Pair induced quenching in high concentration Holmium-doped fiber amplifiers

The spectrum required for future optical communication systems is being extended towards the C-, L- and U-bands, resulting in a significant interest in the spectral region around 2 μm wavelength. Since Holmium doped fiber amplifiers (HDFAs) provide amplification in this spectral region, they have become a focus of researchers working on doped fiber amplifiers. A major factor resulting in the performance degradation of HDFAs is the inhomogeneous energy transfer within Ho3+ ion-pairs in high-concentration Holmium-doped fibers (HDFs), an effect generally known as pair-induced quenching (PIQ). In this paper, we study the luminal and temporal dynamics of pulses of different repetition rates at 2.05 μm in high-concentration HDFs considering the effects of ion-pairs. Input pulses having repetition rates of 25 GHz and 500 kHz are generated using wavelength tunable actively mode-locked Holmium-doped fiber laser (AML-HDFL) based on a single ring cavity and bidirectional pumping. The characteristics of the pulses propagating through high-concentration HDF are analyzed based on different metrics such as average power, peak power, pulse energy, full-width at half maximum (FWHM), and time delay without and with ion-pairs for values of fraction of ion-pairs k = 0 and k = 10%, respectively. The results obtained at optimized length of HDF show that ion-pairs significantly degrade the average power, peak power, and energy of the output pulses for both of the repetition rates. For both k = 0 and k = 10%, the FWHM and shape of the output pulses remain same in the presence of the ion-pairs while, time delay of 4 ps and 19 ns is observed in the output pulses at repetition rates of 25 GHz and 500 kHz, respectively. The effects of increasing the pump and signal power on the average power and energy of the output pulses for k = 0 and k = 10% are also discussed for both repetition rates. This analysis provides important guidelines for designers of 2 μm fiber lasers and amplifiers based on high-concentration HDFs.


Introduction
Rapid progress is being made in research and development of the Holmium-and Thulium-doped fiber amplifiers operating in the eye-safe 2 μm optical window due to various important and emerging applications such as LIDAR, optical communications, deep-space links, coherent lightwave systems, and sensing [1,2].Although the emission band of Tm 3+ exists within the 2-2.1 μm range, its peak emission exists around 1.9 μm which reduces gradually due to low emission cross-section of Tm 3+ beyond 2μm [3].Therefore, Ho 3+ is a suitable rare-earth dopant that can be used to develop efficient light sources and amplifiers operating in the eye-safe 2-2.2 μm optical window.The applications of the amplifiers mentioned earlier require both high repetition rate-low energy as well as low repetition rate-high energy optical pulses at the eye-safe 2 μm wavelength.One of the efficient methods for amplifying such optical pulses is by employing HDFAs.
HDFAs have been extensively researched during the past decade.In [2], the authors enhance the HDFA's performance to attain a substantial small-signal peak gain of 56.5 dB and saturated output power of 3 W. Additionally, they realize added advantages of a simple structure and cost-effectiveness.In [4], a pumping arrangement comprising two successive fiber laser cavities constructed with Erbium-doped fiber (EDF) and Thulium-doped fiber (TDF) is employed.In the study conducted by R. E. Tench and colleagues [5], they present a relatively new HDFA design.This design achieves a slope efficiency (SE) of 64% by employing a non-traditional pump wavelength of 1.84 μm, deviating from the conventional in-band pump wavelength of 1.94 μm.The amplifier demonstrates a gain of 54 dB and an output power of 1 W when subjected to an input signal wavelength of 2.05 μm, utilizing a short 3 m length of HDF.Simakov and collaborators introduced a HDFA featuring a small-signal gain and noise figure of around 25 dB and 46 dB, respectively, at an input signal wavelength of 2.04 μm [6].This was accomplished by employing a laser diode pump with a wavelength of 1.15 μm.The amplifier's performance was further assessed using various pump wavelengths sourced from a Thulium-doped fiber laser (TDFL).The HDFA achieved a peak gain of 41 dB and a noise figure ranging from 7 to 10 dB at a signal wavelength of 2.06 μm, utilizing a pump wavelength of 1.95 μm.In a separate investigation conducted by Simakov and team [7], they successfully developed a wideband HDFA that was pumped at 1.95 μm.This amplifier demonstrated notable characteristics, offering a peak gain of 28 dB and a noise figure ranging from 4 to 9.5 dB within the wavelength range of 2.052.13μm.These features make it suitable for applications in optical communication.In another contribution by R. E. Tench and colleagues [8], they introduced a double-stage-pumped HDFA.This design utilized two forward pumps, resulting in a peak gain and output power of 43 dB and 3.5 W, respectively, at a signal wavelength of 2.051μm.Additionally, this configuration achieved a power conversion efficiency (PCE) of 70%.Walasik and co-authors detailed the design and examination of a dual-stage polarization-maintaining HDFA functioning at 2 μm, under both continuous wave (CW) and pulsed regimes [9].The pumping mechanism for this HDFA involved utilizing a TDFL with a central wavelength of 1.86 μm.
Some studies in the literature have focussed on amplification of low-repetition rate pulses at the wavelength of 1.55 μm in high-concentration EDFs considering the ion-pairs.In the study presented in [10], the investigation focuses on pulse propagation within Erbium-doped fiber amplifiers (EDFAs).The experimental investigation focuses on studying the transmission of pulses with different widths through a heavily doped Erbium fiber.The observed shift in the propagation behavior, transitioning from subluminal to superluminal, is ascribed to the varying pulse widths.This transition is elucidated by the sudden alterations in the signal and pump power profiles along the fiber, which arise due to the pronounced absorption properties of the EDF.In another work, [11], a cluster model is proposed, involving m ions per cluster (where m is greater than 2).The study focuses on the theoretical exploration of the impact of ion number per cluster on the performance of a heavily Erbium-doped fiber laser (EDFL).The findings suggest that an increase in m results in a higher threshold pump power, leading to increased instability in the laser system.The work in [12] involves a numerical solution and analysis of homogeneous and inhomogeneous models, as well as a combined model, for concentration quenching in EDFAs with high Erbium concentration.The numerical analysis involves estimating the Erbium ion cluster in Erbium-doped phosphate fiber and suggests optimal doping concentration and fiber length.In the work by Li et al [13], they delve into the development of closed-form rate and power evolution equations specifically tailored for high-concentration EDFAs.The modeling framework is grounded in considerations of both isolated ions and ion clusters.Utilizing these equations, the study investigates the impact of factors such as the fraction of ion clusters and the number of ions per cluster on signal power, gain characteristics, optimal fiber length, and signal loss.
To the best of our knowledge, there hasn't been any prior exploration into modeling and quantitatively analyzing the propagation and amplification of high and low-repetition rate pulses in HDFs considering the presence of ion-pairs.This paper aims to fill this gap by conducting a comprehensive analysis through numerical simulations, specifically investigating the influence of ion-pairs on the spatial and temporal dynamics of 2.05 μm optical pulses in high-concentration HDFs generated using an AML-HDFL at varying repetition rates.The quantification of ion-pairs is expressed by the parameter k, representing the fraction of ion-pairs in total Ho 3+ population.Our analysis of high and low-repetition rate pulses encompasses key characteristics such as average power, peak power, pulse energy, pulse width, and time delay for values of k = 0 and k = 10%.Additionally, we discuss the effects of increasing both pump and signal power on the average power and energy of the output pulses for k = 0 and k = 10%.The commercial software OptiSystem 21 from Optiwave Inc. [14,15] is employed to design the wavelength-tunable AML-HDFL and HDFA to analyze the spatial and temporal dynamics of the pulses.The paper is structured as follows: Section-2 outlines the theory, while section-3 details the simulation setup and working principle.The results are thoroughly discussed in Section-4 and finally, section-5 presents the conclusions of the paper.

Theory
2.1.Energy Transfer Processes in HDF Doping the core of an optical fiber with Ho 3+ ions does not distribute the ions uniformly.Instead, a large fraction of Ho 3+ ions present in close vicinity of each others form pairs or clusters [2,4].These ion-pairs in heavily doped HDF interact with each other and exchange energy in a way that reduces the gain of the amplifier [2,4].The transfer of energy between Ho 3+ ions involves two distinct upconversion processes.These processes are classified as homogeneous upconversion and inhomogeneous upconversion, the latter also referred to as PIQ as discussed by Li et al [16].In homogeneous upconversion, the ions are evenly distributed and transfer of energy takes place between individual ions that are distant apart [16].On the contrary, the ions are unevenly distributed in the case of PIQ and transfer of energy takes place between excited paired ions [16].Consequently, these two processes result into a wastage of pump power and loss of excited-state population, which reduce the gain of the amplifier.The energy transfer rate between Ho 3+ ions depends upon the distance between the ions i.e. inversely proportional to R 6 [17], where R is the distance between Ho 3+ ions doped into the fiber core.The heavy doping of Ho 3+ ions in HDF results in decreasing the distance between ions to such an extent that creates clusters.As the energy transfer rate between Ho 3+ ions strongly depends upon the distance between them, upconversion between ions within a cluster is faster and detrimental compared to homogeneous upconversion occurring between single ions [17].
The most important energy transitions including upconversion processes of Ho 3+ in Silica host involve the energy level scheme of four energy manifolds ( 5 I 8 , 5 I 7 , 5 I 6 , and 5 I 5 ), as shown in figure 1 [4].Ho 3+ ions can be excited through 1.95 μm pump laser realized by TDFL from 5 I 8 manifold to 5 I 7 manifold [4].The excited Ho 3+ ions rapidly thermalize towards the bottom of this manifold where the ions are stored for relatively longer duration due to long lifetime of around 1.3 ms [18].The excited Ho 3+ ions finally decay towards 5 I 8 manifold due to stimulated emission caused by the photons of input signal centered at 2.05 μm [4].

Rate equations for Single and Paired Ions
The rate equations for single and paired Ho 3+ ions doped in Silica host are very important for understanding the pulse propagation and amplification in high-concentration HDF based on energy transfer processes.It is worth mentioning here that the Schrodinger equations can also be used for transmission characteristics and investigating pulse propagation.This process is done on a large-scale level, but on the micro-level dynamics, the rate equations are more suitable.Rate equations can provide a reasonable approximation for the dynamics of amplification in doped fibers over relatively long time scales compared to the pulse duration, particularly when considering average behaviors such as gain and saturation effects [19].However, they may not capture the full complexity of the interaction between the optical pulses and the gain medium on a picosecond timescale.Generally, mode-locked fiber laser dynamics are solved in the active fiber using rate equations.Therefore, the rate equations are derived for both of the cases i.e. single and ion-pairs under the following assumptions (i) Excited state absorption (ESA) is neglected (ii) Non-radiative decay (NRD) is neglected (iii) Photons emitted from spontaneous emission do not create stimulated emission and (iv) Each cluster consists of two Ho 3+ ions.The unpaired Ho 3+ ions can react with each other through homogenous upconversion with same energy transfer rate for all ions, resulting in identical rate equations for all single ions.On the other hand, the ion-pairs can be considered as a single entity having totally different rate equations.
Figure 2(a) depicts the energy states occupied by single Ho 3+ ions.If N t represents the total Ho 3+ population, then kN t gives the total number of ions that have formed pairs and so 2kN t is the total number of ions that have participated in making ion pairs.Therefore, the concentration of single ions in HDF is The rate equations of carriers for levels 5 I 8 , 5 I 7 , and 5 I 6 are given as [20].
where, N 0 , N 1 , and N 2 are population densities of different levels and W ij represents the different transition rates between i th and j th levels.s are absorption and emission cross sections for the signal, λ p and λ s are the pump and signal wavelengths, h is the Planck's constant, c is the speed of light in vacuum, A is the cross sectional area of the fiber core, and z is fiber length, respectively.The total population of single ions is constant, as shown by equation (4): shows the energy states occupied by all possible combinations in which an ion-pair can exist.The combinations are: both ions in 5 I 8 , one excited to 5 I 7 while other in 5 I 8 , both excited to 5 I 7 , one excited to 5 I 6 while other in 5 I 8 , and one excited to 5 I 7 while other excited to 5 I 6 .Therefore, the rate equations of ion-pairs are given as [20].
Here, N 3 , N 4 , N 5 , N 6 , and N 7 are the population densities of different levels, W pair is the PIQ transition, and W ij represents different transition rates between i th and j th levels with superscripts p and s representing the transitions by pump or signal, respectively.The total population of ion-pairs is constant, as shown by equation (10) below.

Simulation setup
The block diagram of the setup used for conducting this study is shown in figure 3. The setup consists of a pulse generator which is a wavelength-tunable AML-HDFL given as input to a HDFA.The lasing cavity consists of a tunable seed laser used for injection seeding.Two pump diodes, each having a wavelength of 1.95 μm and output power of 0.25 W are coupled with the seed laser by using two wavelength division multiplexers (WDMs).
The coupled signal is passed through a short segment of HDF denoted as HDF-1.The length of HDF-1 is 13.6 m and its Ho 3+ ion concentration is 2 × 10 25 m −3 .An optical isolator (ISO) is used for maintaining light propagation in a single direction inside the cavity.A tunable optical bandpass filter (TOF) having a Gaussianshaped profile and a bandwidth of 2 nm is used to select a particular wavelength in the 2 μm band.A Mach-Zehnder modulator (MZM) driven by an electrical Gaussian-shaped pulse generator (PG) is used to perform the active mode-locking.The repetition rate of the train of optical pulses generated can be varied by varying the repetition rate of the electrical pulse generator shown in figure 3.An optical coupler (OC) is used to extract 10% of the lasing power for further amplification inside the HDF-2, that is pumped by diode P3.The remaining 90% power is circulated in the lasing cavity.The parameters of HDF-1 and HDF-2 used in this setup are similar to the commercial HDF (Model # iXblue IXF-HDF-PM-8-125) [21].Different simulation parameters that have been used in this work are mentioned in table 1.
The act of pumping HDF-1 results in the generation of a broad-spectrum amplified spontaneous emission (ASE) signal within the cavity.This ASE signal, when combined with the seed laser, propagates through the cavity and undergoes filtration by the TOF set at 2.05 μm.It is worth mentioning here that the wavelength of the seed laser and the wavelength of the TOF is same.The selected wavelength from wideband ASE gets amplification while passing multiple times through the TOF and HDF-1.An AML-HDFL is realized by exploiting the loss modulation within the MZM that is driven by an electrical Gaussian PG, resulting in a train of mode-locked pulses at 2.05 μm.The train of mode-locked pulses is extracted from the lasing cavity using an OC.To investigate the effect of amplification, the average power of the input pulses is reduced to -20 dBm using a variable optical attenuator (VOA), as shown in figure 3. The time domain and spectral plots of the train of modelocked pulses centered at wavelength of 2.05 μm for repetition rates of 25 GHz and 500 kHz are shown in figure 4.
It may be observed from figures 4(a) and (b) that the FWHM of the input pulses is 8 ps and 256 ns, while the energy is around 0.4 fJ and 20 pJ, for repetition rates of 25 GHz and 500 kHz, respectively.Also, figures 4(c) and (d) shows that the side-mode suppression ratio (SMSR) of the mode-locked pulses is 75 dB and 70 dB for repetition rates of 25 GHz and 500 kHz, respectively.As mentioned earlier, a part of the lasing wavelength is coupled with another pump diode P3 and the coupled signal is connected to a short segment of HDF-2.The pump diode has a wavelength of 1.95 μm and output power of 0.5 W. After the HDF-2, an optical isolator is used for preventing back reflections.Pumping the HDF-2 excites the Ho 3+ ions from 5 I 8 manifold to 5 I 7 manifold.The excited Ho 3+ ions decay to 5 I 8 manifold, releasing the supplementary photons through stimulated emission induced by the photons of the input pulses that are centered at 2.05 μm.Optical time domain visualizer (OTDV) and optical power meter (OPM) are used for monitoring the resulting signal.

Results and discussion
The input optical pulses pass through HDF-2 and get amplified through the process of stimulated emission.Therefore, it is vital to find the optimal length of HDF-2 required to maximize the average power and energy of the output pulses without and with ion-pairs i.e. for k = 0 and k = 10%, respectively.The length of HDF-2 is varied in steps and its effect on average power and energy of the output pulses is observed for k = 0 and k = 10%, at repetition rates of 25 GHz and 500 kHz, as shown in figure 5.The results are obtained considering doping concentration and pump power of 9.5 × 10 25 m −3 and 0.5 W, respectively, while the power of the input pulses is −20 dBm.
It can be observed from the plots that the maximum average power and energy of the output pulses are obtained for HDF-2 length of 8.33 m and 11.7 m for k = 0 and k = 10%, respectively, at repetition rate of 25 GHz and 500 kHz.The difference between the maximum average powers for k = 0 and k = 10% at repetition rate of 25 GHz and 500 kHz is around 0.157 W and 0.154 W, respectively.Similarly, the difference between the maximum pulse energy for k = 0 and k = 10% at repetition rate of 25 GHz and 500 kHz is around 6.3 pJ and 0.308 μJ, respectively.It can be inferred from these results that for a fixed HDF length, when the fraction of ionpairs increase due to high Ho 3+ ion concentration, the average power and pulse energy decreases.The effect of Table 1.Important simulation parameters.

Parameter Value
Power of P3 0.5 W Wavelength of P3 1.95 μm Core radius of HDF-1 and HDF-2 4 μm Doping radius of HDF-1 and HDF-2 2 μm Numerical aperture of HDF-1 and HDF-2 0.3 Absorption coefficient of HDF-1 and HDF-2@1.95 μm 14.9 dB m −1 Absorption cross-section of Ho 3+ @1.95 μm ion-pairs can be reduced if we increase the length of the fiber.This is the reason for obtaining different optimum lengths of HDF-2 for k = 0 and k = 10%.The optimized lengths of HDF-2 are used in the analysis that follows.
Figure 6 shows the effect of pump power on average power and energy for repetition rates of 25 GHz and 500 kHz, considering the optimized length of HDF-2 and for k = 0 and k = 10%.It is clear from figures 6(a) and (b), that the average power of the output pulses increase by increasing the pump power for both values of repetition rates and fraction of ion pairs k.However, the average power is higher under the same conditions for k = 0 compared to k = 10%.The pump threshold required for lasing is lower for k = 0 than for k = 10%.Similarly, it can be seen from figures 6(c) and (d) that the energy of the output pulses increase by increasing the pump power for both values of repetition rates and fraction of ion pairs k.However, the pulse energy and its rate of increase is higher under the same conditions for k = 0 compared to k = 10%, for both of the repetition rates.
Figure 7 shows the of input pulse power on the average power and energy for repetition rates of 25 GHz and 500 kHz, considering the optimized length of HDF-2 and for k = 0 and k = 10%.It may be observed from figures 7(a) and (b) that average power of the output pulses increase by increasing the power of input pulses and tends to saturate after a certain value for both values of repetition rates and fraction of ion pairs k.However, the average power is higher under the same conditions for k = 0 than for k = 10% for both repetition rates.Similarly, it can be seen from figures 7(c) and (d) that the energy of the output pulses increase by increasing the power of input pulses up to a certain value and then tends to saturate for higher values of input pulse powers.Also, it can be seen that the pulse energy is higher under the same conditions for k = 0 compared to k = 10% for both of the repetition rates.
To investigate the effect of ion-pairs on FWHM of the output pulses, we have obtained the time domain plots of the output pulses at both repetition rates and at the optimized length of HDF-2, as shown in figure 8.It can be seen that the pulse widths of the output pulses for both repetition rates remain same without and with ion-pairs.The peak power and so the energy of the output pulses is higher without ion-pairs compared the case when k = 10%.This analysis indicates that ion-pairs affect only the peak power or energy of the output pulses, while no effect is induced on the pulse width.Moreover, the signal propagating through the The HDF used in this context does not experience significant accumulation of nonlinear phase shift or pulse broadening.This is attributed to the relatively short length of the fiber, measuring 13.6 m.Importantly, this length is less than both the nonlinear length and the dispersion length of the fiber [22].The shape of the output pulse after amplification is an important factor that contributes to timing jitter, hence affecting the system performance in high-speed optical communication.We have investigated the effect of ion-pairs on the shape of the output pulses after amplification. Figure 9 shows the waveform evolution in time domain for the output pulses after amplification for both values of k and repetition rates.Time delay of 4 ps and 19 ns is observed between the output pulses without and with ion-pairs for repetition rates of 25 GHz and 500 kHz, respectively.There are multiple reasons for observing this time delay between the optical pulses.These reasons are based on linear and nonlinear effects and hence do not allow for exact quantification of the time delay.The first and relatively obvious reason is the different optimum lengths of HDF-2 which are 8.33 m and 11.7 m for k = 0 and k = 10%, respectively.The difference of 3.37 m in lengths of HDF-2 will result in a delay between output pulses for both repetition rates.The value of the time delay between the output pulses for different propagation lengths of HDF may be computed using the following relation [23]: Where c is the speed of the light in vacuum, τ is the time taken by optical pulse to travel the length L of the optical fiber and n is the index of refraction of the doped medium at the wavelength of interest.It is important to mention here that the optical pulse is not composed of a single wavelength, instead it is composed of a range of wavelengths.Therefore, all the wavelengths travel at a combined velocity, known as the group velocity [23].The second important reason lies in the extent of nonlinearity induced over the optical pulses for the two repetition rates.The pulse energy for a repetition rate of 500 kHz is higher compared to 25 GHz.Therefore, the amount of nonlinearity due to self-phase modulation induced over the pulses having lower repetition rate is higher [24].This nonlinearity based wavelength chirp results in different propagation speeds of the optical pulses while propagating through the optical fiber.The situation is further exacerbated when the chirped pulses pass through the TOF shown in figure 3. A chirped pulse has a range of wavelengths between its leading and trailing edges.When such a pulse is bandpass filtered at a specific wavelength, the output pulse will be delayed in time, depending upon the amount of chirp induced over the input pulse, as discussed in [24].Since it is very complex to determine the exact amount of chirp induced over each pulse, exact quantification of the resulting time delay is almost impossible.Finally, another contribution to the time delay is observed in figure 9 may come from slightly different refractive indices of the fibers due to different doping concentrations.This time delay would be very small since the difference in refractive indices is also small.Moreover, it can be observed that the peak power of the output pulses is higher for k = 0 compared to k = 10% after amplification for both of the repetition rates while the shape of the output pulses remain undistorted.This analysis indicates that ion-pairs affect the peak power of the output pulses and induces time delay while no effect on pulse shape is introduced.

Conclusions
Luminal and temporal characteristics of optical pulses are studied at different repetition rates propagating through high-concentration Holmium-doped fiber considering the effects of ion-pairs.Input pulses at different  repetition rates are generated using a wavelength tunable actively mode-locked Holmium-doped fiber laser based on a single ring cavity and bidirectional pumping.Different characteristics of the pulses are analyzed in this study without and with ion-pairs for repetition rates of 25 GHz and 500 kHz.The length of the Holmiumdoped fiber is optimized for cases without and with ion-pairs to achieve better performance.The results confirm that ion-pairs significantly affect the average power, peak power, and energy of the output pulses while time delay of 4 ps and 19 ns is observed between output pulses for k = 0 and k = 10% for 25 GHz and 500 kHz repetition rates, respectively.The shape and pulse width of the output pulses remain unaffected.The effect of increasing the pump and signal powers on the average power and energy of the output pulses is also discussed for cases without and with ion-pairs for both repetition rates.

Figure 1 .
Figure 1.Energy level scheme of the four most important manifolds of Ho 3+ in Silica.

Figure 4 .
Figure 4. (a), (b) Time domain and (c), (d) spectral plots of mode-locked pulses having repetition rate of 25 GHz and 500 kHz, respectively.The plots are taken at point A in the setup.

Figure 6 .
Figure 6.Pump power versus average power plots (a), (b) and pump power versus pulse energy plots (c), (d) for k = 0 and k = 10% at repetition rates of 25 GHz (a), (c) and 500 kHz (b), (d).The average power of the input pulses is -20 dBm.

Figure 7 .
Figure 7. Input pulse power versus average power plots (a), (b) and input pulse power versus pulse energy plots (c), (d) for k = 0 and k = 10% at repetition rates of 25 GHz (a), (c) and 500 kHz (b), (d).The pump power is 0.5 W.

Figure 8 .
Figure 8.Time domain plots of output pulses at repetition rates of (a) 25 GHz and k = 0 (b) 25 GHz and k = 10% (c) 500 kHz and k = 0 (d) 500 kHz and k = 10% .The power of pump and input pulses is 0.5 W and -20 dBm at optimized length of HDF-2, respectively.

Figure 9 .
Figure 9.Time domain plots of output pulses without and with ion-pairs for repetition rates of (a) 25 GHz (b) 500 kHz .The power of pump and input pulses is 0.5 W and -20 dBm at optimized length of HDF-2, respectively.
The superscripts p and s represent the transitions by pump and signal, respectively, with the exception of W 11 , which represents the homogenous upconversion rate.The rates are defined as