 (2007). [CrossRef] [PubMed]

17. A. S. Kurkov, E. M. Sholokhov, A. V. Marakulin, and L. A. Minashina, “Effect of active-ion concentration on holmium fibre laser efficiency,” Quantum Electron. 40(5), 386–388 (2010). [CrossRef]

18. S. Jiang and T. Luo, “Thulium and/or Holmium doped silicate glasses for two micron lasers,” US Patent 8265107 (2012).

Variable	Definition
$N_{T m}$	density of thulium ions
$N_{i}$	population density in the ith energy level
$A_{i j}$	spontaneous transition rates
$Γ_{i}$	nonradiative transition rates
$k_{i j}$	cross relaxation rates
$λ_{p / T m}$	wavelength of the pump/Tm signal
$h$	Planck’s constant
$c$	speed of light
$A_{e f f}^{T m / p}$	effective area of the Tm/pump signal
$P_{f / r}^{p / T m}$	forward (f) and reverse (r) propagating pump/Tm power
$σ_{e / a}^{T m} (λ)$	Tm emission/absorption cross section
$Γ_{p / T m}^{T m}$	spatial overlap of the respective signals with the Tm³⁺ ion distribution
$ν_{T m}$	frequency of the Tm signal
$α_{p / T m}$	passive loss

Variable	Definition
$N_{H o}$	density of thulium ions
$k_{57}$	upconversion rate
$λ_{H o}$	wavelength of the Ho signal
$A_{e f f}^{T m / H o}$	effective area of the Tm/Ho signal
$P_{f / r}^{T m / H o}$	forward (f) and reverse (r) propagating Tm/Ho power
$σ_{e / a}^{H o} (λ)$	Ho emission/absorption cross section
$Γ_{T m / H o}^{H o}$	spatial overlap of the Tm/Ho signals with the Ho³⁺ ion distribution
$ν_{H o}$	frequency of the Ho signal
$α_{H o}$	passive loss

Parameter	Value
$A_{10}^{'}$	182.65 s⁻¹ [10–12]
$A_{20}$	181.23 s⁻¹ [10]
$A_{3}^{'}$	775.61 s⁻¹ [10]
$A_{21}^{'}$	2.57 × 10⁸ s⁻¹ [10–12]
$A_{32}^{'}$	1.19 × 10⁵ s⁻¹ [10–12]
$A_{54}^{'}$	63.41 s⁻¹ [10,13]
$A_{74}$	27.97 s⁻¹ [10]
$A_{75}^{'}$	4.55 × 10⁴ s⁻¹ [10,13]
$k_{31}, 12 * k_{13}$	1.8 × 10⁻²² m³/s [7]
$k_{21}, 2 * k_{10}$	3 × 10⁻²⁴ m³/s [7]
$k_{57}$	11.9 × 10⁻²⁴ m³/s [8]
$σ_{a}^{T m} (λ_{p})$	1.45 × 10⁻²⁵ m²
$σ_{a}^{T m} (λ_{T m})$	6.46 × 10⁻²⁷ m²
$σ_{e}^{T m} (λ_{T m})$	1.16 × 10⁻²⁵ m²
$σ_{a}^{H o} (λ_{T m})$	5.62 × 10⁻²⁵ m²
$σ_{a}^{H o} (λ_{H o})$	6.26 × 10⁻²⁶ m²
$σ_{e}^{H o} (λ_{H o})$	4.79 × 10⁻²⁵ m²

Laser type modeled	Pump wavelength (nm)	Power conversion efficiency	Slope efficiency	Measured slope efficiency
Tm:fiber	790	67%	71%	65% [14]
Tm:fiber	805	58%	62%
Ho:fiber	1950	75%	76%	75% [1]
Ho:fiber, core pumped	1950	87%	88%	87% [15]
Tandem	790	50%	54%
Tandem	805	43%	47%
Coaxial fiber	805	54%	57%
Coaxial fiber	790	56%	59%
Co-doped fiber	805	34%	37%
Co-doped fiber	790	43%	46%	42% [16]

Variable	Definition
$N_{T m}$	density of thulium ions
$N_{i}$	population density in the ith energy level
$A_{i j}$	spontaneous transition rates
$Γ_{i}$	nonradiative transition rates
$k_{i j}$	cross relaxation rates
$λ_{p / T m}$	wavelength of the pump/Tm signal
$h$	Planck’s constant
$c$	speed of light
$A_{e f f}^{T m / p}$	effective area of the Tm/pump signal
$P_{f / r}^{p / T m}$	forward (f) and reverse (r) propagating pump/Tm power
$σ_{e / a}^{T m} (λ)$	Tm emission/absorption cross section
$Γ_{p / T m}^{T m}$	spatial overlap of the respective signals with the Tm³⁺ ion distribution
$ν_{T m}$	frequency of the Tm signal
$α_{p / T m}$	passive loss

Tm/Ho-doped fiber laser systems using coaxial fiber

Abstract

1. Introduction

2. Physical Mode

3. Simulation results

4. Discussion and conclusions

References

Cited By

Figures (8)

Tables (4)

Equations (18)

Optics Express