1. Introduction

Nanosecond pulses with high pulse energy and high peak power have many important applications in industry [1] and scientific researches [2]. Fiber lasers and amplifiers are very advantageous in delivering those pulses, for their compactness, robustness, good thermal management, high single-pass gain, diffraction-limited output beams and high optical-to-optical efficiency, compared to other solid-state waveguide lasers [3]. However, in fibers, the tight confinement of the optical field over a considerably long interaction length makes the system susceptible to nonlinear effects [4]. The nonlinear effects such as the stimulated Brillouin scattering (SBS), stimulated Raman scattering (SRS) and the self-phase modulation (SPM) indeed constitute the main performance limitations of fiber lasers and amplifiers. Different from the SBS and SRS which have sharp power thresholds [5], the SPM effect which results in spectral broadening has no threshold, and occurs in fiber amplifiers ubiquitously.

As many applications are sensitive to the bandwidth of pulses, such as pulse compression [6], gravitational wave sensing [7], narrowband and broadband second harmonic generation [8,9], it is necessary to describe the SPM effect quantitatively. Methods have been developed for coherent pulses [10,11] and incoherent pulses [12–15] respectively. In the literatures of SPM mentioned above, the amplification of pulses in the fiber amplifiers is generally simplified to be the small-signal amplification, in which condition the gain is uniform along the fiber. However, to achieve pulses with high pulse energy and high peak power, the fiber amplifiers often work in the condition of gain saturation, where the gain is not uniform along the fiber and pulse shape deformation occurs.

In this paper, the evolution of pulse power in the fiber amplifiers with gain saturation and its influence on SPM effects is analyzed. Experiments are conducted to validate our theoretical analysis for incoherent nanosecond pulses in a super-luminescent diode (SLD) seeded cascaded fiber amplifier system where nanosecond pulses with high peak power and high pulse energy are generated and gain saturation occurs. Qualitative agreements between theoretical analysis and experimental results are achieved. Moreover, at high peak power intensity (> 1 GW/cm²) where higher order nonlinear terms need consideration, we also developed an effective and easy approach, with which the SPM effect can be predicted precisely.

2. Theoretical analysis for SPM with gain saturation

In this section, a summary of the models of the SPM-induced spectral broadening for both coherent and incoherent pulses is presented. Then, based on laser rate equations in fiber amplifiers, the models are generalized to be applicable for pulses with gain saturation.

In typical Yb-doped high power fiber amplifiers, the dispersion of nanosecond pulses can be neglected due to the low dispersion coefficient ($~{20}_{}^{}p{s}^{2}k{m}^{-1}$) and the relatively short fiber length (~several meters). When the peak power of pulses is below the thresholds of SRS and SBS, we take that the evolution of nanosecond pulses in the fiber amplifiers is governed by the gain and the SPM. Thus the nonlinear Schrodinger (NLS) equation that governs propagation of nanosecond pulses in the optical fibers is given by

The solution of Eq. (1) can be written as

For coherent pulses, the output pulse spectrum $I(z,\omega )$ can be obtained directly by taking the Fourier transform of$A(z,T)$ i.e.

As to incoherent pulses, according to the Khinchine theorem [16], the spectral distribution is the Fourier transform of the first-order electric-field correlation function $K(z,\tau )$ of the output pulses. Based on Manassah model [12,13] and its improved form [14], the output spectrum for incoherent pulses can be written as

In most literatures, the amplification in the fiber amplifiers is simplified to be the small-signal amplification, where $g(z,T)$ is a constant, i.e. the small-signal gain $g_0$, which is only dependent on the pump condition, not relevant to $z$ and $T$. In the fiber amplifiers, if the amplified spontaneous emission (ASE) can be neglected, the initial inversion distribution is uniform along the fiber. So, in the condition of small-signal amplification, the power of pulses grows exponentially with the fiber length and the gain is uniform during the pulse duration. Then it is straightforward to obtain$L_{eff}(z)=\frac{G_0(z)-1}{\ln (G_0(z))}z$, where $G_0(z)=\exp (g_{0}z)$ is the power gain of the small-signal amplification.

However, when the pulse energy in the fiber amplifiers approaches and exceeds the saturation energy of the fiber, i.e. the fiber amplifiers work in the condition of gain saturation, the gain $g$ is not uniform, but a function of $z$ and $T$. Thus the power gain and the effective length calculated in small-signal amplification assumption will induce error in predicting the SPM effect with gain saturation.

Based on laser rate equations, the pulse power evolution in fiber amplifiers can be written as [17,18]

Then we can obtain the generalized expression of the power gain and the effective length of pulses as a function of $z$ and $T$,

Different from the condition of small signal input, with a given fiber length of $z$, the effective length and the power gain both are functions of T, which agrees with the fact that pulse deformation emerges in the condition of gain saturation.

With the generalized $G(z,T)$ and $L_{eff}(z,T)$ in Eqs. (8) and (9), Eqs. (5) and (6) can be utilized to calculate the output spectrum of coherent pulses and incoherent pulses with any given temporal and spectral shapes in the fiber amplifiers with gain saturation.

3. Comparisons with experimental results

3.1 Experimental setup

A cascaded fiber amplifier which delivers nanosecond pulses with high pulse energy and high peak power has been built to demonstrate the theoretical analysis for incoherent pulses above. The experimental setup is shown in Fig. 1. The seed is provided by an electric-pulse-driven SLD emitting a rectangular 200-ns-duration signal centered at 1062 nm. The seed diode pulses are firstly amplified in the Yb-doped fiber preamplifiers. After pre-amplifiers, the pulse energy of output pulses can achieve 0.840 μJ. And no pulse shape deformation and spectral broadening is observed.

 

Fig. 1 Experimental setup

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Then the pulses are launched into a 3.0 m-long Yb-doped fiber with a 15-μm-diameter, 0.087 NA core and 130-μm-diameter, 0.46 NA cladding. The output pulse energy of 180 μJ can be achieved in this amplifier stage. The amplification in this amplifier stage is no longer in the small-signal amplification condition, and pulse shape deformation emerges which implies gain saturation [see the inset of Fig. 2(a)]. At the meantime, spectral broadening is observed which will be used to demonstrate our theoretical analysis [see Fig. 2].

 

Fig. 2 (a) The power gain evolution along the fiber calculated with gain saturation (solid) and with small-signal (dashed) model for leading edge and trailing edge of the pulse respectively; the inset is input (blue dashed) and output pulse shape (red solid) of the 15/130 μm fiber amplifier stage, and the predicted output pulse shape (red dot-dashed) with Eq. (7); (b) the effective length as a function of T with (red dot-dashed) and without gain saturation (blue dashed) in consideration, and the difference between two models (black solid); (c) the measured and calculated output spectrum with small-signal model and gain saturation model together with the measured input spectrum; (d) the measured spectral broadening factors as a function of peak power, compared with the calculated results from our extended model and small-signal model.

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The amplified pulses are then coupled into the main amplifier stage by two lenses. The fiber used in the main amplifier stage is a 3 m-long photonic crystal fiber (PCF) with 40-μm-diameter, 0.03 NA core and 200-μm-diameter, 0.55 NA cladding. The main amplifier is end backward pumped with a 975-nm laser diode. Moreover, a 10-nm 3-dB-bandwidth centered at 1062 nm filter is inserted between the two lenses to filter the residual pump, ASE and SPM-induced spectral broadening in the 15/130 μm stage. The output pulses can achieve pulse energy of 0.940 mJ and peak power of 50 kW. In this amplifier stage, both spectral broadening and pulse shape deformation are observed which can also be utilized to analyze the influence of gain saturation on SPM effect [see Fig. 3].

 

Fig. 3 (a)At pulse energy of 0.324 mJ, the RMS bandwidth of the output spectrum as a function of $\gamma \text{'}$; the inset is experimental results (red star) compared with the predicted results of our original model (black dot); (b) the measured output spectrum at peak power of 25 kW, compared with the predicted spectrum calculated by gain saturation model and the small-signal model with $\gamma \text{'}$; (c) the measured output spectrum at peak power of 50 kW, compared with the predicted spectrum calculated by gain saturation model and the small-signal model with $\gamma \text{'}$; (d) the measured and calculated broadening factors as a function of peak power, and compared with the calculated results in small-signal model.

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3.2 Comparisons in the 15/130 μm fiber amplifier stage

In this section, we report the experimental spectral broadening of the pulses amplified in the 15/130 μm Yb-doped double-cladding fiber amplifier stage and compare them with our theoretical analysis in Section 2. In this amplifier stage, the input pulse energy is 0.840 μJ, and the pulse shape is rectangular, as mentioned above.

The amplification of pulses in the fiber amplifier at the output pulse energy of 100 μJ is analyzed in details below. The output pulse shape is shown in the inset of Fig. 2(a), where significant pulse distortion due to gain saturation is observed. Equation (7) is used to predict the output pulse shape distortion due to gain saturation with the saturation energy $E_{sat}$ of 85 μJ for the fiber ($A_{eff}$~170 μm² and $({\sigma }_{es}+{\sigma }_{as})$~0.38 pm²). To calculate the small-signal gain $g_0$ in Eq. (7), the Frantz-Nodvick model$E(L)=E_{sat}\ln \left\{\exp (g_{0}L)[\exp (E(0)/E_{sat})-1]+1\right\}$ [17] is utilized, where$E(0)$, $E(L)$are the input and output pulse energy respectively.

To gain more insight, we can take that the pulse is formed by many intervals with duration $\Delta \tau$. The pulse deformation implies that the power gain for each interval is different, obviously the power gain evolution along the fiber for each interval is also different. We can use Eq. (8) to predict the actual evolution of power gain along the fiber for each interval in the condition of gain saturation. We demonstrate the evolution of power gain along the fiber for leading edge and the trailing edge of the pulse calculated by Eq. (8) in Fig. 2(a).

To make a comparison, the evolutions of power gain estimated by small-signal model for the two intervals are also shown in Fig. 2(a). It is easy to find that at the leading edge of pulses, the difference between the gain saturation and small-signal model is negligible. However, at the trailing edge of the pulse, there is a drastic difference between the two calculative results, which is consistent with the origin of gain saturation.

As the effective length is calculated by integrating the power gain over the fiber length, the effective length is calculated by Eq. (9) and demonstrated in Fig. 2(b), which is a function of T. To make a comparison, we also use the small-signal model to estimate the effective length for each interval, i.e.$L_{eff}(T)=\frac{G(T)-1}{\ln (G(T))}L$, which is also shown in Fig. 2(b). We can see the difference (defined as the differences between the two models divided by the calculative results of the gain saturation model) can be as large as 30%. Thus, in fiber amplifiers with gain saturation, utilizing small-signal model to calculate the effective length will induce large error. It is necessary to take the gain saturation into consideration and use Eq. (9) in calculating the effective length.

With Eq. (6), we can predict the output spectrum, at the output pulse energy of 100 μJ, as shown in Fig. 2(c), with the nonlinear coefficient of 0.67 W⁻¹km⁻¹, which is obtained from the formula $\gamma =2\pi n_2/(\lambda A_{eff})$ [10] where $n_2$ is the nonlinear index (2.0$\times$10⁻²⁰ m²/W), $A_{eff}$ is the effective area for the fiber (~170 μm²) and $\lambda$ is the center wavelength of signal pulses (1.062 μm).The measured output spectrum and the input spectrum are also shown in Fig. 2(c). The input spectrum in our system is broadband whose shape is nearly rectangular, with some ripples at the top which mainly comes from the interference in the components used in the seed and pre-amplifier system. Due to the SPM, the measured output spectrum shows broad tails at two sides of the central part. The calculated spectrum has the same characteristics. To make a comparison, the predicted spectrum with the small-signal model is also shown in Fig. 2(c). From the figure, we can find that our extended model can achieve a better agreement with the measured spectrum than the small-signal model does.

To evaluate the agreement of the measured and the calculated spectrum, the root-mean-square (RMS) bandwidth $\Delta {\lambda }_{rms}={(\frac{{\displaystyle \int {(\lambda -\overline{\lambda })}^{2}I(\lambda )d\lambda }}{{\displaystyle \int I(\lambda )d\lambda }})}^{1/2}$ where $\overline{\lambda }=\frac{{\displaystyle \int \lambda I(\lambda )d\lambda }}{{\displaystyle \int I(\lambda )d\lambda }}$ is utilized here. The measured RMS bandwidth of 3.24 nm with our extended model has a good agreement with the calculated RMS bandwidth of 3.22 nm, with a broadening factor (the output spectrum bandwidth divided by the input spectrum bandwidth) of 1.2 compared to the input RMS bandwidth of 2.70 nm.

In this amplifier stage, the spectral broadening factors at different output peak power are also measured, which are shown in Fig. 2(d). The predicted spectral broadening factors with our extended model with gain saturation, compared with the results calculated by the small-signal model, are also shown in Fig. 2(d). We can find the spectral broadening factor grows with the peak power, and our extended model provides a more precise prediction than the small-signal model.

However, it is noticeable that at the maximum output pulse energy of 180 μJ (with the corresponding peak power of 2.4 kW), both our extended model and the small-signal model fail in predicting the spectrum broadening, the error can be as large as 14%. The similar failure are also reported in Ref [14], but without explanations. In our analysis, the most likely reason is that due to the high peak power intensity in the fiber (~1 GW/cm²), the higher order nonlinear terms of the NLS equation will influence the nonlinear process in the fiber [10]. We will demonstrate an efficient approximation of our extended model in predicting the SPM effects of pulses with high peak power intensity in Section 3.3.

3.3 Comparisons in the PCF amplifier stage

The pulses are coupled into the PCF and amplified further. The shapes of the amplified pulses deform seriously as the amplified pulse energy increases, which implies that the amplification in the PCF also fits the gain saturation condition.

However, the predicted results with our extended model in Section 2 are always larger than the experimental results at the same peak power, as shown in the inset of Fig. 3(a). And the differences between the measured broadening factor and the predicted values increase with the peak power. The main reason might be that the peak power intensity in the PCF can achieve as high as 7 GW/cm², where the higher order nonlinear terms need to be considered in the NLS equation. A recommended approach is replacing the nonlinear coefficient $\gamma$ in Eq. (1) by $\gamma \text{'}=\gamma (1-b_s{\left|A\right|}^2)$, where $b_s$ is the saturation parameter governing the power level at which the nonlinearity begins to saturate [10]. However, to obtain the exact solution of the NLS equation with the replacement above (i.e. the cubic-quintic or quintic NLS equation) is complicated.

In our experiments, we notice that limited to the extraction ability of our experimental system, the peak power of pulses delivered by our PCF amplifier lays in a relative narrow range (from 8 kW to 50 kW). So, we attempt to utilize a modified constant $\gamma \text{'}$ to replace the nonlinear coefficient $\gamma$ in our extended model to predict the SPM-induced spectral broadening in our experimental condition.To validate the $\gamma \text{'}$ in our experimental condition, at the output pulse energy of 0.324 mJ, we show the calculated output bandwidth as a function of the value of $\gamma \text{'}$ utilized in the calculations in Fig. 3(a). We can find that the bandwidth increases with the value of $\gamma \text{'}$ monotonously. The calculated value is the closest to the measured RMS bandwidth of 3.15 nm, when $\gamma \text{'}$ is around 0.05 W⁻¹km⁻¹. This value is smaller than the original nonlinear coefficient calculated by the formula $\gamma =2\pi n_2/(\lambda A_{eff})$.

With the $\gamma \text{'}$ value of 0.05 W⁻¹km⁻¹, we calculated the output spectrum of at different peak power of output pulses, which are shown in Figs. 3(b) and 3(c), along with the measured spectra. The figure shows good agreements of the calculated and the measured spectra at different output peak power. The differences between the RMS bandwidths of the measured and calculated spectrum are 0.8% and 0.5% at the peak power of 25 kW and 50 kW respectively.

To make a comparison, the calculated spectra with the small-signal model are also shown in Figs. 3(b) and 3(c). From the figures, we can find that our extended model have a clear advantage in predicting the SPM-induced spectral broadening. The differences between the RMS bandwidths of the measured and calculated spectrum with the small-signal model are 13.1% and 21% at the peak power of 25 kW and 50 kW respectively, which are far beyond the calculative results with our extended model.

The corresponding broadening factors as a function of peak power are shown in Fig. 3(d). The SPM-induced spectral broadening increases with the peak power of the output pulses, and an excellent agreement of the calculated and the measured results is shown in the figure. This implies that in our experimental condition, with the $\gamma \text{'}$ approximation our extended model can be utilized to predict the SPM effect of pulses with peak power intensity beyond 1 GW/cm².

The spectral broadening factors calculated by the small-signal model at different peak power are also shown in Fig. 3(d), with the same $\gamma \text{'}$ approximation method. The difference between the experimental results and the results calculated by the small-signal model grows with the peak power and can be as large as 21% at peak power of 50 kW. It indicates that the small-signal model can’t predict the SPM-induced spectral broadening at high peak power intensity, even with the $\gamma \text{'}$ approximation method.

In this section, with our model, we obtain an effective nonlinear coefficient $\gamma \text{'}$ for SPM-induced spectral broadening in pulses with high peak power intensity (> 1 GW/cm²) in our experimental condition, and with $\gamma \text{'}$, the output pulse spectra at different peak powers are predicted precisely with our extended model.

4. Summary

In this paper, we take gain saturation into consideration in analyzing the SPM-induced spectral broadening for coherent and incoherent nanosecond pulses in fiber amplifiers. Based on the laser rate equations and the NLS equation, the extended models of SPM effects with gain saturation are obtained. Then we validate our theoretical analysis for incoherent nanosecond pulses in a SLD seeded cascaded fiber amplifier system where nanosecond pulses with 50 kW peak power and 1 mJ pulse energy are generated and gain saturation in the amplification occurs. The excellent agreement of the calculated and experimental results are obtained at low peak power intensity (< 1 GW/cm²). At high peak power intensity (> 1 GW/cm²) the influence of higher order nonlinear terms on SPM is not negligible any more. An efficient and easy approach is also demonstrated with our extended model to predict the SPM-induced spectral broadening precisely at high peak power.