1. Introduction

Currently, there is growing demand for robust, compact and low-cost ultrashort pulse laser sources for scientific, medical [1] and industrial applications [2]. Fiber architecture is an attractive candidate for fulfilling these requirements due to its inherent compactness and monolithic design features. However, there are still issues which have to be solved in order to make ultrashort-pulse fiber oscillators widespread either as stand-alone lasers or as seed sources for different applications. Normally, pulse generation in fiber oscillators is based on passive mode-locking by using a saturable absorber element, which induces peak-power-dependent losses. The most widely used are “real” saturable absorbers based on saturable absorption in specific materials such as semiconductors [3,4], carbon nanotubes [5], or graphene [6], as well as effective saturable absorber configurations such as nonlinear amplifying fiber loop mirror (NALM) [7,8], and the nonlinear polarization rotation (NPE) [9]. Unfortunately, each of these mode-locking configurations has its drawbacks. Material-based saturable absorbers, especially semiconductors, are prone to long-term degradation [10] and this significantly limits the lifetime of such laser source. NPE requires non-polarization-maintaining fibers and thus is sensitive to environmental perturbations. NALM configuration requires precise control of the power splitting ratio and lacks tuning flexibility. Moreover, both NPE and NALM are susceptible to pulse destabilization and multi-pulsing because of their characteristic transmission-intensity curves [11].

An attractive alternative to conventional mode-locking techniques is a method based on self-phase modulation (SPM) induced spectral broadening and offset spectral filtering, effectively producing pulse peak-power-dependent transmission. Utilizing the pulse peak-power-dependent transmission of pulse re-amplification and spectral re-shaping method (2R) for optical data regeneration was first proposed by Mamyshev [12]. A decade later, based on this concept, self-pulsation in the telecommunication wavelength range was demonstrated [13,14]. However, the performance achieved was significantly below that of typical mode-locked fiber oscillators. In our previous work [15], we first demonstrated an ultrashort pulse generator, based on SPM-induced spectral broadening and offset-alternating filtering, and providing stable picosecond pulse train with moderate energies. Subsequently, research on fiber oscillators, based on nonlinear spectral re-shaping, gained a lot of momentum with different groups demonstrating further improvement [16–19]. In particular, record-breaking results were demonstrated by Liu et al. [18] and recently Sidorenko et al. [20]. At the time of writing this paper, the best results achieved were 190 nJ pulse energy and 35 fs pulse duration after dechirping from single-mode fiber [20]. Nevertheless, the capabilities of this pulse generation method have not yet been fully explored. There are unanswered questions about how the characteristics of generated pulses depend on the parameters of the pulse generator circuit. Most of the recent works were focused on the operation regime where chromatic dispersion of the fiber has a strong influence [16–20]. However, this is not the only option, as we will show in this work. Numerical investigation of eigen pulses generated in concatenated Mamyshev regenerators was earlier carried out by Pitois et al. [21]. They showed that the characteristics of the pulses and the operation regimes are strongly dependent on circuit parameters, however they did not carry out a systematical evaluation of the influence of circuit parameters for generated pulses (mainly 3 fixed parameter sets were investigated). In this research, by using the simplified model without chromatic dispersion, we were able to “scan” nearly the whole realistic parameter space and identify circuit parameters leading to stable pulse generation, as well as the characteristics of these pulses. Our investigation showed that by selecting the correct parameters, stable pulse generation is possible in such a circuit, even when there is no significant chromatic dispersion influence. On the other hand, operation under normal dispersion conditions has certain advantages which we will distinguish as well.

Another subject of such pulse generators, which is not fully clear, are the conditions required for self-starting operation. From a practical standpoint, self-starting is an important feature required for most of the applications. In our previous work, we used an additional back-reflecting mirror and acousto-optic modulator (AOM) to form a gated coupled cavity and to start pulse generation [15]. Liu et al. [18] reported achieving self-starting by forming constant coupled cavity supporting CW radiation, Samartsev et al. [17] used pump modulation and Sidorenko et al. [20] proposed using a switchable sub-cavity with a saturable absorber to start pulse generation. In this paper, we will show that in a linear fiber generator configuration, self-starting can also be achieved by simply setting the correct filter overlap.

In summary, this work presents both numerical and experimental results, investigating pulse generator operation at different fiber dispersion conditions and the relationship between circuit parameters and generated pulse characteristics. The work is organized as follows. In section 2 we will describe the numerical models and experimental setups which we used in this investigation. In section 3 we will analyze pulse generator operation and the characteristics of the generated pulses under conditions of zero chromatic dispersion. In section 4 we will extend our analysis to include a chromatic dispersion of the fiber. The main influence of normal and anomalous dispersion for pulse generation will be identified. Finally, in section 5 we will provide our experimental observations regarding conditions allowing for self-starting operation.

2. Description of the numerical model and experimental setups

In this work, we consider a linear pulse generator setup, consisting of a single gain medium, passive fiber elements and two spectral filtering elements [Fig. 1]. Compared to the ring setup, the linear configuration is simpler and is characterized by a reduced set of parameters. This allows for more general analysis. Moreover, results of this analysis can also be applicable to a ring configuration which contains ideally symmetrical gain and passive fiber elements.

 

Fig. 1 Schematic diagram of pulse generator setup described in the numerical model. Main elements of the setup: S_p1, S_p2 – passive fiber segments, S_a – active fiber segment (point amplifier), F₁, F₂ – spectral filter elements (spectral characteristics shown in corresponding insets).

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Pulse propagation in the proposed setup requires a description of three main effects: pulse propagation in the passive (active) fiber, pulse amplification (point amplifier) and pulse spectral filtering. Generally, we described pulse propagation in fiber by numerical integration of the nonlinear Schrödinger equation [22]:

With the reduction of pulse bandwidth and fiber lengths, chromatic dispersion influence for pulse propagation diminishes. The influence of chromatic dispersion can be neglected when the fiber length is much shorter than the dispersion length for pulses of corresponding bandwidth [Fig. 2]. In such case, the nonlinear Schrödinger equation can be reduced to a simplified form (containing only the SPM component) which can be integrated, thus providing an analytical expression of the SPM effect. This simplified description was applied to the analysis and experiments described in section 2.

 

Fig. 2 Dispersion length versus spectral width of Gaussian pulses. Group velocity dispersion parameter of fused silica at 1064 nm was used in calculations (${\beta }_2=16.5$ ps²/km).

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We described pulse amplification in the gain medium by using a constant gain factor. Such simplification can be justified for narrow bandwidth pulses and steady-state operation regime. Spectral filtering was described using the Gaussian spectral distribution filter function with variable filter bandwidth ($\text{Δ}{\lambda }_{f1/2}$, see Fig. 1) and spectral separation between filters ($\text{Δ}{\lambda }_{sp}$, see Fig. 1).

The results of the numerical analysis were compared with experiments. We performed a series of experiments by investigating two different pulse generator setups [Fig. 3] distinguished mainly by characteristics of spectral filtering (see Table 1). In the first setup (setup 1), narrowband fiber Bragg gratings (FBG) were used for spectral filtering. These FBG were designed for nearly Gaussian reflectance spectrum but low reflectance efficiency of ~45%. In the second setup (setup 2), broadband interference filters were used with approximate super-Gaussian transmission spectrum. The narrow bandwidth of the filters used in setup 1 allows this configuration to be considered as nearly dispersionless for significant total lengths of fiber. For example, if we choose the case of 0.08 nm bandwidth filters and consider that the pulse spectrum broadens twice during propagation in the fiber, the calculated dispersion length is more than 1 km as it can be seen from Fig. 2. Alternatively, if we apply the same consideration for setup 2 with ~3.5 nm filters, we get a dispersion length of ~1 m, and this value can be easily exceeded in a typical fiber laser setup. Therefore, setup 1 represents the case when the chromatic dispersion is negligible and the simplified model including only SPM can be applied. In contrast, setup 2 represents the case when the influence of chromatic dispersion on pulse formation is moderate and must be included in the model.

 

Fig. 3 Schematic diagram of the experimental setup with FBG filters (setup 1, a) and interference filters (setup 2, b). WDM – wavelength-division multiplexer, SM PM-Yb – single-mode polarization-maintaining Ytterbium-doped fiber, Pump LD – pump laser diode (976 nm).

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Table 1. Main parameters of experimentally tested configurations

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3. Pulse generation at zero group velocity dispersion conditions

The analysis presented in this section is restricted to dispersionless pulse propagation and perfectly Gaussian spectral distributions of the filter bands. Such restrictions allow describing pulse generator operation by using a small set of simple circuit parameters. These parameters are introduced below:

To initiate pulse generation in the modeled circuit, we used the initial pulse envelope described as a random irradiance value array with a uniform distribution (white noise) and normalized to fixed total energy in the calculation window. The chosen total energy was approximately an order of magnitude higher than the pulse energy of the generated pulses. Next, after a fixed number of roundtrips (typically 400), we checked whether the pulse formed in the circuit was stable, by calculating the standard deviation of pulse energy in subsequent (100) cycles. We define stable pulses as pulses which are identical after each roundtrip, so energy deviation for such pulses should approach an infinitesimal value limited by calculation precision. In the following paragraphs, we will present the result of such numerical analysis indicating what role each of the circuit parameters plays in the characteristics of generated (stable) pulses.

First of all, the analysis conducted showed an interesting dependence of stable pulse generation on gain parameter (when other parameters are fixed). In the case when fiber lengths fulfill condition L₁ = L₂, only two parameters govern whether pulse generation is stable – this is gain and spectral separation of the filters. Therefore, all configurations leading to stable pulse generation can be summarized in the single plot shown in Fig. 4. In this plot, stable pulse generation configurations are represented by regions colored in blue. At first glance, it might seem surprising that stable pulse generation does not depend on fiber length. Actually, with changing fiber length, the energy of the generated pulses adapts so that the total acquired nonlinear phase stays the same and there is no change in other pulse characteristics. Stability regions, as shown in Fig. 4, can be grouped according to the way the pulse spectrum broadens and which part of the spectrum is filtered. Regions marked by the letter A correspond to configurations when both filters are aligned at spectral peaks (of the characteristic for SPM pulse spectrum) displaced symmetrically [Figs. 5(a), 5(c), 5(e) and 5(f)] with respect to distance from the corresponding marginal spectral peak. Regions marked by the letter B represent configurations where one of the filters is aligned with a marginal spectral peak, and the other filter is aligned with a spectral peak situated farther from the edge of the spectrum [Fig. 5(b)]. Such configurations can also be distinguished by differences in pulse spectra between both outputs. Finally, for regions marked by the letter C, one of the filters is aligned with a marginal spectral peak, and the other filter is situated at a spectral minimum farther from the spectral edge [Fig. 5(d)]. Numbers near region letters describe the positional number of the filtered spectral peak (or minimum) when counting from the edge of the spectrum [Fig. 5].

 

Fig. 4 Pulse energy standard deviation (color) dependence on the gain (G) and normalized spectral separation of the filters (S). Blue color corresponds to stable pulse generation, green/yellow – unstable pulse generation, white color – no pulse generation, dark yellow – marks region where CW lasing threshold is exceeded in the spectral range where filter bands slightly overlap.

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Fig. 5 Calculated pulse spectra before filter (black trace) and rejected by filter to the output (red trace) at filter separation parameter S = 4.5 and gain value equal to: 7.2 (a), 15.2 (b), 19.4 (c), 24 (d), 29.8 (e), 40.38 (f). At each graph, upper spectra correspond to filter 1, and lower spectra correspond to filter 2. Visible parts of black trace indicate which part of the pulse spectrum was filtered and directed back to the circuit for the next cycle.

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To make this classification of stability regions clearer, we will now analyze the pulse stability map from the perspective of the fixed spectral separation parameter at S = 4.5 and varied gain. When starting at zero gain and gradually increasing the gain, the first stable pulse generation occurs when the pulse spectrum broadens enough to allow the marginal spectral peaks to be filtered for the next cycle [Fig. 5(a)]. This operation region (A1 in Fig. 4) is also characterized by the broadest stable operation gain interval (especially at low S parameter). When increasing gain further, a non-symmetrical stable operation regime (region B3 in Fig. 4) can be achieved when there are filtered spectral parts situated at a different distance from the edge of the pulse spectrum [Fig. 5(b)]. When increasing gain even further, a stable operation regime is achieved (region A2 in Fig. 4) when both filters are centered at the second spectral peak from the edge [Fig. 5(c)]. Further increase of gain allows reaching a stable operation regime (region C3 in Fig. 4) when one of the filters is situated at a spectral minimum [Fig. 5(d)]. Finally, stability regions appearing at even higher gain values (regions A3, A4, A5 in Fig. 4) are of the same type and differ in the distance the filtered part of the spectrum is away from the spectrum edge (Figs. 5(e) and 5(f)). For these types of stability regions, the width of the stability region becomes narrower with the increase of region order, which means that it would become increasingly difficult to achieve stable pulse generation in an experimental setup under such conditions. Eventually, the highest operational gain value is limited by CW generation (marked as dark yellow in Fig. 4) at the spectral region where filter bands overlap. This limiting gain value can be increased by increasing spectral separation of the filters. However, we will restrict our further analysis to the lowest-gain stability region (A1), mainly because of the broadest gain value interval of stable operation and lowest filtering losses. It should also be noted that for some gain values in between stable operation regions (mainly at low S and high G, as shown in Fig. 4), pulse generation still takes place, but generated pulses change their parameters either randomly or periodically after each roundtrip.

If we restrict ourselves to a single stability region, we can conclude that the gain value corresponding to stable pulse generation depends on the normalized spectral filter separation parameter S. For A type stability regions, with an increase of parameter S, the gain value G, corresponding to stable operation, increases and its interval decreases (stability region becomes narrower). Furthermore, it should be noted that the absolute gain value also depends on losses in the circuit (such as output losses). For the data presented in Fig. 4, no output-coupling or any other additional losses were included. However, these results can easily be extended for any configuration in which equal losses are induced symmetrically with respect to the gain element. In such cases, gain values presented here have to be increased proportionally to induced total single-trip losses.

The presented numerical analysis shows that in the dispersionless case, stable pulse generation can be achieved, but only within narrow gain value intervals. This was also confirmed by results of experiments conducted using setup 1. In Fig. 6, output pulse energy versus pump power is shown for a circuit configuration with L₁ + L₂ = 51.5 m total fiber length and two different filter separation parameters. Pulse generation in such an experimental setup was possible only within narrow pump value intervals as shown in Fig. 6. Increasing or reducing pump power outside of these boundaries resulted in the disappearance of pulse generation. From numerical modeling (when including optical-frequency-dependent point amplifier description [23]) we can evaluate approximately that these pump values correspond to gain value intervals of the A1 stability region shown in Fig. 4. Although the operational pump intervals shown in Fig. 6 for S = 1.7 and S = 2.8 are nearly equal, according to the modeling, gain intervals corresponding to these pump values differ by a factor of ~1.9. This is essentially in accordance with Fig. 4. Moreover, the experimentally recorded feature that pulse energy at the output tends to saturate before reaching maximum gain was predicted by our numerical model of pulse generation in the dispersionless case.

 

Fig. 6 Experimental measurements of output pulse energy and average power versus pump power in setup 1. Measurements were carried out at both outputs of the setup marked in Fig. 3(a). Other parameters of the setup: $\Delta {\lambda }_{f1/2}$ = 0.08 nm, L₁ + L₂ = 51.5 m.

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We will now analyze in more detail the influence of normalized spectral separation of the filters (parameter S) for the operation of the circuit. This parameter governs how much the pulse spectrum should be broadened to sustain pulse generation. This sets conditions for the required SPM-induced nonlinear phase, which determines pulse peak power and energy (at other fixed parameters). Thus, with an increase of the S parameter, the pulse spectrum broadens and the energy of the generated (stable) pulses increases [Fig. 7(a), blue traces]. Obviously, when increasing the S parameter, gain must be increased accordingly, as previously mentioned. To verify these numerical results, we have conducted experiments using setup 1, in which spectral separation of the filters was gradually increased after stable pulse generation was started. In order to sustain pulse generation, we had to increase the pump power slightly. We were able to increase the spectral separation of the filters up to S = 4.75 in the configuration with 222 m total fiber length. Attempts to sustain pulse generation while increasing filter separation further than described were not successful. We then measured pulse spectrum and pulse energy at the outputs and compared them with calculated values for the corresponding setup. As can be seen in Fig. 7(b), the spectrum of generated pulses changes as predicted by the numerical modeling results. Moreover, pulse energy increases in agreement with modeled dependencies [Fig. 7(a)]. Note that numerical results are presented in Fig. 7(a) by two traces corresponding to the maximum and minimum gain values of the A1 stability region. This is because pulse energy depends slightly on gain and precise gain value in an experimental setup was not known. In addition, numerical calculations showed that at the maximum gain value of the A1 stability region, pulse energy (and spectrum) before output 1 and output 2 can slightly differ, despite the fully symmetrical circuit configuration, so in Fig. 7(a), the maximum pulse energy case is displayed as a trace corresponding to the maximum gain parameter.

 

Fig. 7 a) Pulse energy dependence on the normalized filter separation parameter S. Numerical calculation results (traces) and experimental measurements (points). Maximum and minimum gain correspond to maximum and minimum gain values of the A1 stability region, for which pulse energy standard deviation <10⁻²⁰ J. b) Comparison of calculated (blue trace) and experimental (red trace) pulse spectra at output 2. Parameters of the setup: $\Delta {\lambda }_{f1/2}$ = 0.08 nm, L₁ + L₂ = 222 m, S = 1.97-4.69 (exact values are shown in the graphs).

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The next parameter we will address in more detail is the spectral bandwidth of the filters. As expected, our analysis showed that spectral bandwidth of the filters is mainly responsible for the pulse duration of generated pulses. The generated pulses are nearly as short as permitted by filter bandwidth. Thus, the pulse duration is inversely proportional to filter bandwidth (in the frequency domain). However, the proportionality factor depends slightly on filter separation parameter S and gain G. For the experimental investigation of filter bandwidth influence for pulse duration we used FBG’s with two different bandwidths. We have evaluated pulse duration from measured autocorrelation traces of generated pulses. These results were qualitatively in agreement with modeled dependencies [Fig. 8].

 

Fig. 8 Pulse duration dependence on filter bandwidth. Calculation results (traces) and experimental measurements (points). Inset – corresponding experimental pulse autocorrelation traces. Maximum and minimum gain correspond to maximum and minimum gain values of A1 stability region, for which pulse energy standard deviation <10⁻²⁰ J.

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Keeping in mind that pulse peak power is determined by the required amount of spectral broadening, it is evident that filter bandwidth influences pulse energy as well. Because the duration of generated pulses is inversely proportional to filter bandwidth, their energy is also inversely proportional to the filter bandwidth.

Lastly, we will analyze pulse generation dependence on fiber length. Total fiber length (L₁ + L₂) influences total SPM-induced nonlinear phase-shift accumulated by pulses and hence their spectral broadening. As already mentioned, the required phase-shift to sustain stable pulse generation is governed by spectral separation of the filters. So, at fixed spectral separation and filter bandwidth, when fiber length is decreased, pulse peak power and energy have to increase proportionally in order to sustain the required amount of spectral broadening. This property was evident from numerical modeling and supported by experimental data [Fig. 9]. However, in the experiments we were unable to acquire data at short fiber length (<35 m) because of difficulties encountered in exciting pulse generation in the circuit with short total fiber lengths. So, the question of how far one can proceed with decreasing total nonlinearity in the system to increase output pulse energy still remains open.

 

Fig. 9 Output pulse energy dependence on total fiber length of the circuit. Calculation results (traces) and experimental measurements (points). Other parameters of the setup: $\Delta {\lambda }_{f1/2}$ = 0.08 nm, S = 2.8 (self-starting configuration), S = 3.1 (external excitation), filter reflectance: 45%. Red point corresponds to experimental configuration when pulse generation was excited by using picosecond pulses from external source. Maximum and minimum gain correspond to maximum and minimum gain values of the A1 stability region, for which the pulse energy standard deviation is <10⁻²⁰ J.

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Numerical modeling also showed interesting results when fiber lengths L₁ and L₂ of the circuit are not equal: L₁ ≠ L₂. In such case, conditions allowing for stable pulse generation additionally depend on fiber length ratio k_L = L₂ / L₁. Calculated pulse energy standard deviation dependence on fiber length ratio and gain at fixed spectral separation of the filters S = 2 is shown in Fig. 10. Using the previously introduced classification, we can distinguish A, B, and C types as well as mixed-type stability regions. The A1 stability region appears at the lowest gain value and features mostly symmetrical operation when the parameters of pulses at both outputs are nearly identical. However, the width of this region with respect to gain reduces when the fiber length ratio parameter decreases. At higher gain, B-type stability regions appear which feature asymmetrical operation, when pulses generated at output 1 and output 2 can differ strongly in spectral width and pulse energy [Fig. 11]. Difference in pulse parameters (pulse energy) from different outputs increases with the order of the B-type region. Furthermore, in the stability map, C-type regions and even mixed type B-C regions can be distinguished. Latter describe the operation when pulse spectrum characteristics gradually change between two types with an increase of gain.

 

Fig. 10 Calculated pulse energy standard deviation (color) dependence on the gain (G) and fiber length ratio parameters (k_L). Other parameters of the circuit used in calculations: S = 2, Δλ_f = 0.4 nm. Blue color corresponds to stable pulse generation, green/yellow – unstable pulse generation, white color – no pulse generation. Inset graphs show typical pulse spectra before (black) and after (red) filtering at stability regions marked by corresponding arrows.

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Fig. 11 Dependence of the pulse energy (color) before filter 1 (a) and filter 2 (b) on fiber length ratio and gain. Other parameters of the circuit used in calculations: S = 2, Δλ_f = 0.4 nm, L₁ = 5 m.

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4. Pulse generation at normal and anomalous group velocity dispersion

Normal group velocity dispersion, which is characteristic for typical fused silica fibers for wavelengths <1.3 µm, is responsible for pulse broadening in time by inducing linear chirp which has the same sign as the chirp generated by SPM at the main part of the pulse [22]. It can be easily predicted that because of the pulse broadening and reduction of pulse peak power, the energy of generated pulses, in a fiber generator circuit at normal dispersion conditions, should increase. However, numerical modeling shows that the influence of chromatic dispersion has significantly broader influence for pulse generation. In particular, normal dispersion of the fiber allows pulse generation to be stabilized. This is manifested by a broader interval of gain values which allow for stable pulse generation as can be seen in Fig. 12. It is evident from the plot presented that the stability region (indicated by the blue color) broadens with an increase of total induced normal dispersion. Moreover, with an increase of normal dispersion, pulse energy parameters tend to equalize between both outputs. In our calculations we managed total induced dispersion by changing the length of a typical fiber with fused silica dispersion parameters (β₂ = 0.0165 ps²/m, β₃ = 4.43x10⁻⁵ ps³/m @ 1064 nm center wavelength [24]), however dispersion can also be managed by changing intrinsic properties of the fiber (material composition or waveguide structure) [25,26]. This produces the same results with respect to the stability map but influences other properties of generated pulses such as pulse peak power and energy. Pulse energy increases with decreasing fiber length when other parameters and total induced dispersion are fixed. Therefore, it could be beneficial to use a shorter fiber with significantly higher chromatic dispersion as this would allow achieving a broader stability region with respect to gain together with higher pulse energy.

 

Fig. 12 Calculated pulse energy standard deviation (color) dependence on gain and total fiber length (which governs total dispersion) parameters. Fiber dispersion parameters used in calculations: β₂ = 0.0165 ps²/m, β₃ = 4.43x10⁻⁵ ps²/m. The blue color corresponds to stable pulse generation, green/yellow – unstable pulse generation, white color – no pulse generation. Circuit parameters used in these calculations: Δλ_f = 0.4 nm, S = 2 (a), S = 3 (b). Dashed grey lines indicate boundaries of stability region A1 in dispersionless case.

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Numerical modeling results also showed that spectral separation of the filters influences the relationship between total induced normal dispersion and the achieved gain interval of stable generation. With the increase of the S parameter, gain interval broadening shifts to the region of higher total dispersion (compare Figs. 12(a) and 12(b)).

The property that the stability region broadens significantly with respect to gain under the influence of normal dispersion was also confirmed by experiments accomplished using setup 2. After starting pulse generation in this setup, pulse generation was sustained when changing pump power from the lowest value of 108 mW up to maximum pump power available from a pump laser diode (300 mW). Even though the energy of the pulses generated was very stable when pump power was within two broad intervals (black and red points in Fig. 13), for some pump values in between these intervals, pulse energy fluctuated. Nevertheless, the pump power interval of operation (ΔP_pump ≈192 mW) was more than an order of magnitude broader than we were able to achieve in the dispersionless case (ΔP_pump ≈7 mW)(recall Fig. 6).

 

Fig. 13 Experimental measurement of output average power and pulse energy versus pump power in the setup 2. Filter spectral separation was as shown in Fig. 17(b).

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Further numerical analysis, addressing the properties of generated pulses (in case of normal dispersion conditions) showed that pulses broaden in time because of normal dispersion influence, and the amplitude of the pulse spectrum modulation decreases [Fig. 14]. Moreover, the dependence of pulse energy on total fiber length [Fig. 15] is worth noting. Starting with fiber lengths much shorter than dispersion length, pulse energy initially decreases with the increase of fiber length (which corresponds to the dispersionless operation regime, compare the blue trace with the dashed trace in Fig. 15). Pulse energy then reaches the minimum, and from that point, it increases with increasing fiber length up to the point where the chosen gain value becomes too low to sustain pulse generation. Additional corrections to achievable pulse energy arise from the feature of operation in a normal dispersion regime that stable generation can be achieved at a broad interval of gain values (in contrast to the dispersionless case). So, by increasing gain, the energy of generated pulses can be greatly increased (compare blue and violet traces in Fig. 15). This increase is, of course, accompanied by a spectral broadening of the pulse spectrum (an increase of SPM-induced nonlinear phase) and eventual increase of pulse duration. However, such generated pulses are mostly linearly chirped and can be compressed to high-quality ultrashort pulses. By using setup 2, we were able to experimentally generate pulses with energy up to 3.8 nJ and compress them to 145 fs duration. Pulse energy, in this case, was mainly limited by available pump power.

 

Fig. 14 Numerical calculations of pulse properties (before filters) when circuit parameters are: Δλ_f = 0.4 nm, S = 2. a) Pulse duration versus total fiber length. b) Typical pulse temporal profiles when total fiber length is 1 m (black trace) and 60 m (red trace). c) Typical pulse spectra when total fiber length is 1 m (black trace) and 60 m (red trace).

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Fig. 15 Calculated pulse energy (before filters) dependence on total circuit fiber length. Continuous traces correspond to operation when fused silica chromatic dispersion (@λ_c = 1063 nm) is included, S = 2, G = 6 and filter bandwidths are: Δλ_f = 0.4 nm (blue trace), Δλ_f = 0.8 nm (green trace), Δλ_f = 1.6 nm (orange trace), Δλ_f = 3.2 nm (red trace). Violet trace corresponds to the same configuration as represented by the blue trace but with increased gain to G = 14. Dashed trace corresponds to the dispersionless case with other parameters the same as for a blue trace: S = 2, G = 6, Δλ_f = 0.4 nm.

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We have additionally performed numerical calculations investigating pulse generator operation under anomalous dispersion conditions. In particular, we used fused silica dispersion parameters at 1554 nm center wavelength (β₂ = −0.0284 ps²/m, β₃ = 1.53x10⁻⁴ ps³/m). Our investigation showed that stable pulse generation is still possible under small anomalous dispersion conditions. However, the stability region for such configurations is narrow [Fig. 16(a)], and it further narrows with an increase of anomalous dispersion contribution.

 

Fig. 16 Calculation results of operation under anomalous dispersion: a) Calculated pulse energy standard deviation (color) dependence on gain and total fiber length (which governs total dispersion) parameters. Fiber dispersion parameters used in calculations: β₂ = −0.0284 ps²/m, β₃ = 1.53x10⁻⁴ ps³/m. Blue color corresponds to stable pulse generation, green/yellow – unstable pulse generation, white color – no pulse generation. b) Pulse spectra before filter 1 when fiber length is 1 m (black trace) and 60 m (orange trace). c) Pulse temporal profiles before filter 1 when fiber length is 1 m (black trace) and 60 m (orange trace). Circuit parameters used in these calculations: Δλ_f = 0.4 nm, S = 2.

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SPM and anomalous dispersion have the opposite influence on pulse chirp, and from calculated pulse spectra, it can be seen that spectrum bandwidth, registered before one of the filters, reduces with the increase of anomalous dispersion contribution [Fig. 16(b)]. Eventually, spectral broadening becomes too small, and pulse generation stops. On the other hand, anomalous dispersion also causes pulses to compress during propagation in fiber, and so their duration decreases Fig. 16(c).

We did not perform an experimental investigation of such pulse generator configuration, so we will not go into more details regarding operation under anomalous dispersion.

5. Self-starting operation

Our numerical results indicated that specific circuit parameters lead to the formation of stable pulses even when starting from random noise-like pulses. However, in the experimental setup, the self-starting operation was more difficult to achieve (in our previous work we could not even achieve it without additional means). One of the reasons for this is that the gain of the fiber amplifier is not constant during the transitional start-up process, it is affected by gain saturation effects. Nevertheless, we found out that self-starting is possible when the correct overlap of filter spectral bands is set precisely, and fiber amplifier gain is increased above normal operating conditions. Because of filter overlap and high gain, CW operation for some wavelengths becomes possible and pulsating radiation is generated, which triggers self-starting. Self-starting was confirmed in both experimental setups when using filters with bandwidths of 0.08 nm, 0.04 nm (setup 1), 1.7 nm, 3.5 nm (setup 2) and different total fiber lengths: 35–222 m in setup 1, 5–15 m in setup 2. However, conditions allowing for self-starting in most cases had to be found empirically.

For setup 1 with FBG Gaussian filters, the optimal (for self-starting) the filter overlap parameter value was in the range of S = 1.7–2 [Fig. 17(a)]. Such self-starting conditions can be partially explained by analyzing the pulse generation stability map shown in Fig. 4. As can be seen from the figure, at small filter overlap and high gain, the non-stable pulse generation region extends from very high gain down to the stable pulse generation region. This allows starting some non-stable pulse generation at high gain and then gradually reducing gain downto the lowest operation value. In contrast, when parameter S > 2.5, operation at higher gain and lowest stable pulse generation gain region becomes separated by an interval in which pulse generation is not sustained at all. Our experiments with setup 1 showed compatible behavior. When we increased the pump power significantly (by a factor of 3–4) above the lowest operational value (~100 mW), multiple non-stable pulses per roundtrip were spawned [Fig. 18(a)]. All these excess pulses were terminated, and stable pulse generation was achieved [Fig. 18(b)] when we gradually reduced pump power down to the lowest operational value. If we did not reduce pump power after self-starting, these randomly spaced excess pulses changed their position in time, some disappeared, and some new ones appeared. However, the pulses, which corresponded to a roundtrip of the circuit (it is the pulse with the highest amplitude in Fig. 18(a)), stayed mainly stable.

 

Fig. 17 Typical filter overlap leading to self-starting operation: a) Narrowband FBG filters in setup 1; b) Interference filters in setup 2.

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Fig. 18 a) Typical pulse “forest” spawned in setup 1 during start-up. Measured using a 15 ps pulse response photodiode and 20 GHz real-time oscilloscope. b) Typical pulse train during normal single-pulse operation in setup 1. Measured using 1 GHz bandwidth photodiode.

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In the case of setup 2, the spectral bands of the filters could not be described by Gaussian distribution (closest approximation is super-Gaussian), so the introduced normalized spectral separation parameter cannot be applied to describe (and compare) filter overlap for this setup. Instead, for these filters, we can specify a maximum relative transmission value in the region where filter bands overlap (marked by red points in Fig. 17). Our experiments with setup 2 showed that self-starting was possible when this parameter was between 6 and 20% [Fig. 17(b)]. This interval was broader compared to the interval evaluated for setup 1 with FBG filters (8-12%).

Overall, most reliable self-starting was experimentally achieved in setup 2 using interference filters with a 3.5 nm bandwidth. For this configuration, we have also arranged a start-up time measurement setup. Measured distribution of 400 data points is shown in Fig. 19. Although all start-up attempts were successful and median start-up time was around 0.5 s, the measured distribution exhibits long tail characteristic for log-normal distribution. That is, for some of the attempts self-starting time was significantly longer.

 

Fig. 19 A typical histogram of start-up time distribution for setup 2. Other parameters of the setup: L₁ + L₂ = 9.5 m.

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It should also be noted that self-starting operation, with low filter separation, opposes to high-energy operation achievable in case of normal dispersion, high spectral separation of the filters and high gain. However, our experiments showed possibilities for increasing spectral separation of the filters after pulse generation has been started. Therefore, by using some means for smooth filter tuning, it should be possible to arrange a circuit which would self-start and then tune its pulse parameters for the highest pulse energy possible.

6. Conclusions

In this work, we present the results of a thorough investigation of features of a linear pulse generator setup. We demonstrate both numerically and experimentally that stable pulse generation in such circuits is possible even under conditions of negligible chromatic dispersion (that is under the influence of only SPM). For such configurations, different types of operation regimes were indicated, and relationships between circuit parameters and parameters of generated pulses were identified. Next, we analyzed the influence of chromatic dispersion (of fiber) for pulse generation. Our numerical calculations showed, and experimental results confirmed that normal dispersion (of fiber) not only affects pulse parameters (duration and chirp) but also broadens the interval of stable operation with respect to amplifier gain. This allows for more reliable operation and higher energy of generated pulses. In contrast, according to numerical calculations, operation under conditions of anomalous dispersion is characterized by an even reduced stability region with respect to gain. Finally, we have shown that self-starting in such circuits can be achieved by simply setting the correct filter overlap and increasing the gain of the amplifier. Experimentally we were able to confirm self-starting operation both in the setup operating under conditions of normal dispersion and in the setup operating under conditions of negligible chromatic dispersion.