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Abstract
Coaxial fiber lasers, consisting of a Ho-doped core surrounded by a Tm-doped ring, are studied via experiments and numerical simulations. Previous simulations indicated that coaxial fiber lasers have the potential to reach power conversion efficiencies of up to 54%, but experiments have yielded much lower efficiencies. To understand this difference, a wavelength dimension is added to the previous model. The new simulations explain the discrepancies with the experiments via spectral gain competition and lead to a path to optimize experimental efficiencies. Specifically, an output coupler is spectrally designed to optimize the efficiency of coaxial fiber systems, and a path towards realizing the predicted 54% efficiency is presented. The results indicate that the Tm/Ho coaxial fiber laser has the potential to be a compact and efficient means of producing 2100 nm radiation.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Power scaling of holmium fiber lasers is limited by the unavailability of efficient, high-power pumps at 1950nm. Consequently, holmium fiber lasers are therefore typically realized in the form of tandem laser systems [1,2]. However, this methodology complicates the physical system by requiring two separate lasers. An alternative method of producing 2100 nm radiation is through a Tm/Ho co-doped dual-clad fiber laser, where the holmium and thulium are doped into the same core and high-power 800-nm pump light is injected into the cladding. Such Tm/Ho co-doped fiber lasers are typically less efficient than the tandem configuration, as their performance is limited by energy transfer upconversion [3].
An alternative approach is to use coaxial fibers [4–7], where a holmium-doped core is surrounded by a thulium-doped ring, which in turn is surrounded by an undoped pump cladding, as depicted in Fig. 1. The Tm ring is cladding pumped, and the resulting signal is reflected back into the cavity to pump the Ho. In contrast to the co-doped case, the Tm and Ho dopants to not occupy the same physical space, allowing for higher efficiency in both thulium and holmium lasers while simultaneously mitigating energy transfer upconversion between the Tm and Ho ions. While the initial experimental demonstration of such Tm./Ho coaxial doped fiber lasers resulted in only 10% slope efficiency [4], a recent theoretical study of coaxial fiber lasers established the potential for coaxial fiber lasers to reach slope efficiency of 57%, and a system efficiency of 54%, potentially exceeding the efficiencies of the other two methods [7].
The significant difference between the experimental and theoretical results can be explained in part by the suboptimal fiber and output coupler designs. To remain consistent with the high efficiency simulation results in [7], the thulium laser must lase at 1950nm, near the holmium absorption peak. However, the spectral data shown in [4] indicates that the Tm ions are lasing at wavelengths longer than 1950nm, in the range 1960-2000nm. Examination 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 these longer wavelengths.
It is clear that to truly benefit from Tm/Ho coaxial fiber laser systems, each cavity must be optimized in terms of not only selecting the optimal lasing wavelengths, but also in suppression of unwanted wavelengths that detract from the efficiency of either embedded laser cavity. In this paper, experiments and simulations are used together to both (a) explore the physics of these thulium-holmium systems, and (b) define specifications for both cavities in order to enable highly efficiency lasing of the joint system. In Section 2, the physical model and the experimental setup are presented. Both experimental and theoretical results are presented in Section 3, while a discussion of the results and concluding remarks are presented in Section 4.
2. Physical models
2.1 Simulations
The model used to study Tm/Ho-doped silica coaxial fiber lasers includes all of the kinetics of both Ho and Tm species, the spatial geometry of the coaxial and dual clad geometries, and the wavelength dependence of the emission and absorption. In this way, both systems are modeled simultaneously, including both “lasing” emission and amplified spontaneous emission (ASE). Specifically, there is no mathematical distinction between the two; all wavelengths are modeled in an identical fashion with no predisposed assumption of the output emission wavelength. In this way, the operational wavelength of the lasing emission will be naturally selected by the cavity configuration and spectral properties of the rare-earth ions, output couplers, and other reflectors.
To develop the model, rate equations for Tm3+ and Ho3+ were defined. The rate equations for the four lowest energy levels of Tm3+[8] are described by Eqs. (1)-(4):
Fig. 2. Energy level diagrams for (a) thulium and (b) holmium ions, demonstrating the cross relaxation (CR) and upconversion (UC) processes.
The rate equations for the four lowest energy levels of Ho3+ [9] are described by Eqs. (5)-(7):
Next, the power propagation of each signal must be defined for the coaxial fiber. In the model developed for this fiber geometry, the thulium laser equations in particular must be modified from typical fiber laser models. 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. The power propagation in the coaxial fiber is then described by Eqs. (8)-(10):
For each simulation, the optical power is subject to certain boundary conditions. The pump power boundary conditions are described in Eqs. (11)-(12) and the Tm/Ho power boundary conditions are described in Eqs. (13)-(14):
The parameters used in the simulation are listed in Table 1. The input mirror was assumed to reflect 1% of the pump light and 99% at all of the Tm and Ho signal wavelengths. The measured spectrum of the output coupler used in the experiment was input into the model, and was subsequently varied to investigate its impact on the laser performance. The absorption and emission cross sections were measured from samples of Tm- and Ho-doped silicate glass, and are shown in Fig. 3. The absorption cross section was measured using absorption and emission spectroscopy [10]. The emission lifetimes of the samples were also measured, and the emission cross section was derived using reciprocity [11] and the Füchtbauer–Ladenburg method [12]. For the coaxial fiber, 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 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 [13]. 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, defined as a change of less than 1% in output power between subsequent iterations.
Fig. 3. Ground state absorption (GSA) and stimulated emission (SE) cross sections for Tm (a) and Ho (b) ions.
Table 1. Simulation parameters
Certain parameters in the simulation were chosen to best match the experimental configuration of the coaxial fiber laser in laboratory experiments, which will be described in the next subsection. The pump wavelength was set to the experimental value of 805 nm, rather than the 790 nm typically used for pumping Tm:fiber, and the pump power was set to 60 W. The length of the fiber was set to 400 mm, and the dopant densities were set to 4 wt% for Tm and 1 wt% for Ho as per the fibers used in the experiment. The performance of the fibers was compared using the power conversion efficiency (PCE), defined by $\eta = {P^{Ho}}/{P_{launch}}$. 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.
2.2 Experiment
A prototype coaxial fiber was manufactured by AdValue Photonics. The fiber had a 1 wt% Ho doped silicate core with a 23 µm diameter and 0.10 NA, surrounded by a 4 wt% Tm doped silicate ring with an outer diameter of 75 µm and 0.13 NA. A graphical representation of the refractive index profile design of the manufactured fiber is shown in Fig. 4.
Fig. 4. Designed refractive index difference over cladding index as a function of radial position for the coaxial fiber manufactured by AdValue Photonics.
The performance of the prototype fiber was evaluated using the setup shown in Fig. 5. The pump laser was a quasi-CW 805 nm diode laser capable of powers up to 400 W, though only powers up to 60 W were used for the experiment in order to stay below the damage threshold for other optical components. The fiber length was 400 mm and was held straight due to its short length. An aperture was used to strip all pump light that cannot be coupled into the coaxial fiber. The focal lengths of the collimating lens and focusing lens were chosen to maximize the coupling efficiency from the pump laser to the coaxial fiber. After emerging from the output coupler, the laser light was either focused into an optical spectrum analyzer (OSA) to evaluate the spectral output of the laser, or to a power meter to measure the output power and evaluate the efficiency. A Ge filter was placed in front of the power meter to filter out any residual pump light.
Since these fibers had high doping density in silicate glass, intrinsic (scattering) loss can be a concern, especially since the experimentally measured efficiencies were much lower than the original simulations. In order to understand how the passive loss of the fibers impacts the overall efficiency of the laser, the passive losses in the fiber were measured using the setup shown in Fig. 6. The measurement is complex since there are three regions of interest: the pump cladding, the Tm-doped ring, and the Ho-doped core. Therefore, two sources were used in aligning the input fiber to the coaxial fiber. A broadband SLED was used for speckle-free imaging for alignment into the desired region of interest, while a 976 nm laser diode was used for the loss measurement. The wavelength of the laser diode was chosen to be 976 nm because there is minimal absorption by both Ho and Tm ions at that wavelength. The fiber-coupled diode laser was spliced to a photonic crystal fiber (PCF) with a 20 µm/0.05 NA core. The specifications for this fiber were chosen to meet the core specifications of the prototype fiber such that optimal alignment to the core of the coaxial fiber could be achieved without light spilling into the Tm-doped ring. Near the output end of the PCF, the protective coating was stripped off, and the fiber was placed in index-matching gel to strip any remaining cladding light before injection into the coaxial fiber, thereby ensuring precision alignment into the region of interest. The end of the delivery fiber was then butt coupled to the fiber under test, and index-matching gel was placed between the ends of the fibers to minimize the Fresnel reflections. The input end of the fiber under test was placed on a stage so the fiber could be moved with respect to the PCF to illuminate the different regions of interest. The output end of the fiber under test was imaged into a camera using the lens to easily determine through which fiber component the light is propagating and optimize its coupling in that region. A power meter was placed at the output end of the fiber under test to measure the output power, and was easily removed for imaging purposes.
3. Results
First, the wavelength-dimensioned model presented in Section 2 was benchmarked against the results of [7]. When the optimal fiber parameters and output coupler were input to the simulation, the efficiency was 54%, matching the previous results. However, the experimental conditions differed notably from optimum in dopant density, Tm ring and Ho core diameters, and fiber length. In addition, the pump power used in the experiment was significantly less than the pump power used in [7]. With the laser operating much closer to threshold, the power conversion efficiency is expected to decrease. In order to understand the efficiency limit of the experimental setup, the fiber parameters and pump power from the experiment were input to the simulation. The output coupler was set at the ideal (non-physical) configuration, with zero reflectivity at all wavelengths except at 1950 nm and 2100 nm. The output coupler transmission and intracavity power spectrum for the manufactured fiber are presented in Fig. 7. The intracavity power spectrum shows the Tm ions lasing only at 1950 nm and the Ho ions lasing only at 2100 nm.The resulting PCE was 41%, which is close the optimal efficiency in [7] given the differing fiber parameters. Note that under these ideal conditions, only 54% of the thulium cross relaxation is required to produce the 41% efficiency achievable by the fiber used in experiments. To produce the ideal coaxial efficiency of 54% [7], 70% of the thulium cross relaxation is required.
Next, we adjusted the simulation to best replicate the experimental conditions described in Section 2.B. The passive loss in each region of the prototype fiber was measured and the results, which are listed in Table 2, were added to the simulation. In the experiment, there is an air gap between the pump side dichroic mirror and the fiber. While the output coupler is butt coupled to the fiber, there is potential for an air gap to exist in error. To account for both these errors, Fresnel reflections off each end of the fiber as well as both mirrors were included in the simulation. When these factors were considered, the efficiency of the laser dropped to 31%. Finally, the transmission of the output coupler used in the experiment was measured and input to the simulation. The resulting PCE, from pump to Ho emission, was 2.4%. The results of this simulation are shown in Fig. 8. Here, the Tm ions are lasing near 1920 nm and 1980 nm, but there is no Tm lasing signal at 1950 nm. The Ho ions are lasing near 2100 nm as desired, but with low efficiency.
Table 2. Measured passive loss
The results in Fig. 8 can be compared to the experimental results for the same conditions, which are presented in Fig. 9. The measured efficiency of this laser was 1.6%. For the experiment, the signal powers generated from the Tm and Ho ions could not be separated, so the spectral and efficiency data represent the combined signal. However, there is a clear spectral separation between the Tm signal and the Ho signal. The Tm signal ranges from 1990-2050 nm, while the Ho ions are lasing at 2060-2110 nm and above.
The experiment and simulation were repeated for several other output couplers, with little or no improvement in efficiency. When the passive loss of the fiber and loss due to Fresnel reflections was considered, all configurations had much lower efficiency than the configuration shown in Fig. 7. The key to understanding this difference in efficiency can be found by examining the intracavity power. In both the simulated and experimental intracavity power spectra, it can be seen that the Tm ions are not lasing at 1950nm, but closer to 1980nm where absorption by Ho is at a local minimum. The Tm ions avoid lasing at 1950 nm because of the loss due to absorption by the Ho ions, which results in a higher threshold. Lasers will always try to lase in the lowest threshold state, which in this case means the Tm ions will lase at a longer wavelength despite the smaller emission cross section. However, this negatively impacts the Ho laser, as the Ho ions absorb less efficiently at wavelengths longer than 1950 nm.
To improve the efficiency of the system, it is important to choose an output coupler that will force the Tm ions to lase at 1950 nm while prohibiting lasing at any other wavelength. The laser model was next applied to design a manufacturable output coupler that will enable similar efficiencies to the ideal output coupler shown in Fig. 7. The ideal output coupler is nonphysical, with reflection at only two wavelengths over a broad spectrum. The new output coupler design will define tolerances on the width and depth of the reflection peaks near these two wavelengths. Moreover, a manufacturable output coupler must at minimum have two characteristics: non-ideal efficiencies (i.e. no absolute 0% or 100%), and spectral transition regions with reasonable (nm-scale) width. The resulting output coupler, along with the simulation results, are shown in Fig. 10. Under ideal conditions, this laser was able to reach efficiencies of 40%, only a small decrease from the 41% achieved with the ideal output coupler. The intracavity power spectrum shows that the Tm ions lase at the long wavelength edge of the bandpass, at 1950 nm. The Ho ions lase at the short wavelength edge, at 2100 nm.
The optimal output coupler, shown in Fig. 10, is designed to be manufacturable while still allowing the system to achieve near optimal efficiency. This is just one example of an output coupler that will work, but it incorporates several features that are essential for the coaxial fiber laser to lase efficiently. First, the short wavelength edge of the bandpass must not extend below 1940 nm. If this occurs, the Tm ions will emit at the short wavelength end of the band, where the Ho ions cannot absorb as efficiently. Similarly, the long wavelength edge must not extend above 1950 nm, or the Tm ions will emit at the long wavelength edge of the band. At these wavelengths, there is less efficient emission by the Tm as well as less efficient absorption by the Ho. For the Ho ions, the edge must be at 2100 nm or the Ho will emit at shorter wavelengths, closer to the peak of the emission cross section. The suppression of signal between 1950-2100 nm is critical for efficient operation of the coaxial fiber laser. If only 50% of the signal in this region is allowed to get through, the Tm ions will avoid lasing around 1950 nm.
Suppression in this region will also prevent gain competition between Tm and Ho ions in the 2020-2050 nm range. The gain analysis on the thulium population with a 1% reflector is shown in Fig. 11(a). With a 1% reflector, the Tm population only requires 5% inversion to have gain in the 1980–2040 nm region, though the gain is relatively small, especially when compared to the gain in the 1940-1960 region. If a reflector with 99% reflectivity at 1940-1960 nm and 1% reflectivity above 1960 nm is applied, the resulting gain can be seen in Fig. 11(b). With this reflector, the gain in the 1960-2050 nm region is still slightly positive for a 5% and 6% inversion. However, the gain in the 1940-1960 nm region is two orders of magnitude larger than the parasitic gain in the 1980-2040 nm region and will dictate the Tm lasing wavelength.
Fig. 11. Tm gain in the (a) 1980-2050nm region with a 1% reflector and (b)1940-2050nm region, with 99% reflectivity from 1940-1960nm and 1% reflectivity above 1960nm.
4. Discussion and conclusions
Comparing Figs. 8 and 9, it is obvious that there are still some differences between the simulation and experiment. The simulated efficiency was higher than the measured efficiency, and there were differences in the output spectra of the lasers. This is due in large part to the modal properties of the simulation. The simulation assumes perfect mode mixing in the highly multimode Tm ring and single mode output. In the real fiber used for the experiment, perfect mode mixing does not occur, due to the fiber’s circular geometry and the fact that the fiber is held straight. In addition, the model does not consider scattering off the fiber end face if the cleave is less than perfect, which can be the case for non-silica fibers. Nevertheless, there are several similarities. Both the experiments and simulations using experimental parameters resulted in efficiencies less than 3%, much lower than the ideal 40% efficiency. In both cases, the Tm ions avoided lasing at 1950 nm, instead lasing near 1980 nm. In addition, the Ho ions lased at wavelengths longer than 2100 nm in both the experiment and the simulations. Most importantly, the matching of experiments and simulations led to the understanding of the underlying physics that caused the low efficiency and pointed to the path for designing high-efficiency coaxial Tm/Ho fiber lasers.
The spectral output shown in Fig. 9 displays temporal instability, as did the first experimental demonstration of coaxial fiber lasers [4]. It is common for fiber lasers heavily doped with ions such as Tm, Er, and Yb to self-pulsate. It is also well known that by ensuring proper cavity design, self-pulsations can be eliminated. This and prior [4] experimental demonstrations had unexpected low efficiency, resulting in unoptimized cavity design. Much work has been done, both theoretically and experimentally, to understand the mechanisms behind self-pulsations and to find ways to eliminate it [18,19]. Some methods include adding passive fiber to the laser cavity [20] and using bi-directional pumping [21]. The same bi-directional pumping technique has been used to create a stable CW Tm/Ho co-doped fiber laser [22], which has similar dynamics to the Tm/Ho coaxial system. Therefore, it is expected that the output of the properly engineered coaxial fiber laser can operate CW without self-pulsations.
In many cases, the spectrum of fiber lasers is stabilized using fiber Bragg gratings (FBGs). In the case of the coaxial fiber laser, the output coupler design can be simplified by incorporating an FBG into the Ho core. The highly multimode nature of the Tm ring would make it challenging to use an FBG to stabilize the spectrum of the Tm emission. However, FBGs can be manufactured for multimode operation provided the allowed optical bandwidth of the Tm emission is sufficiently large.
In conclusion, a wavelength dimensioned model was developed to aid in understanding the difference between experiments and simulations of Tm/Ho-doped coaxial fiber lasers and to optimize the performance of coaxial fiber lasers for future experiments. Many factors contributed to the low efficiency found in experiments, primarily the output coupler spectrum, but also including fiber geometry, passive fiber loss, and Fresnel losses in the system. A realistic output coupler was designed to optimize the output of the coaxial fiber lasers. Using a low-loss fiber with the optimal geometry and the optimized output coupler, the full potential of coaxial fiber lasers can be realized.
Disclosures
The authors declare no conflicts of interest.
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