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Abstract
The design of a stable and reliable passively mode-locked laser cavity with Thulium doped fiber is a challenge especially due to its large saturation power owing to a shorter metastable lifetime. In this work, we investigate the influence of various cavity parameters on the stable mode-locked operation of a semiconductor saturable absorber mirror (SESAM) based passively mode-locked Thulium doped fiber laser through numerical modelling. Specifically, we present a detailed study on the effect of the repetition rate, modulation depth, saturation fluence, and damage threshold of the saturable absorber, quality factor of the laser cavity, bandwidth of the passive reflector, and gain medium characteristics on stable operation of a passively mode-locked fiber laser. Experimental results are also presented to validate the key aspects of the simulation.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Thulium doped fiber lasers (TDFL) with emission around 2000 nm wavelength have received considerable attention in the recent past owing to their applications in multiple domains such as medicine, defence and spectroscopy [1]. The inherent broad gain bandwidth ($>300$ nm) makes Thulium doped fiber (TDF) an attractive laser gain medium for generating ultra short pulses. Passively mode-locked TDFLs with sub-100 fs pulse duration and bandwidth $>$100 nm have been demonstrated in the past using various kinds of saturable absorbers (SA) such as semiconductor saturable absorber mirrors (SESAM) [2], carbon nanotubes [3,4], graphene oxide paper [5] and copper thin films [6]. Tm$^{3+}$ doped in silica has a richer spectroscopy with possibilities of cross relaxation and energy transfer upconversion [7]. The metastable lifetime is also relatively smaller compared to Er$^{3+}$ and Yb$^{3+}$ in silica, thus making the realization of a stable Thulium doped passive mode-locking more challenging.
There are various regimes of operation of a passively mode-locked laser, which include Q-switched mode-locking, continuous wave (CW) mode-locking, multipulsing and pulsating operation. In Q-switched mode-locking, the mode-locked pulses appear within a Q-switched pulse envelope and is typically observed for pump power levels closer to the laser threshold [8]. In contrast, a relatively stable continuously mode-locked operation can be achieved at higher pump powers. Depending on the cavity dynamics, when the pump power exceeds a certain value, a CW mode-locked cavity supports more than one pulse per cavity roundtrip time in accordance with the area theorem [9], and results in an undesirable regime of operation commonly known as multipulsing. Cheng et al. have reported a pulsating operating regime, which involves oscillation of pulse energy rather than multipulsing at higher pump power levels [10]. Apart from pump power, several other cavity parameters including cavity roundtrip time, modulation depth, saturation fluence and damage threshold of saturable absorber, Quality factor (Q factor) of the cavity and the time dynamics of the gain medium itself are important in deciding the bounds of the gain values corresponding to stable single pulse operation of a passively mode-locked laser. In order to attain the desired single pulse CW mode-locked operation, it is critical to understand the influence of these parameters on the operating regime of the laser.
A closed form analysis for mode-locking with fast and slow saturable absorbers has been reported previously [11,12]. However, the analytical formulation is valid specifically for homogeneously broadened bulk laser media, with approximations valid for near threshold operation. A closed form analysis is challenging in the case of an inhomogeneously broadened fiber laser, especially when it is operating far from threshold. Thus a numerical study is inevitable to track the exact evolution of pulse in the laser, and to study the independent contribution of each cavity parameter on stable single pulse operation of the system. Pulse evolution in passively mode-locked TDFLs has been demonstrated in the past [13–15]. However, several of the previous numerical studies are focused on pulse shaping in a specific cavity configuration and are aimed at supporting the experimental results focused on achieving single pulse CW mode-locking. In a previous work, multipulsing in a TDFL and its dependence on pump power, cavity length and output coupling ratio has been investigated [16]. Unfortunately, this study does not consider continuous tuning of the repetition rate, the dependence on several key cavity parameters such as saturation power of gain medium, properties of saturable absorber and the range of gain values corresponding to stable mode-locking.
Saturable absorption is another key element in the operation of a passive mode-locked laser, and it is essential that the numerical model also considers the impact of the saturable absorber characteristics. The effect of modulation depth of the saturable absorber in pulse shaping in passively mode-locked TDFL is reported in [17], however the nonlinearity and group velocity dispersion of the optical fiber is not taken into account in this study. The damage threshold of saturable absorber is another very important factor that has to be considered while modelling a practical system. Cheng et al. have theoretically and experimentally investigated the region of CW mode-locking in TDFL with GHz repetition rate, in which Tm$^{3+}$ doped barium gallo-germanate glass fiber is considered as the gain medium [10]. The region of CW mode-locked operation is identified to be between Q-switched mode-locking (lower bound) and pulsating operation (upper bound) and it is predicted that the region of CW mode-locking almost vanishes below 0.5 GHz.
Passively mode-locked TDFLs exhibiting CW mode-locking at repetition rate $<$ 0.5 GHz have been demonstrated experimentally using Thulium doped silica fiber as the gain medium [3,4,18]. There have been a lot of successful demonstration of passively mode-locked fiber lasers using gain media consisting of various kinds of rare earth dopants such as Erbium, Ytterbium etc. These gain media differ in their intrinsic parameters such as metastable lifetime, saturation energy, dispersion, which could affect the stable region of mode-locking. Such a comparative study of different gain media is also not reported in the past. The key point from the above discussion is as follows: even though the influence of several independent parameters on the stability of mode-locking have been reported previously, a single comprehensive study on the influence of the cavity parameters on the stable mode-locked operation of a passive mode-locked laser is yet to be reported. Such a study will help to choose the cavity parameters appropriately and design the cavity in order to attain the desired single pulse CW mode-locked operation in a reliable manner.
In this paper, we numerically investigate CW mode-locked single pulse operation of Thulium doped mode-locked fiber laser with a saturable absorber mirror (SAM) in the cavity as the mode-locking element. Different cavity configurations are considered to study the impact of various parameters such as cavity roundtrip time, characteristics of SAM (modulation depth, saturation fluence, damage threshold), Q of the cavity and bandwidth of passive reflector on stable single pulse operation of a passively mode-locked TDFL. We present a generic simulation model that includes the nonlinearity and group velocity dispersion (GVD) of the fiber. We compare the results with that in case of a passively mode-locked Erbium doped fiber laser to study the impact of metastable lifetime and saturation power of the gain medium on CW mode-locked single pulse operation. The paper is organised as follows. Section 2 describes the details of the simulation model, the corresponding experimental setup, along with the parameters of the SAM under study. Section 3 discusses the influence of repetition rate, SAM parameters, cavity parameters and the gain medium on the stability of mode-locked laser. This is followed by the conclusion from the study. The details of experimental results to corroborate the simulation results are also discussed wherever applicable.
2. Theoretical modelling and simulations of the Tm-doped fiber laser
We study the dynamics of passive mode-locking in a TDFL in a Fabry-Perot cavity through numerical modelling and simulations. The schematic of the laser cavity is shown in Fig. 1, which is chosen to be the same as the one used in the experiments. The gain medium of the laser is an 80 cm long Thulium doped double clad silica fiber (absorption coefficient 3 dB/m at 793 nm). The pump power from a multi-mode pump diode at 790 nm is coupled into the gain fiber using a multi-mode pump combiner. Among the various permissible pumping wavelengths - 790 nm, 1210 nm and 1630 nm, pumping at 790 nm is chosen because of the "two for one" cross relaxation mechanism associated with it, despite the large quantum defect [7]. The Fabry-Pérot cavity is constructed using a SAM on one end and a passive reflector on the other. Two types of passive reflectors are considered: (i) fiber Bragg grating (FBG) with a centre reflection wavelength of 2000 nm and bandwidth 0.1 nm (reflectivity : 50 %, 95 %), and (ii) gold-coated mirror (GCM), which is a broadband reflector with an average reflectivity of 62 % over the amplified spontaneous emission (ASE) band of the gain medium. In the former case, the laser output is obtained from the transmission port of the FBG, while in the latter case, the output is tapped from the 10% port of a 90/10 splitter, as illustrated in Fig. 1. A mode-field adapter (MFA) is used for ensuring the mode-matching between the double clad doped fiber and the single clad fiber of the passive reflector as well as to dissipate any residual pump power.
Fig. 1. Schematic of the passively mode-locked Thulium doped fiber laser with SAM and (i) FBG (ii) GCM
Numerical studies are carried out with various combinations of commercially available SAMs and passive reflectors to study the effect of different parameters on output characteristics of a passively mode-locked laser. FBGs with different values of reflectivity are considered in the cavity to study the effect of Q of the cavity. Replacing the FBG with a GCM helps to study the effect of bandwidth of passive reflector. The critical parameters of the SAM, such as modulation depth, saturation fluence and damage threshold have an important role in deciding the output characteristics of a passively mode-locked laser. The relevant parameters of some commercially available SAMs at 2 $\mu$m listed in Table 1 are considered in the simulation. The relaxation time of each of these SAMs is 10 ps.
Table 1. Characteristic parameters of commercially available SAMs at 2 $\mu$m, considered for this work
The temporal and spectral evolution of the pulse with each roundtrip in the cavity is tracked by solving the generalized nonlinear Schrodinger equation [11] that describes pulse propagation through fiber given as,
As an initial condition, the electric field corresponding to the intracavity optical radiation is initialised with white noise defined over the simulation time window, and is propagated through the various components over a roundtrip [20]. The time window is chosen to be equal to the cavity roundtrip time. The evolution of electric field in the fibers - both active and passive, is simulated by solving Eq. (1) using the split-step Fourier method. Please note that all the numerical simulations reported here are performed using our own codes and executed using Matlab. The value of $g_c$ is updated for every roundtrip through Eq. (2) and used for solving the propagation through the gain fiber. The passive components are modelled to provide constant loss values, and the reflection of FBG is modelled as a wavelength dependent loss. The intensity-dependent absorption ($q(t)$) of the saturable absorber is derived by solving the following equation,
where $q_0$ is the modulation depth of the SAM, $P_{sig}(t)$ is the instantaneous value of signal power, $E_{sat}$ is the saturation energy and $\tau$ is the relaxation time of the saturable absorber. It has to be noted that the total loss due to saturable absorber comprises of the intensity-dependent absorption and the non-saturable loss. Under fast saturable absorber conditions, where the relaxation time of absorber is small compared to the pulse width, the saturable absorber response is expected to be instantaneous and hence Eq. (3) reduces to, where $P_{sat}(=\frac {E_{sat}}{\tau })$ is the saturation power of the SAM.The electric field envelope is evaluated over several subsequent roundtrips in an iterative manner until a steady state in terms of pulse power and width is obtained. Table 2 lists the parameters of the gain medium and passive fiber considered in the simulation. An insertion loss of 0.6 dB is additionally included to consider the loss of the pump combiner and the MFA.
Table 2. Characteristic parameters of gain medium and passive fiber used in simulation
3. Simulation results
Pulse evolution in a mode-locked fiber laser has a strong dependence on the pump power and in turn on the initial small signal gain, $g_0$. A threshold value of $g_0$, corresponding to a specific pump power, is required for the initialization of mode-locking and the consequent pulse formation in the cavity. As the small signal gain increases, mode-locking becomes stronger, resulting in pulses with shorter width and higher energy. In a mode-locked system, the pulse width and the peak amplitude are related through the area theorem [9] which states that the product of peak amplitude and the pulse duration is a constant determined by the group velocity dispersion and the Kerr nonlinearity of the gain medium. Thus, when the pulse width is limited due to the available bandwidth, the peak power is also clamped. Eventually, beyond a certain value of $g_0$, the residual cavity gain increases, leading to the multipulsing regime.
Fig. 2. Intracavity pulse evolution, starting from noise, at g0 values (a) below threshold where a stable pulse is not formed (b) in the stable region resulting in a single stable pulse in one roundtrip (c) beyond stable region resulting in multipulsing. Colour indicates pulse power in Watts.
Figure 2 shows the pulse evolution in a cavity at a repetition rate of 100 MHz for different values of $g_0$. In case of $g_0$ below the threshold, the noise does not build up as a pulse, even after multiple roundtrips as shown in Fig. 2(a) whereas for $g_0$ value above threshold the noise input eventually builds up as a single pulse after a few roundtrips, as evident from Fig. 2(b). In the case of larger $g_0$, multiple pulses are seen to converge to a stable state after a few roundtrips, as seen in Fig. 2(c). We define the range of $g_0$ values over which single pulse operation is obtained as the stable region of CW mode-locking.
The stable region of mode-locking is further dependent on various cavity parameters such as repetition rate, characteristic parameters of SAM, bandwidth of passive reflector, Q of the cavity and characteristic parameters of the gain medium. As mentioned in Sec. 2, the cavity is realized in multiple configurations depending on the choice of the SAM and passive reflector. For each of the cavity configurations, we determine the minimum (lower bound) and maximum (upper bound) values of $g_0$ for mode-locking with single pulse output in order to define the stable region of mode-locking. Simulations are carried out for different cavity configurations to study the effect of each of these cavity parameters on the stable region of mode-locking and is presented in the following sections. The relevant experimental results are also presented along with the simulation results in these sections wherever applicable.
3.1 Effect of the repetition rate
In a passively mode-locked laser, the repetition rate is decided by the roundtrip length of the cavity. The repetition rate is varied in the simulation by changing the length of the passive fiber in the cavity to study its effect on stable region of mode-locking. The effect of dispersion of the passive fiber is also considered in the simulations. Figure 3(a) shows the stable region of mode-locking at various repetition rates for a cavity consisting of SAM 2 and FBG with reflectivity of 50 %.
Fig. 3. (a) Region of mode-locking at different repetition rates from a cavity with SAM 2 and FBG of reflectivity of 50 %. The stable region of mode-locking is shown in the shaded region. The upper limit of gain, limited by the available pump power and that by the length of the gain medium are also shown as dashed lines (b) Pulse width and peak power at the bounds of the stable region of mode-locking
The lower bound (shown in blue dots in Fig. 3(a)) corresponds to the value of small signal gain ($g_0$) below which no pulse is formed, and the upper bound (shown in blue right triangle) corresponds to the value of $g_0$ above which multipulsing occurs (similar to that shown in Fig. 2(c)). It is evident from the simulation results that the stable region of mode-locking expands over a wider range of $g_0$ with increase in the repetition rate. The long time interval between pulses (roundtrip time) in the case of smaller repetition rate allows for the growth of noise within the roundtrip time, which is more profound at higher value of $g_0$, and results in narrower region of mode-locking at lower repetition rates. The simulation is repeated multiple times, with different realisation of noise input in each trial. The noise average power is maintained at 1 $\mu$W for generating these different realisations. The standard deviation in the values of upper bound of $g_0$ from these multiple trials over the range of repetition rates shown is found to be less than 0.4 Np/m; thus proving that the largest error in the estimate of upper bound of $g_0$ through simulations is $<$ 0.4 Np/m.
Figure 3(b) shows the pulse width and peak power at the lower and upper bound of $g_0$ for different repetition rates. For a given repetition rate, the pulse width decreases and the peak power increases with increase in $g_0$ value. Certain other parameters such as available pump power and available gain from the given length of gain fiber further limit the region of mode-locking in a practical system. The green dashed line in Fig. 3(a) represents the maximum gain (marked as gain bound) that can be extracted from an 80 cm length of TDF (for an arbitrarily high pump power), whereas the purple dashed line (referred to as pump bound) represents the $g_0$ value corresponding to the maximum available output power (4 W) from the pump diode employed in the experiments. The black star corresponds to the result obtained from an experiment with repetition rate (13 MHz) and $g_0$ (estimated numerically corresponding to the operating pump power) for the relevant configuration. From Fig. 3(a) it is evident that the experimental operating point lies within the stable region of mode-locking established through the simulation. It should be noted that the value of $g_0$ bounded by the available pump and the possible gain is specific to the implementation of the fiber laser. However, this analysis serves as a design guideline to determine the stable region of mode-locking. The nature of impact of repetition rate on stable region of mode-locking is the same irrespective of the characteristic parameters of the SAM chosen for the cavity. However, the exact bounds of $g_0$ for stable CW mode-locking at a particular repetition rate will be affected by the characteristic parameters of the SAM in the cavity. Thus the influence of SAM parameters should also be considered while designing a practical system.
3.2 Effect of SAM parameters
The critical parameters of a SAM that need to be considered while designing a mode-locked laser are its modulation depth, saturation fluence, damage threshold and relaxation time. In order to analyse their impact on the stable mode-locked operation, these parameters are varied according to the values listed in Table 1. The reflectivity of FBG is fixed to 50 % for these simulations.
3.2.1 Influence of the modulation depth and saturation fluence of SAM
Figure 4(a) compares the stable region of mode-locking of cavities with SAM 1 and SAM 2. On account of its higher modulation depth and saturation fluence, the stable region of mode-locking is marginally wider when SAM 1 is deployed in the cavity. The discrete points (in black) shows the corresponding experimental operating points with SAM 1 and SAM 2 in the cavity, which are at lower repetition rate owing to longer length of the experimental cavity. Figures 4(b) and (c) respectively show the region of mode-locking at the lower repetition rates with SAM 1 and SAM 2. It is evident that the experimental points lie within the region of stable mode-locking established through the simulations. Fig. 5 shows the experimental result of output pulse in time domain for cavity configurations with SAM 1 and SAM 2 at a repetition rate of 13 MHz. The measured pulse width from both the configurations is $\sim$ 500 ps at a pump power $\sim$ 800 mW. The corresponding simulation results from cavity configurations with SAM 1 and SAM 2 predict shorter pulse widths of 170 ps and 218 ps respectively. Considering the effective experimental measurement bandwidth (1 GHz) in the simulation with an appropriate low pass filter, the simulation predicts pulse widths of 417 ps and 446 ps respectively, which roughly agree with those measured through experiments.
Fig. 4. (a) Comparison of stable region of mode-locking of cavities with SAM 1 and SAM 2 for repetition rates from 10 MHz - 100 MHz. The stable region of mode-locking (marked as shaded region) at lower repetition rates with (b) SAM 1 (c) SAM 2 in the cavity. Experimental operating points are shown in black discrete points
Fig. 5. Pulse shape observed from a cavities with SAM 1, SAM 2 and with FBG of reflectivity value of 50 %. Effective measurement bandwidth is 1 GHz
In order to understand the effect of different SAM parameters on output characteristic of mode-locked laser more clearly, the above simulations are repeated at a repetition rate of 100 MHz. Table 3 shows the simulation results in which different cavity configurations are realised by employing the various SAM parameters listed in Table 1. As it is evident from Table 1, the non-saturable loss of the SAM increases with increase in modulation depth. The increase in non-saturable loss increases the background loss in the cavity, which results in an increase in the lower bound of stable region of mode-locking with increase in modulation depth of SAM. The stable region of mode-locking is wider for SAM 1 on account of higher modulation depth. The improvement in stable region of mode-locking is in accordance with the increase in modulation depth.
Table 3. Simulation results with different SAMs, at a repetition rate of 100 MHz and FBG with reflectivity of 50% on the other end of the cavity.
The saturation fluence/saturation energy of SAM determines the pulse fluence/energy that is required to increase its reflectivity by $1/e$ ($\sim$ 37 %) of the initial value. However, for pulse energy very large compared to the saturation energy, the SAM reflectivity is strongly saturated and this could lead to multipulsing [21]. The region of stable mode-locking in case of SAM 3 and SAM 5 have significantly decreased owing to their lower value of saturation fluence. If we closely examine the results given in Table 3, it is evident that shorter pulse width and higher peak power are possible with SAM having higher modulation depth.
3.2.2 Influence of the damage threshold of SAM
Apart from the constraint introduced by the gain bound and pump bound, the intracavity power is further limited by the damage threshold of the SAM, which imposes a limitation on the practically achievable value of $g_0$. Figures 6(a) and (b) show the stable region of mode-locking for cavities with SAM 1 and SAM 2 respectively, where the upper bound corresponds to the value of $g_0$ for which the intracavity pulse energy is equal to energy corresponding to the damage threshold of the SAM in the cavity. It must be noted here that the upper bound imposed by the damage threshold of the SAM is much lower than the bounds of single pulse operation evaluated in the previous section. Hence, these results (shown in Fig. 6) indicate that the primary constraint on stable region of mode-locking at a particular repetition rate in a practical system is imposed by the damage threshold of SAM used rather than the pump or gain limitations, which in turn limits the achievable pulse width and pulse energy from a mode-locked laser as shown in the last two rows of Table 3. It is also noticeable from Fig. 6 that the intracavity pulse energy at the lower bound becomes close to or above the SAM damage threshold as the repetition rate decreases, thus limiting the operation to higher repetition rates for long term and stable performance. It has to be noted that, for all these simulations, the reflectivity of the FBG on one end of the cavity is fixed to 50 %. Replacing the FBG with another having a different value for reflectivity will change the Q factor of the cavity, the effect of which is discussed in the next section.
3.3 Effect of the Q of the cavity
The Q factor, which is a measure of energy stored in the cavity is expected to have a profound effect on the mode-locking characteristics. Higher Q values can be attained by reducing the losses per roundtrip. For this study, while the SAM in the cavity is fixed to SAM 2, the FBG on the other end of the cavity is replaced with an FBG with reflectivity of 95 %, thus realising a cavity with higher Q. Figure 7(a) compares the regions of mode-locking of cavities with low Q (with 50 % FBG) and high Q, with all other cavity parameters being identical. The lower bound of stable region of mode-locking increases marginally with decrease in Q of the cavity. As it is evident from Fig. 7, the region of mode-locking increases with decrease in Q of the cavity. As discussed in Sec. 3.2, the primary constraint on stable region of mode-locking in a practical system is the damage threshold of SAM. The stable region of mode-locking with respect to the damage threshold of SAM is narrow as shown in Fig. 7(b). For the same value of pump power (or $g_0$), the intracavity power is higher for a cavity with a higher Q. Thus the intracavity energy corresponding to the damage threshold of the SAM is reached at a lower pump power levels in the case of cavity with higher Q compared to the case of a cavity with lower Q, leading to a wider range of stable region of mode-locking for cavity with lower Q.
Fig. 7. Comparison of stable region of mode-locking for cavity with low Q and high Q (a) bounds with respect to multipulsing (b) bounds with respect to damage threshold
To better analyse the influence of Q of the cavity, the background loss of cavity with 95 % reflectivity FBG is increased to achieve a continuous variation of Q, at a repetition rate of 100 MHz. Figure 8 shows the influence of cavity Q on output characteristics of mode-locked laser output. As it can be seen from Fig. 8(a), where the red asterisk points represent the upper bound of $g_0$ with respect to multipulsing, the range of gain values corresponding to the stable region of mode-locking has increased initially when the Q of the cavity is reduced (moving right to left on the graph), which is similar to the observation in Fig. 7. However, it further reduces as the Q is decreased beyond a certain value. It is also noticed that the pulse starts experiencing energy oscillation (pulsating operation) before the onset of multipulsing in the case of cavities with low Q, which are shown in Fig. 8(a). Figure 8(b) shows the pulse energy as a function of number of roundtrips at values of $g_0$ within and higher than the stable region of mode-locking when the Q of the cavity is $\sim$ 5 $\times$ 10$^6$. Such pulsating operation further reduces the stable region of mode-locking. The reduction in stable region of mode-locking below certain value of Q is probably because of the relatively smaller value of photon lifetime of the cavity (as shown in top x-axis of Fig. 8(a)) with respect to the cavity roundtrip time. Such analysis has to be repeated for the specific cavity parameters while designing a practical mode-locked laser.
Fig. 8. (a) Stable region of mode-locking at different Q of the cavity (b) Pulse energy as a function of number of roundtrips
3.4 Effect of the bandwidth of the passive reflector
In a mode-locked laser, the minimum attainable pulse width is inversely proportional to the gain bandwidth. All the results that are discussed in the preceding sections are based on a cavity with a FBG (bandwidth = 0.1 nm) as the passive reflector. In the presence of an FBG in the cavity, the minimum attainable pulse width is limited because of the bandwidth of the FBG. In order to study the effect of bandwidth of the passive reflector in the cavity, the FBG in the cavity is replaced with a GCM. GCM serves as a broadband reflector with an average reflectivity of 62 % in the wavelength range of interest. It must be noted here that the broad reflection band of the GCM can support a wide spectrum, and thus results in a short pulse with width smaller than the relaxation time of the SAM. The assumption of fast SAM would not hold good in such a condition, and hence the SAM characteristics are simulated by solving Eq. (3) instead of Eq. (4).
TDFLs operate in the anomalous dispersion regime, and in such cases, it is plausible to have the chirp due to self phase modulation (SPM) and GVD balancing each other, resulting in the formation of chirp-free soliton pulses with secant hyperbolic ($sech$) pulse shape. In soliton mode-locking, the pulse formation is dominated by the balance between GVD and SPM, however a slow saturable absorber is essential to stabilize the soliton as well as to start the pulse formation [22]. A soliton pulse in a laser cavity undergoes perturbation because of gain dispersion and cavity loss, which results in a co-propagating dispersive wave as the soliton tries to regain its shape. These dispersive waves build up at certain specific frequencies where a phase matching condition is satisfied, and hence result in the formation Kelly side bands [23,24]. Figure 9 shows the features of output pulse from a cavity with GCM, at a repetition rate of 100 MHz and at a $g_0$ value of 1.7 Np/m. SAM 2 is considered for this simulation. The time domain trace shows $sech$-pulse with a full width at half maximum (FWHM) pulse width ($\tau _{FWHM}$) of 360 fs (time resolution in simulation = 10 fs), and the corresponding spectra exhibits the characteristic Kelly side bands of a soliton pulse at frequencies as predicted theoretically [24]. It should be noted that the center frequency of the frequency axis corresponds to 150 THz (2000 nm).
Fig. 9. Output pulse characteristics of a passively mode-locked fiber laser with gold-coated mirror (a) in time domain (b) Pulse spectrum for a repetition rate of 100 MHz
Figs. 10(a) and (b) respectively show the stable region of mode-locking at different repetition rates with SAM 2 and SAM 3 in the cavities. It is evident from Figs. 10(a) and (b) that the stable region of mode-locking is narrow in the presence of a broadband reflector in the cavity. The reduction in stable region of mode-locking in case of soliton mode-locking is expected because of the high nonlinearity experienced by short soliton pulses, which restricts the operation to lower pulse energy. Figure 10(c) shows the pulse width and peak power values at the lower and upper bound in case of soliton mode-locking with SAM 3 in the cavity. As observed in Sec. 3.1, the pulse width decreases and peak power increases as the $g_0$ value increases, for a particular repetition rate. It should also be noticed that the cavity is multipulsing for all values of $g_0$ for repetition rates below 30 MHz. Even though the theory predicts that stable single pulse formation is not possible at repetition rates below 30 MHz, the experimental results shown in Fig. 11 shows single pulse formation at different pump powers at a repetition rate of 11 MHz from a cavity with GCM and SAM 2. In this case, the laser output is measured using a photodetector of bandwidth 10 GHz and sampling scope of bandwidth 30 GHz. A minimum pulse width of $\sim$ 84 ps is measured at a pump power of 668 mW. The time domain trace from the experiment shows an asymmetry in the pulse shape as the $g_0$ value is increased. Furthermore, the pulse width increases with increase in pump power, which is an opposite trend than that observed from the simulation. Comparing with the simulation results, it is plausible that the cavity used in our experiments is indeed operating in the multipulsing regime as the repetition rate is below 30 MHz and the measurement system is unable to resolve the individual pulses. This also explains the asymmetry in the pulse shape as well as an increase in pulse width with respect to the pump power.
Fig. 10. Stable region of mode-locking for cavity with GCM and (a) SAM 2 (b) SAM 3 (c) Pulse width and peak power at the lower and upper bound at different repetition rate, with SAM 3 in the cavity
Fig. 11. Experimental observation of pulse shape from mode-locked laser cavity consisting of GCM and SAM 2 as the cavity mirrors.
While the repetition rate is fixed to 100 MHz, the SAM in the cavity with GCM is changed to see the effect of SAM parameters on stable region of mode-locking in case of soliton mode-locking. Similar to that observed in the case of mode-locking with fast SA, the lower bound increases with increase in non-saturable loss of SAM, however the effect of modulation depth of SAM is insignificant as shown in Table 4. To observe the effect of Q of the cavity in soliton mode-locking, the background loss in the cavity is increased at a repetition rate of 100 MHz. In this case, the SAM in cavity is fixed to SAM 2. A trend similar to the case of mode-locking with fast SAM is observed, i.e, the stable region of mode-locking increases initially with decrease in Q of the cavity and then decreases when the Q is decreased beyond certain value.
Table 4. Simulation results with different SAMs, at a repetition rate of 100 MHz
3.5 Effect of the gain medium
The characteristics of passively mode-locked Erbium doped fiber lasers have been studied extensively in the past [25–27]. It is well known that the spectroscopy of Thulium doped silica system differs from that of Erbium doped system in various aspects such as energy transitions and metastable lifetime. In order to quantify the influence of spectroscopy of gain medium, simulations are repeated for a similar cavity discussed in Fig. 3(a) with SAM 2, but with Erbium doped fiber (EDF) as the gain medium. The results are compared with an equivalent cavity based on the Thulium doped fiber. The equation for a fast saturable absorber (Eq. (4)) is used for the simulation. Figure 12 shows the comparison of regions of stable mode-locking for Erbium and Thulium systems observed through our simulations. The upper bound in Fig. 12 is defined with respect to the damage threshold of the SAM. In the case of mode-locking with a fast saturable absorber, the operation in single pulse regime requires $\frac {P_L}{P_A}< 1$, where $P_L$ is the saturation power of the gain medium and $P_A$ is the saturation power of the SAM [28]. Owing to the longer metastable lifetime of Erbium (10 ms) compared to that of Thulium (334.7 $\mu s$), the ratio $\frac {P_L}{P_A}$ is smaller for an Erbium system than Thulium for a given SAM. This could potentially be the reason for a broader stable region of mode-locking for Erbium based mode-locked laser as shown in Fig. 12(b). It can also be ascertained that stable mode-locking operation at much lower repetition rate is achievable with an Erbium system compared to Thulium.
Similarly, passively mode-locked Ytterbium (Yb) doped fiber lasers are also been demonstrated in the past [29,30]. Our simulation model can be used to model passively mode-locked Yb doped fiber laser as well, in which case mode-locking is realised from an all normal dispersion (ANDi) cavity. In an ANDi mode-locked laser, the pulse is highly chirped, in which case increasing the value of $g_0$ (pump power) leads to the formation of dissipative soliton that can support higher pulse energy and a broader range for stable region of mode-locking.
3.6 Simulation of the ring configuration
The generic model presented in this paper can be further extended to simulate a fiber ring laser as that shown in Fig. 13(a). Figure 13(b) shows the stable region of mode-locking for the ring configuration, in which case SAM 2 is considered in the cavity. In the absence of any filtering element such as FBG, the ring configuration is operating in the soliton mode-locking regime and hence the stable region of mode-locking is narrow. For a given repetition rate, the pulse passes through the gain medium once per roundtrip in a ring configuration whereas it is twice in case of FP cavity. In other words, the results observed related to gain values for stable operation will occur at approximately twice the repetition rate, for a given length of the cavity. It has been noticed from the results shown in Fig. 10(a) that the mode-locked laser is not stable below 30 MHz repetition rate in case of soliton mode-locking in FP cavity. Similarly, the ring cavity is not stable below $\sim$ 80 MHz in case of soliton mode-locking in a ring configuration, as predicted by Fig. 13(b). The practical challenge that cannot be overlooked is that the cavity length required to achieve 80 MHz is about 2.5 m, and would require some specialised splicing.
Fig. 13. (a) Schematic of ring configuration (b) Stable region of mode-locking for ring configuration
4. Conclusion
In conclusion, the effect of several key cavity parameters on pulse evolution in a SESAM-based passively mode-locked TDFL is investigated through numerical modeling and simulations. Further, some of the simulation results are also supported by experimental results. The stable region of mode-locking is defined with respect to the value of small signal gain ($g_0$) over which the cavity supports a stable single pulse per roundtrip. Because of the inherent anomalous dispersion of silica fibers at 2 $\mu$m, TDF systems typically tend to operate under soliton mode-locking regime. However, in the presence of an FBG with limited bandwidth, the output pulse is broader compared to the relaxation time of the SAM and hence the cavity is considered to be mode-locked by a fast SAM. In such a case, the stable region of mode-locking is observed to be enhanced with decrease in cavity roundtrip time, increase in the modulation depth of SAM, and higher saturation fluence of SAM. The decrease in Q of the cavity can also potentially support a broader range of stable region of mode-locking, however an enormous increase in background loss in the cavity (thus realising system with lower Q) leads to pulse instabilities and thus reduction of stable region of mode-locking.
In a practical system, the bounds of stable region of mode-locking are restricted because of the available pump power as well the available gain from the gain medium, however, the primary constraint is imposed by the damage threshold of SAM. It is also observed that the gain medium with higher relaxation time and hence smaller saturation power supports a much broader range of gain values corresponding to the stable region of mode-locking. Similarly, an all normal dispersion laser cavity can also support broader range of gain values for mode-locking with higher pulse energy. On the other hand, soliton mode-locking leads to much shorter pulse width, although with lower pulse energy and narrower range of gain values for the stable region of mode-locking. In this case the influence of cavity parameters such as repetition rate, and the properties of the SAM in determining the stable region of mode-locking is minimal. It is also observed that the results from a ring configurations are similar to that from a FP configuration, but at repetition rate almost twice of that of FP configuration for a given length of the cavity. The influence of various cavity parameters on stable region of mode-locking are summarised in Table 5.
Table 5. Summary of influence of various parameters on stable region of mode-locking
The simulation model could be further extended to model an inhomogeneously broadened system by considering the gain spectrum consisting of multiple Lorentzian profiles with different centre frequencies. The results presented here form a practical guideline to achieve stable and reliable passive mode-locking under different operating conditions.
Funding
Ministry of Human Resource Development; Uchchatar Avishkar Yojana (UAY, Project No: 37ID.IITM_012).
Acknowledgments
We acknowledge Suparna Seshadri for initiating the simulations.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon request.
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