1. Introduction

Supercontinuum (SC) generation directly from a nonlinear fiber amplifier, especially from a nonlinear ytterbium-doped fiber amplifier (YDFA), has become a hot research topic owing to its low peak power requirement of pump pulse, all-fiber structure, high output power and high optical to optical conversion efficiency [1–4]. However, the related theoretical research has been rarely discussed because the mechanism of SC generation in an YDFA is complex due to the combined function of laser gain amplification and nonlinear effects. Dynamic rate equations are widely used to calculate the time-dependent population density and light intensity in fiber amplifiers which can obtain the gain distribution with time and fiber length. Unfortunately, the light intensity is not able to demonstrate nonlinear effects and the interference of different frequencies [5]. In [6], the Ginzburg–Landau equation, which combined the non-linear Schrödinger equation (NLSE) with the famous Maxwell-Bloch equation for describing the dynamic response of two-level system, is given to simulate the pulse evolution in YDFA. However, the Ginzburg–Landau equation does not take the higher-order dispersion and nonlinear effects into account, which cannot be served as the numerical model for SC generation in fiber amplifier. In [7], the complex Ginzburg–Landau equation is used to simulate the propagation and SC generation of pulses with different initial chirp in the fiber amplifier. Even though the higher-order dispersion and nonlinear effects are included in the calculation, the small signal gain coefficient and the gain profile of the doped fiber are artificially set and are assumed to be identical along the fiber length, which are not in agreement with the real situation. Meanwhile, the dynamic amplification process cannot be analyzed and the pump power cannot be taken as a changeable parameter directly. In [5], a combined model of the laser rate equations and the Ginzburg–Landau equation for simulation of the amplification of the ultrashort pulses in an YDFA is demonstrated. Similarly, the higher-order dispersion and nonlinear effects are ignored and the nonlinear effects in YDFA cannot be calculated precisely. From above, we can find that an appropriate theoretical analysis method should be come up with for the purpose of further understanding the dynamic physical process of SC generation in an YDFA, which is of great importance to optimize the output power, spectrum range and flatness of the SC.

In this paper, we combine the laser rate equations and generalized non-linear Schrödinger equation (GNLSE) to simulate the evolution of laser pulses in the YDFA. After describing the basic theory of the combined model and the feasibility of this model in Section 2, we show in Section 3 a detailed description of the mechanism of SC generation in an YDFA. After that, the pump power, the fiber length and the pulse width dependence of the SC are investigated theoretically and experimentally.

2. Theoretical model

2.1 Combined model of rate equations and GNSLE

Ultrashort laser pulse propagation in an optical fiber can be modeled by using GNLSE [8],

2.2 Solution method for numerical simulations

Adaptive split-step Fourier method (SSFM) [9] has been used to solve the GNLSE. Linear and nonlinear steps are calculated in the frequency and time domain respectively on the condition that the step dz is small enough that the dispersive and nonlinear effects can be assumed to act independently. Figure 1 shows the schematic steps of numerical simulation of propagation of pulses in the YDFA. An iterative process can be described in several steps:

 

Fig. 1 Schematic steps of numerical simulation of propagation of pulses in YDFA

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1. Divide the fiber into N steps and the length of every step is H. 2. Solve the rate equations (Eqs. (2) - (6)) to calculate the population density distribution, pump power and signal power, gain factor at every step. 3. Divide the H-long-fiber into n steps with a length of h of every step. The selection principle of h along the fiber was proposed in [9]. 4. Solve GNLSE in N steps respectively by using adaptive SSFM. Note that the initial complex amplitude of every next step is given by Eq. (11). 5. Iterative calculation: Solve the rate equations to calculate the population density distribution, pump power and signal power of the next pulse, as well as solving GNLSE. The temporal interval between two seed pulses is given by Eq. (8). Here t is the reciprocal of the repetition rate and z is the propagation distance. 5. Calculate the output energy and average power of every pulse by using

2.3 Feasibility of the model

Since the physical process of SC generation in an YDFA is sophisticated, with a close relation to the combined function of power amplification and a number of nonlinear effects, the parameters of signal pulse and the performance of the amplifier, we carry out a comparison of experimental and simulation results about the SC generation from an YDFA in order to verify the feasibility of the model.

The selected gain fiber in the amplifier is a piece of double-clad YDF with 15 μm and 130 μm as the diameters of fiber core and inner cladding respectively. The effective core/clad numerical aperture (NA) is 0.08/0.46 and the highest cladding absorption is about 5.4 dB/m at 976 nm. The calculated dispersion curve is shown in Fig. 2(a). It can be seen that the zero dispersion wavelength (ZDW) is located at 1273 nm. The loss curve for silica fiber is shown in Fig. 2(b) which is given by${10}^{-3}{e}^{4.67/\lambda }+6\times {10}^{11}{e}^{47.8/\lambda }$, where $\lambda (\mu m)$ is the light wavelength [10]. Table 1 gives the values of other parameters used in the simulation. Note that the effective absorption coefficient can be given by

 

Fig. 2 (a) The dispersion curve and (b) the loss curve of the YDF

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Table 1. Parameters Used in the Simulation

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Figure 3(a) and 3(b) indicate the simulation results. As shown in Fig. 3(a), the output power of every pulse increases with the number of pulses and come to be stable when the pulse number is larger than 50. The final average output power is 14.66W (25W 976 nm pump power). Figure 3(b) shows the output spectrum of the last pulse in the numerical calculation, from which we can see a significant spectrum broadening and the first-order Stokes peak.

 

Fig. 3 The simulation results of SC generation from an YDFA (5 m gain fiber) (a) the output power for different pulses (b) the output spectrum for the 68th pulse

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The scheme of the experimental setup is shown in Fig. 4. The parameters of the seed pulse (pulse width and repetition rate) are same with the numerical calculation. The average output power of the fiber laser seed is 35 mW. After the single-mode YDFA, the output average power rise up to 205.6 mW, which corresponds to the 2kW peak power of the pulse in the numerical simulation. A 3 nm band pass filter is used to reduce the forward and backward amplified spontaneous emissions (ASE). The last stage is a double-clad YDFA (15/130 μm core/cladding diameter with 0.08/0.46 NA and 5.4 dB/m absorption at 976 nm) pumped by a 25 W laser diode pump. The output passive fiber end is angle polished for the purpose of reducing back reflection and preventing end facet damage. The output spectrum is shown in Fig. 5 (red line) and the average output power is 18.9 W when the 976 nm LD pump power is 25 W. For comparison, we choose the average output spectrum of 60th-68th pulse with the stable output pulse energy (the blue line in Fig. 5) as the final calculated spectrum. This is reasonable because the tested spectrum from optical spectrum instrument in the experiment is also time-averaged.

 

Fig. 4 The experimental setup: Isolator (ISO), bandpass filter (BPF), laser diode (LD, 976 nm), and wavelength division multiplexer (WDM)

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Fig. 5 The comparison between the simulated and experimental result

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From the comparison in Fig. 5, we can conclude that the shape of the calculated spectrum is generally agreed with the experimental result with the exception at 976 nm. This may be caused by the unabsorbed pump power at 976 nm in the experiment. Likewise, the output power of the simulation result is smaller than the experimental results because of the residual pump power in the experimental output. In conclusion, the experimental and simulated results verify that this numerical model is correct and feasible for the theoretical analysis of SC generation in an YDFA.

3. Simulation results

3.1 Simulation results in pulse iterative calculation process

The parameter setup in this section remains the same with section 2 except the fiber length is 8 m in this part. As shown in Fig. 6(a), the final average output power is about 13.7W. Figure 6(b) illustrates the residual pump power at the end of the gain fiber for every pulse. With the increase of the pulse number, the residual pump power at the end of fiber length keeps increasing until the pulse number is over 50. Meanwhile, the average output power becomes stable. Figure 6(c) shows the peak intensity in the time domain for different pulses at the end of the gain fiber. The peak intensity grows up gradually and regularly for 1st −35th pulse, which means that the dominant effect in this period is gain amplification rather than pulses fission due to nonlinear process. But afterward the peak intensity appears to fluctuate evidently, indicating that the amplified pulses breakup remarkably into a number of disordered ultra-short pulses during the propagation. The evolution of gain factor $\exp (dz\times G_0/2)$ (as shown in Eq. (12)) at 1060 nm along the propagation distance is given in Fig. 6(d), where dz is the step length H in the calculation. Generally, for every pulse, the gain factor drops down with the growth of distance, which means the intense pulse amplification usually happens at the beginning of the propagation distance. Similarly as shown in Fig. 6(a), the distribution of the gain factor for every pulse seems to be stabilized with the rise of iterative calculation times. Especially for the 50th and 60th pulse in Fig. 6(d), the two curves do not deviate from each other a lot. Correspondingly, in Fig. 6(a), the output power is stable when the pulse number is larger than 50, which verifies that the output power and the distribution of the gain factor come into equilibrium simultaneously.

 

Fig. 6 The calculation results in pulse iterative calculation process (8 m gain fiber) (a) the output power (b) the residual pump power (c) the peak intensity in time domain at the end of the gain fiber (d) the gain factor along the fiber length

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3.2 Mechanism of SC generation

The evolution of the spectrum along the fiber length and the time domain of the last calculated pulse are illustrated in Fig. 7. Obviously, the initial main effect is the pulse amplification during the propagation. When the pulse arrives at 3 m, the first-order Stokes peak and a weak anti-stokes peak occurs at 1116 nm and 1016 nm respectively. As the peak intensity locates at the center of the pulse and is easily affected by Raman scattering, the center of the pulse in time domain appears to split up. With the amplification of the pulse and the energy transferring to Stokes components, a fourth-order stimulated Raman scattering (SRS) has been achieved at 3.5 m and the center of the pulse further breakup. Most part of the pulse envelope lies in the front part. This can be explained by the fact that the pulse propagates in the normal dispersion region of the gain fiber till now and the longer wavelength has a faster propagation speed. As shown in the spectrum at 5 m, after the long wavelength edge of the spectrum reaches and above the ZDW of the YDF (1273 nm), modulation instability (MI) and soliton self-frequency shift (SSFS) take place of SRS to be the dominant nonlinear effects. The pulse split up to several ultra-short pulses due to MI and these ultra-short pulses give birth to higher-order solitons. These higher-order solitons evolve to lowest-order solitons under the function of nonlinear dispersion effects. These solitons experience SSFS and thus lead to the spectral broadening at the long wavelength direction. At the end of the fiber length, the spectrum reaches 2000 nm at long wavelength edge. The blue-shift dispersive wave, derived from red-shift soliton, extends the short wavelength edge to 800 nm. This presented mechanism of SC generation in an YDFA is well-corresponding to our previous experimental results and analysis [2, 3].

 

Fig. 7 The evolution of the spectrum and the time domain along the fiber length (the 68th pulse - the last calculation pulse)

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3.3 Fiber length dependence of SC

As mentioned in section 2 and section 3.1, the final average output power for 5 m and 8 m gain fiber length is 14.6 W and 13.7 W respectively. Figure 8 (a) and 8(b) shows the evolution of the spectrum along the fiber length for 5 m and 8 m gain fiber length. SRS occurs earlier in 8 m-long-fiber due to the longer fiber length and lower SRS threshold, which gives the solitons a longer distance for SSFS and the dispersive wave for blue-shift. The self-frequency shift of solitons increases with the propagation distance [8], thus leading to a broader spectrum and a larger energy loss at long wavelength. Accordingly, as shown in Fig. 8(c), the longer gain fiber, the broader the final output spectrum, which results in a lower output power.

 

Fig. 8 The calculation results for different gain fiber lengths (a) the evolution of the spectrum along the 5 m fiber length (b) the evolution of the spectrum along the 8 m fiber length (c) the final output spectrum comparison between two fiber lengths

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In order to compare the numerical model analysis with the experimental results we carry out an experimental investigation on the SC generation with different gain fiber lengths. The scheme of the experimental setup is also shown in Fig. 4. We fix the setup of input pulse from the single-mode amplifier and vary the length of the gain fiber at the last-stage amplifier. Figure 9(a) shows the output power of the SC source versus the incident pump power under different fiber length. The output power increases almost linearly with the growth of the pump power on the condition of 5 m gain fiber. The maximum power is 18.9 W, which is higher than the one in the situations when the gain fiber is 10 m and 15 m. These results appear no difference with the numerical results. The output spectra of the SC source with maximum output power are shown in Fig. 9(b). Obviously, when the gain fiber is 5 m, the spectrum only extends to 1600 nm and the spectral range is much narrower than the other two spectra. This can be explained by the fact that the short fiber length limits the propagation distance of solitons and the range of their self-frequency shift. Those two spectra generated by 15 m-length and 10 m-length gain fiber in Fig. 9(b), both extending to 2000 nm, are similar with each other, except for a weak blue-shift dispersive wave around 800 nm in the situation of 15 m-length. This dispersive wave is an independent peak while in Fig. 8(c) (the red curve) the blue-shift dispersive wave does not depart from the broadband spectrum. This is because in the experimental setup, the output fiber is a double-clad passive fiber (about 1 m long), which gives the dispersive wave a more sufficient blue-shift distance. However, this is not considered in the simulation. Since a longer gain fiber length leads to a more dramatic long wavelength broadening caused by SSFS and more energy transfers to over 2000 nm, which is attenuated due to the significant loss of the silica fiber (as shown in Fig. 2(b)), the output power of 15 m gain fiber is only 13.9 W and lower than that of 10 m length.

 

Fig. 9 Experimental results for different gain fiber lengths (a) the output power of the SC source versus the incident pump power under different fiber lengths (b) The output spectra of the SC source with maximum output power (The legend is the maximum average output power when the 976 nm pump power is 25 W)

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From above, we can conclude that the fiber length is one of the most significant elements that influence the efficiency of Stokes components energy conversion and the frequency shift of solitons. The optimization of the fiber length plays an important role both in improving the spectral range and controlling the output power when generating SC in an YDFA. Supercontinuum generation dominantly by generation of Stokes components and their fusion in the normal dispersion region was earlier studied in passive fiber such as single-mode silica fiber [11], P₂O₅-doped [12] and germanium-doped silica fiber [13].These cascaded-SRS-dominated spectra were flat and broad due to long fiber length and high Raman gain coefficient, which leads to a low Raman threshold, significant Raman conversion and large frequency red-shifts. However in an YDFA, Raman conversion efficiency is lower in an YDFA because of the limited Yb-doped fiber length and lower Raman gain. Taking place of cascaded SRS, SSFS and MI dominate the spectral broadening in anomalous dispersion region. Compared with Raman-dominated supercontinuum, although it will take more efforts to flatten the spectrum generated directly from an YDFA, the superiority of this approach is the simple structure and high output power (even up to a hundred of watt [2]).

3.4 976 nm pump power dependence of SC

In order to find out the dependence of the spectral range of the SC upon the 976 nm pump power, the calculated spectra under different pump powers are given in Fig. 10(a) and the experimental results are shown in Fig. 10(b) and 10(c). When the pump power is up to 10W, two distinct stokes peaks are observed at 1118 nm and 1175 nm while the second-order Stokes wave cannot be clearly recognized when the pump power up to 25 W. The similar phenomenon is observed in Fig. 10(b) (when the pump power is 6.2 W). As Raman gain spectrum in silica fibers extends over a large frequency range (up to 40 THz) [8], the first-order Stokes peak is relatively broad so that the second-order Stokes peak is broader and less distinctly recognized. As shown in Fig. 10(a), with the increase of pump power, the spectrum has extended to anomalous region (14.66 W and 19.58 W spectra) and 1060 nm peak decreases than before (see in the sub graph of Fig. 10(a)). This reduction can also be observed in Fig. 10(b). It illustrates that the efficiency of the energy transferring from 1060 nm to longer wavelength through SRS is relatively high and the 1060 nm signal energy cannot be accumulated. In addition, a distinct peak at 1400-1600 nm can be found both in the simulation and the experiment due to SSFS, which is called soliton peak. Meanwhile, a wave trough around 1370 nm is observed due to the absorption of hydroxide ion. With the further growth of the pump power, the spectral components in the gain bandwidth of Yb are amplified and transfer the energy to longer wavelength through SRS (in normal dispersion region), SSFS and MI (in anomalous dispersion region). As a result, the peak power of soliton is also amplified. Since the frequency shift of solitons $\Delta \nu \propto P_{0}z$ (P₀ – peak power of solitons, z – propagation distance), the rise of peak power of solitons with the rise of pump power makes a larger frequency shift and a broader spectrum may be obtained. However, deviating from the simulation in Fig. 10(a), the spectrum does not further broaden with the increase of pump power in Fig. 10(c), when the spectrum has already extended to 2000 nm. The loss of silica fiber above 2000 nm reduces the efficiency of energy transferring and more power accumulates at 1060 nm and Stokes peaks (1100 nm – 1300 nm). That’s why these components experience an energy growth in Fig. 10(c) rather than a decline in Fig. 10(b).

 

Fig. 10 (a) The calculated spectra of the SC source when the length of the gain fiber is 5 m (b) (c) The experimental output spectra of the SC source when the length of the gain fiber is 15 m (Legend: output power (pump power))

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3.5 Pulse width dependence of SC

The pulse width (full-width at half-maximum, or FWHM) is also one of the most significant parameters that influence the quality of SC generation. We simulate the SC generation with three different pulse widths. The three pulse widths are 30 ps, 10 ps and 1 ps and the corresponding repetition rates are 0.83 MHz, 2.5 MHz and 25 MHz. The average power and peak power of the 1064 nm signal pulse is 50 mW and 2 kW respectively. Other parameters keep the same with those in Section 2.3 except the pump power is set to be 10 W in this section. As shown in Fig. 11, the average output powers for 30-ps, 10-ps and 1-ps pulse are 5.59 W, 5.75 W and 6.02 W respectively. The shorter the pulse width is, the more pulse number it needs. Because of the higher repetition rate, the temporal interval between two pulses (given by Eq. (8)) in iterative calculation is shorter and it needs more iteration times to achieve a stable output. Figure 12 shows the time and frequency domains of the last calculated pulse at the end of fiber length. Generally, a longer pulse will be more beneficial to the broadening of SC under the same condition. As mentioned in Section 3.2, the broadening of long pulses are dominated by SRS in normal dispersion region. The longer pulses have higher pulse energy and can split into more ultra-short pulses due to MI, which can generate more new frequency components in the long wavelength direction. However, for ultra-short pulse (pulse width< 1ps), the dominant nonlinear effect is self-phase modulation (SPM) rather than SRS in normal dispersion region, which results in a relatively narrower spectral broadening. Therefore, it is difficult to reach ZDW and obtain a wider SC. In conclusion, in order to generate SC with wider spectral range in a fiber amplifier when the wavelength of signal light is located in the normal dispersion region of the gain fiber, one should use a longer pulse.

 

Fig. 11 Output powers for different pulse number with (a) 30 ps, (b) 10ps and (c) 1ps pulse width.

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Fig. 12 The (a)-(c) time domain (a) and (d)-(f) frequency domain of the last calculated pulse at the end of fiber length.

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We investigate the influence of pulse width experimentally as well. As shown in Fig. 13, the experimental setup is a four-stage fiber master oscillator power amplifier (MOPA) configuration. The first stage is the fiber laser seed. Lasers with three different pulse width are used in this stage. YDFAs in the second and the third stage are identically set with those in Fig. 4. In the last stage, a 15 m long YDF, which has a 25 μm/400 μm core/cladding diameter with 0.06/0.46 NA and a 1.8 dB/m absorption at 976 nm, is used as a gain fiber. A (6 + 1) × 1 fiber combiner is used to couple the pump light of two laser diodes (220 W, 200 /220 μm output fiber). A mode field adapter is used to realize the transformation of the 15/130 μm output fiber of the isolator and the 25/400 μm input fiber of the (6 + 1) × 1 fiber combiner. An angle polished fiber end cap is utilized at the output to avoid end facet damage and reduce backward reflection. Table 2 illustrates parameters of three pulses used in the experiment. The central wavelengths of three pulses are all around 1060 nm. Meanwhile, the product of the pulse width and the repetition rate of these pulses are very close, which means the peak power of them are also close on the condition that the average powers are identical. Therefore, the influence of peak power on SC generation can be ignored. The temporal and spectral properties of seed A, B and C are shown in Fig. 14.

 

Fig. 13 The experimental setup: Isolator (ISO), bandpass filter (BPF), laser diode (LD, 976 nm), wavelength division multiplexer (WDM) and mode field adapter (MFA).

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Table 2. Parameters of Three Pulses

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Fig. 14 The temporal (a)-(c) and spectral (d)-(f) properties of seed A, B and C

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Figure 15(a) indicates the spectra and output powers from the amplifiers in the third stage of the three seed signals. The corresponding peak power of seed A, B and C are 17.6 kW, 18.3 kW and 12 kW respectively. The first-order Raman peaks are observed in these three spectra. However, a distinct peak only appears in spectrum-A while in spectrum-B and spectrum-C this Raman peak cannot be clearly identified due to the amplified ASE. Even though a band pass filter is used in the single-mode amplifier to reduce ASE pedestal of signal B and C (as shown in Fig. 13(e) and 13(f)), a weak ASE component can also be amplified in the second amplifier. Moreover, the relatively low repetition rate of signal B and C also cause difficulty of suppressing ASE. The output powers of three seed signals under different power of the last-stage amplifier are shown in Fig. 15(b), which indicates that the slope efficiencies are identical with each other. The output power is finally up to 300 W in three situations. In Fig. 15(c) and 15(d) the spectrum of signal A at different output powers are depicted. Similarly with Fig. 10(b) and 10(c), a weak 1060 nm peak at low output power (shown in Fig. 15(c)) and a distinct 1060 nm peak at high output power (shown in Fig. 15(d)) are observed due to the same reason. Finally, the spectrum ranging from 1000 nm-2000 nm is obtained when the output power is up to 300W. The spectral variation trend of signal B and C are similar with signal A, so herein we only show the spectrum of three signal pulses under the maximum output power in Fig. 15(e). As shown in the sub graph of Fig. 15(e), the signal peak around 1060 nm of seed C is the highest among three signals. This can be explained by the fact that signal C has the lowest repetition rate and the most intense ASE among three signals. As ASE source is a continuous wave with low peak power, despite of its power amplification in the last-stage amplifier, it cannot achieve SRS threshold and will results in the energy accumulation around 1060 nm as well as the difficulty of transferring energy to long wavelength.

 

Fig. 15 Experimental results of different signals: (a) the spectrum and output power of three seed signals after two preamplifiers (b) the output power of three signals versus pump power from the last-stage amplifier (c) (d) output spectra from the last-stage amplifier pumped by seed-A with corresponding output powers (e) the comparison of the final spectra with maximum output power pumped by seed A, B and C.

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Above all, pulse width influences little on SC generation from an YDFA experimentally, which is not in accordance with simulation results. The reasons are as follow: firstly, we only investigate the long pulse owing to the limitation of experimental conditions while the physical mechanisms of SC generation pumped by long pulses are identical. Secondly, the seed signal A with 9-ps width is amplified in two preamplifiers, which have led to an initial pulse broadening before entering the last-stage amplifier. Thirdly, all of the spectra have already extended to 2000 nm, which is almost the upper limit because of the loss of silica, so the difference at longer wavelength among three signals cannot be indicated. Besides, the measurement error can also leave out some details. For example, an inapparent dispersive wave is observed around 850 nm. Generally,the longer pulse width leads to the higher intensity at short wavelength, which agrees with the simulation results.

4. Conclusions

We have numerically investigated SC generation in a nonlinear YDFA with a combined model of the laser rate equations and GNLSE. The mechanism of SC generation by this model corresponds well with our previous experimental investigation. The dominated spectral evolution mechanism is SRS in normal dispersion region and MI, higher-order soliton fission and SSFS in anomalous dispersion region. In addition, our model cannot only predict the output power and spectral broadening of the SC but also describe the dynamic process of pulse amplification. The simulation results also indicate that the length of the gain fiber, the pump power and the pulse width play key roles in SC generation. All of these parameters can be well-designed before experimental operation by using our numerical method. Further understanding the mechanism of SC generation in a nonlinear YDFA should allow us to design a more proper and stable fiber amplifier system to generate near-infrared SC for various applications.