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Abstract
Using numerical simulation, we investigate the formation of noise-like pulses with an extremely broadband spectrum in passively mode-locked fiber lasers. The mechanism of spectral broadening is due to the Kerr nonlinearity of the fibers making up the laser cavity. It is found that optimization of the nonlinear-dispersion parameters of the fiber cavity can lead to the generation of noise-like pulses with a spectral width exceeding the gain bandwidth by a factor of more than 10.
© 2021 Optical Society of America
1. INTRODUCTION
Passively mode-locked fiber lasers are widely used in various areas of science and technology. These lasers have unique capabilities that provide a basis for further improvement and development of sources of light pulses. An important advantage of these lasers for various applications is the great variety of their lasing regimes. In addition to the regular single-pulse mode-locking generation, these lasers can operate in various multiple-pulse regimes and exhibit hysteresis phenomena, bistabilities, and multistabilities (bistability between a continuous wave and mode-locked or $Q$-switched regimes, a multistable dependence of the number of identical solitons in the laser cavity on the initial lasing conditions, a multihysteresis dependence of the number of solitons on pump power and other laser parameters, etc.) [1–8].
Lasers operating in a noise-like pulse regime [9–17] occupy a special place among fiber lasers with a pulsating radiation pattern. Noise-like pulses have good space–time localization and are stable structures despite the instability of the solitons of which they consist. The stabilization mechanism of noise-like pulses was determined in [18]. These lasers are of interest as sources of high-energy pulses with a broad emission spectrum, whose width exceeds the spectral width of the gain band. The widest spectra of noise-like pulses with a width of 203 nm have been obtained in an erbium fiber laser [19]. Such lasers have potential applications in optical coherent tomography, super-continuum generation, laser machining, etc. [20–22]. The question of the potential of fiber lasers of noise-like pulses for generating extremely wide emission spectra remains open.
The aim of this study is to determine an optimal model for the formation of noise-like pulses in fiber lasers to obtain the widest spectra of the generated radiation. In the model analyzed here, the broadening mechanism of pulse spectra is associated with the Kerr nonlinearity of the fibers forming the laser cavity. Our model is based on the Gaussian spectral dependence of the gain, since the approximation of the quadratic gain dispersion is incorrect for this problem.
This paper is organized as follows. In Section 2, we present the physical model and the master equations used to study the lasing dynamics of noise-like pulses. Section 3 is devoted to numerical simulation of the formation of noise-like pulses and a discussion of the obtained results. The main conclusions are given in Section 4.
2. PHYSICAL MODEL AND MASTER EQUATIONS
The investigated ring laser is schematically represented in Fig. 1. It includes a gain fiber (Amplifier), a fiber with anomalous dispersion (An Disp) for obtaining a noise-like pulse regime, a dispersion-free fiber (${\rm Disp} = {0}$) for producing a significant frequency chirp and hence a wide spectrum of generated pulses, an output coupler (OC) for extracting part of the radiation from the laser, and a fiber with nonlinear losses (NL) to form pulses. The connections of these fibers are labeled with numbers 1 to 4.
To describe the evolution of radiation in the anomalous-dispersion and dispersion-free fibers, we use the following normalized nonlinear Schrödinger equation [23,24]:
The fiber with NL (of length ${L_l}$) provides passive mode locking in the investigated laser. These losses are given by the relation [18]
where ${\sigma _1} = {p^2}/4{p_2}$. The dependence of the NL ${\sigma _n}$ on the intensity $I$ is shown in Fig. 2. At low intensity, the NL decrease with increasing intensity, reach zero at $I = {I_m}$, where ${I_m} = p/2{p_2}$, and then increase. This dependence of the NL ${\sigma _n}$ on the intensity $I$ is qualitatively similar to the corresponding dependence in the nonlinear polarization rotation technique [7]. The total cavity linear losses $\sigma$ are determined by the linear losses ${\sigma _1}$ and the linear losses associated with the output of radiation from the cavity ${\sigma _0}$ ($\sigma = {\sigma _0} + {\sigma _1}$). The field evolution in the fiber with NL ${\sigma _n}$ is described by the equation Equation (3) also takes into account the decrease in the amplitude $E$ due to the linear losses ${\sigma _0}$.When the width of the radiation spectrum becomes comparable to the gain bandwidth, the analysis of the field amplification requires a more accurate model than the model with quadratic gain dispersion. To analyze the evolution of the field in the gain fiber of length ${L_g}$, we use a model with a Gaussian spectral gain profile. The change in the spectral components of the field ${E_\omega} = \int E(\tau)\exp (i\omega \tau){\rm d}\tau$ in the gain fiber is described by the following equation [25]:
where ${F_\omega}$ is the Gaussian spectral gain profile and $g$ is the gain for the central spectral component with $\omega = 0$. For the gain-saturation model [see Eq. (6)] used, the gain $g$ is independent of $\tau$, but depends on $\zeta$. The dimensional frequency ${\omega ^\prime}$ is determined from the normalized frequency $\omega$ by the following relation: ${\omega ^\prime} = \omega /\delta \tau$. Expression (5) implies that the carrier frequency ${\omega _0}$ corresponding to $\omega = 0$ is indeed the maximum of the gain spectral profile. For profile (5), the full gain bandwidth $\Delta {\omega _0}$ at half maximum ${F_\omega}(0)/2$ is given by the expression $\Delta {\omega _0} = 2 \sqrt {(\ln 2)/{D_r}}$. Taking into account the influence of the spectral contour ${F_\omega}$ on the gain saturation leads to the following equation for the gain [26]:When the width of the generated radiation spectrum is small compared to the gain bandwidth (${D_r}{\omega ^2} \ll 1$), expression (5) becomes the equality ${F_\omega} \approx 1 - {D_r}{\omega ^2}$, and after the Fourier transform of Eq. (4), we obtain the well-known equation
In the case of generation of radiation with a broadband spectrum $\delta \omega \ge \Delta {\omega _0}$, the model based on Eqs. (7) and (8) yields an underestimated spectral width $\delta \omega$ of the generated radiation and an underestimated value of the gain $g$. Indeed, in this case, Eq. (4) transforms into the following equation:
and for the spectral components $|\omega | \gt 1/\sqrt {{D_r}}$ (i.e., $|\omega | \gt 0.6\Delta {\omega _0}$), the amplifying medium acts as an absorbing medium, which is certainly not correct.The proposed generation model described by Eqs. (1)–(6) allows us to reveal the role of individual intracavity laser elements in the evolutionary dynamics of noise-like pulses and to adequately estimate the broadening of their spectrum due to the nonlinearity of the refractive index of the cavity fibers and other intracavity elements. All characteristic changes in the noise-like pulse in each of the laser elements (Fig. 1) are presented in Fig. 4.
3. RESULTS OF NUMERICAL SIMULATION AND DISCUSSION
The following values of the laser parameters were used in the numerical simulation: ${L_a} = 11\;{\rm m} $, ${L_g} = {L_a}$, ${L_l} = {L_a}$, ${L_f} = 4{L_a}$, ${D_a} = 0.5$ (${\beta _2} = - 0.022\;{{\rm ps}^2}\,{{\rm m}^{- 1}}$), ${D_f} = 0$, ${q_a} = 0.5$, ${q_f} = 2$, $\gamma = 3 \times {10^{- 3}} \;{{\rm W}^{- 1}}{{\rm m}^{- 1}}$, ${I_0} = 3.8 \;{\rm W}$, $a = 20$, $b = 0.2$, ${\sigma _1} = 5$, ${\sigma _0} = 0.05$, ${I_m} = 5$, and ${D_r} = 0.05$ ($\Delta {\omega _0} = 7.4$). The dimensionless time interval $\delta \tau = 1$ corresponds to 1 ps. The intensity interval $\delta I = 1$ corresponds to 3.8 W. The dimensionless spatial interval $\delta \zeta = 1$ is one round-trip period. The choice of the laser parameters is due to the requirement that they should be as close as possible to those of real erbium-doped fiber lasers operating in the noise-like pulse regime [15,28]. With a change in laser parameters, the units of time $\delta t$ and intensity ${I_0}$ are easily recalculated.
For numerical simulation, we used the standard split-step Fourier method based on splitting the nonlinear-dispersion problem into nonlinear and dispersion parts [29]. The functions $E(\tau)$ and ${E_\omega}$ were determined through each other by means of the fast Fourier transform. After sequential passage of a noise-like pulse through the fiber with anomalous dispersion, the dispersion-free fiber, the fiber with NL, and the gain fiber, the change in the pulse was determined using Eq. (1) with parameters of the fiber with anomalous dispersion, Eq. (1) with parameters of the dispersion-free fiber, Eq. (3), and Eq. (4), respectively. The time size of the computational box was chosen to be much larger than the width of the noise-like pulse. The amplitude $E(\tau)$ was subject to periodic boundary conditions. In this way, we excluded the edge artifact effects associated with the uncontrolled movement of the noise-like pulse along the time axis $\tau$. At each spatial step $d\zeta$ for the input temporal distribution $E(\tau)$, we determined the change in this distribution. The computed field was used as a new input temporal distribution of the field for the next spatial step $d\zeta$. The temporal size of the numerical box was 204.8, i.e., $0 \le \tau \le 204.8$. The parameters of the equations were chosen so as to make possible the generation of a noise-like pulse. The initial distribution of the field was a Gaussian pulse with varying pulse duration and with a peak amplitude that exceeds the level of zero transmission for the NL. For such initial conditions, the steady-state solution did not depend on the variations of duration of initial pulse. The steady state is an attractor for a wide range of initial conditions.
Figure 3(a) shows the noise-like pulse regime established after the transient process. In other fragments of the computational time box, which are not shown in the figure, there are no solitons. As the numerical simulation shows, the noise-like pulse is a very stable formation despite the permanent chaotic motion, change, creation, and disappearance of the solitons forming this pulse. As can be seen in Fig. 3(b), the time distribution of the intensity in the noise-like pulse has a pedestal. From this pedestal, new solitons are formed instead of disappearing solitons, resulting in spatial–temporal localization of noise-like pulses [18].
Fig. 3. (a) Time distribution of intensity $I(\tau)$ versus number of round-trips $\zeta$ for the steady-state noise-like pulse regime. (b) Time distribution of intensity $I(\tau)$ on the logarithmic scale $\lg (I(\tau)/{I_{{\max}}})$ for $\zeta = 1000$. ${I_{{\max}}}$ is the maximum intensity. The intensity distributions are shown for cavity point 4 in Fig. 1.
With increasing pump power $a$, the duration of the noise-like pulse in Fig. 3(a) and its energy increase linearly without changing the average peak intensity of the solitons forming the pulse. Therefore, theoretically, the energy of a noise-like pulse can be infinitely increased, which provides an effective way to obtain a high-energy pulse without its destruction. Previously, this behavior of a noise-like pulse was demonstrated experimentally and by numerical simulations using a quadratic-dispersion–gain model [11]. These properties of a noise-like pulse are similar to those of a rectangular pulse under the dissipative soliton resonance: with increasing pump power, the duration of a rectangular pulse and its energy increase monotonically, while its amplitude remains unchanged [30,31].
The investigated laser consists of interconnected fibers with different nonlinear-dispersion parameters. As a result, the pulses propagating in the laser cavity are subjected to a periodic impact, which leads to stochastization of the generated radiation and the formation of noise-like pulses [32,33]. The obtained results on the properties of the transient process and the structure of noise-like pulses are similar to the corresponding results for a laser with a distributed medium [18] in which the impact mechanism of the formation of noise-like pulses does not occur, and their formation is due to the quintic nonlinearity of the refractive index. That is, despite the different mechanisms of formation of noise-like pulses, they have similar properties.
Fig. 4. Time dependence of intensity $I(\tau)$ at the end points of the four fibers of the laser cavity through which the pulse passes successively with a change in $\zeta$ from 200 to 201. 1: end point of the fiber with nonlinear losses; 2: end point of the gain fiber; 3: end point of the fiber with anomalous dispersion; 4: end point of the dispersion-free fiber (see Fig. 1).
Figure 4 shows the change in the intensity distribution $I(\tau)$ of a noise-like pulse as it moves along the cavity. The distributions are presented for the end points of each of the four fibers through which the pulse passes sequentially (points 1, 2, 3, and 4 in Fig. 1). When the noise-like pulse passes through the fiber with the NL shown in Fig. 2, low-intensity solitons decay, and the intensities of powerful solitons decrease and get close to ${I_m}$ (distributions for points 1 in Fig. 4). After passing through the gain fiber, the intensities of solitons become much greater (distribution for point 2), and they are converted to multisoliton pulses. In the fiber with anomalous dispersion, multisoliton pulses are split into individual solitons of lower intensity [34]. Here stochastization of the multisoliton ensemble occurs, which is associated with the complicated interaction and oscillations of solitons (distribution for point 3 in Fig. 4). The duration of solitons decreases due to the anomalous dispersion and nonlinearity of the refractive index. At points 2, 3, and 4, the noise-like pulse has the same energy. In the dispersion-free fiber, the distribution of the pulse intensity $I(\tau)$ remains unchanged. As a consequence, the distributions $I(\tau)$ at points 3 and 4 coincide. This result also directly follows from the solution to Eq. (1) with ${D_i} = 0$. However, in this fiber, a significant frequency chirp of solitons occurs due to the nonlinearity of the refractive index. As a result, the spectrum width of the noise-like pulse also increases significantly [spectral dependence 4 in Fig. 5(a)]. The wings for the spectral distributions at cavity points 1, 2, and 3 turn out to be approximately the same.
Fig. 5. (a) Averaged spectral distributions on the logarithmic scale $\lg ({\bar I _\omega})$ for the noise-like pulse at cavity points 1–4. Averaging was performed over 1000 passes of the field through the cavity. (b) Central parts of the distributions on an enlarged scale. The dashed curves describe the spectral profile of the gain [see Eq. (5)].
Figure 5(b) shows the changes in the central part of the spectral distribution of the noise-like pulse as it passes through the cavity fibers. In the gain fiber, distribution 1 is transformed into distribution 2. In this case, the spectral components of the radiation lying in the spectral gain band increase significantly. The spectral components lying outside the spectral gain band remain unchanged. As can be seen from curves 2 and 3, in the steady-state lasing regime, the averaged spectral distribution of the noise-like pulse in the fiber with anomalous dispersion is unchanged. In this fiber, solitons undergo periodic oscillations. As a result, the soliton characteristics averaged over the ensemble of solitons forming the noise-like pulse and over various statistical realizations of this ensemble over 1000 cavity passages turn out to be unchanged. In the dispersion-free fiber, an increase in the frequency chirp leads to the transformation of a significant part of the energy of the field spectral components lying in the spectral gain band to the energy of the spectral components corresponding to the wings of spectral distribution 4. In the fiber with NL, the intensities of the spectral components of the field decrease and, as a result, spectral distribution 4 is transformed into distribution 1.
Fig. 6. (a) Instantaneous spectral distribution ${I_\omega}$ of the noise-like pulse at $\zeta = 1100$ [see Fig. 3(a)], (b) averaged spectral distribution ${\bar I _\omega}$, and (c) averaged spectral distribution on the logarithmic scale $\lg ({\bar I _\omega})$; the black circles correspond to the maximum spectral intensity of the broadband spectrum and its half value. The thin solid curve shows a parabolic dependence that approximates the spectral distribution of the broadband fraction of radiation in the vicinity of its maximum. The averaging area is $\delta \zeta = 1000$. The dashed curves describe the spectral profile of the gain.
Figure 6(a) shows an instantaneous spectrum of the noise-like pulse in point 4, which stochastically changes from one passage of the pulse through the cavity to another. As can be seen in Figs. 6(b) and 6(c), the total output radiation consists of two fractions: broadband radiation and the radiation filling the spectral profile of the gain band. The latter is associated with the usual amplification of broadband radiation in the gain fiber due to the stimulated emission of photons by atoms of the amplifying medium. Conversion of this radiation to broadband radiation is associated with the frequency chirp of noise-like-pulse solitons in the dispersion-free fiber. The occurrence of the chirp is due to the Kerr nonlinearity of the fiber. The spectrum of a significant part of the radiation of the noise-like pulse (80% of the total pulse energy equal to 71) lies outside the spectral gain band shown by the dashed curve in Figs. 6(b) and 6(c). Approximating the upper part of the spectral profile of the broadband radiation in Fig. 6(c) by a parabolic function, we find the maximum value of the spectral intensity, its half value, and the spectral width (FWHM) for this part of the radiation. For broadband radiation, the spectral width is $\Delta \omega \approx 90$. Accordingly, this width is 12.2 times greater than the spectral gain bandwidth $\Delta {\omega _0} = 7.4$.
Let us give an estimate for the possible spectral width of radiation $\Delta \omega$ in the studied laser. This width is determined mainly by the change in frequency along each of the solitons forming the noise-like pulse as they pass through the dispersion-free fiber. The frequency change is associated with phase modulation due to Kerr nonlinearity. Thus, we get $\delta \omega \approx 2\delta \varphi /({\tau _p}/2)$, where ${\tau _p}$ is the soliton duration, and $\delta \varphi$ is the phase difference between the soliton peak and its distant wing after passing the radiation through the dispersion-free fiber. For the soliton in the form of a hyperbolic secant, the width of its spectrum is related to its duration by the ratio $\delta {\omega _p} \approx 2/{\tau _p}$. Estimating the maximum value for the spectral width of an input soliton $\delta {\omega _p}$ as equal to the gain bandwidth $\Delta {\omega _0}$, we obtain $\Delta \omega \approx 2\delta \varphi \Delta {\omega _0}$ or $\Delta \omega \approx 2{q_f}{I_{{\max}}}{l_f}\Delta {\omega _0}$; here, ${I_{{\max}}}$ is the peak intensity of the soliton, and ${l_f}$ is the dimensionless length of the dispersion-free fiber. For ${q_f} = 2$, ${l_f} = 0.25$, ${I_{{\max}}} \sim 10$ [see Fig. 3(a)], we get $\Delta \omega \sim 10\Delta {\omega _0}$, which is consistent with the corresponding result of numerical simulation. For dimensional physical quantities, we obtain $\Delta {\omega ^\prime} \approx 2\gamma I_{{\max}}^\prime {L_f}\Delta \omega _0^\prime $, where the prime denotes the dimensional value of the corresponding physical quantity.
Figure 7 shows the change in the spectrum of the output noise-like pulse when the dispersion of the dispersion-free fiber deviates from zero. As in the case of normal and anomalous dispersion, the width of the spectrum decreases. Thus, both normal and anomalous dispersion lead to a decrease in the spectral width of the pulse. In the first case, the frequency chirp associated with the nonlinearity of the refractive index leads to spreading of the solitons forming the noise-like pulse and hence to a decrease in their peak intensity. As a result, spectral broadening occurs only on the initial part of the fiber, where the peak intensity of the solitons is quite high. In the case of anomalous dispersion, the frequency chirp leads to oscillations of solitons and, accordingly, to oscillations of their frequency chirp. Thus, a monotonic increase in the frequency spectrum of the soliton also occurs only in a part of the fiber. Fiber with an average dispersion close to zero can be obtained by alternating sufficiently short fibers with dispersions of opposite signs. When the quadratic dispersion of the refractive index is close to zero, the resulting significant broadening of the emission spectrum may lead to the need to take into account high-order frequency dispersion. This problem requires special consideration.
Fig. 7. Averaged spectral distributions for the noise-like pulse when the dispersion of the dispersion-free fiber deviates from zero. ${D_f} = 0$ for curve 1, ${D_f} = - 0.2$ (normal dispersion) for curve 2, and ${D_f} = 0.2$ (anomalous dispersion) for curve 3. The dashed curve describes the spectral profile of the gain.
We also investigated the formation of noise-like pulses with different arrangements of the fibers forming the laser. As shown by numerical simulation, the variant presented in Fig. 1 is most optimal for the formation of noise-like pulses with a wide radiation spectrum. Conditions for the formation of limited wide spectra of noise-like pulses, including conditions due to the Raman effect [19,35,36], and the factors impeding this process will be studied in future research.
4. CONCLUSION
Using numerical simulation, we investigated the formation of noise-like pulses with a wide radiation spectrum in a passively mode-locked fiber laser. The widest emission spectra were obtained in a ring laser that sequentially included a fiber with NL, a fiber amplifier, an anomalous-dispersion fiber, a dispersion-free fiber, and an OC to remove part of the radiation from the laser resonator. It was found that the broadest spectra occurred in the resonator configuration where a dispersion-free fiber is located before the OC for extracting part of the radiation from the laser. With an increase in the length of the dispersion-free fiber, the frequency chirp of the solitons forming a noise-like pulse grows, resulting in an increase in the width of the spectrum of this pulse. We found that the spectral width could exceed the gain bandwidth by a factor of more than 10. Fiber lasers operating in the noise-like pulse regime are of interest as sources of high-energy pulses with a broad emission spectrum. Such lasers have potential applications in optical coherent tomography, optical radar, and fiber optic sensor systems.
Funding
Russian Foundation for Basic Research (19-52-53002); National Natural Science Foundation of China (11674133, 11911530083); Ministry of Science and Higher Education of the Russian Federation (FSUN-2020-0007).
Disclosures
The authors declare that there are no conflicts of interest related to this paper.
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