Soliton control in fiber lasers with a semiconductor optical amplifier by off-set filtering

Sergei K. Turitsyn

doi:10.1364/OL.492015

Employment of a semiconductor optical amplifier (SOA) to provide both gain and gain modulation for mode-locking in fiber lasers is a viable alternative to using an active rare-earth-doped fiber (see, e.g., [1–16] and references therein). The SOA, initially developed for optical communication and signal processing applications, has many attractive properties including compact size, broadband gain, and the possibility of direct gain modulation by controlling the injection current. The combination of a fiber resonator with an SOA to generate optical pulses can take advantage of both fiber-optic and semiconductor technologies.

However, the nonlinear properties of the SOA [17] make stability and control of the generated pulses a non-trivial physical and engineering problem. Adding an SOA to the optical fiber cavity where pulse dynamics is determined by the interplay of the Kerr effect and dispersion further increases the total number of parameters and degrees of freedom in the resulting nonlinear system. Management of the complex evolution of light in SOA-based lasers requires understanding of the underlying nonlinear dynamics and study of the corresponding mathematical models. It is well known that under certain conditions, simplified path-average soliton models, including dissipative solitons, can be applied to characterize light evolution with small amplitude and phase changes over one round trip [18]. These models are generic for different types of lasers and, moreover, the underlying Ginzburg–Landau equation describes various phenomena and a number of applications in different areas of physics [19]. Soliton lasers have the advantage of relative simple construction, well-defined transform-limited (un-chirped) pulse shape, well-understood features, and a possibility to use powerful theoretical methods in design (see, e.g., [20–23] and references therein). In more general terms, nonlinear coherent structures in lasers and resonators are the key to optical frequency comb technology [24–29]. Soliton theory [20,21,30,31] includes various developed techniques for soliton control—stabilization of pulse parameters in the presence of perturbations or reduction of soliton jitter due to noise or nonlinear interactions. This can be especially interesting in the case of SOA-based fiber lasers where, without control elements, the generated optical pulses continuously change their central frequency and time position. Soliton theory also might be useful to quantify the dependence of fluctuations of pulse characteristics on system parameters.

In general, an accurate description of the cyclic transit intra-cavity dynamics of radiation in a laser resonator can be provided by Poincaré mapping [32] of the light field over a round trip. The overall modification of the properties of a field $A(t)$ over the round trip is given by the operator that is a product of consecutive transformations of the field by the elements of the laser cavity. These elementary transformations can be both a point action (e.g., filter or coupler) or distributed (e.g., nonlinear evolution in an optical fiber). For instance, in the fiber laser context, these basic elements may include: different fiber spans, amplifier(s), saturable absorber, optical filter, out-coupler, and so on. When resonator components have a nonlinear response, in general, these operators are non-commuting. Here, we introduce a path-average model for the laser system that comprises the following basic elements: optical fiber with the anomalous dispersion, SOA as a gain medium, and spectral filter with the frequency off-set from the central frequency of the SOA. We assume here that the saturable absorber, responsible for the generation of pulses from noise, does not impact the final pulse shaping.

Note that even when it is possible to accurately describe a single nonlinear element, it is not always feasible to define the overall transformation over one round trip in a simple way. For instance, in the laser system considered here, the operator corresponding to the transfer function of the semiconductor optical amplifier reads [17]:

A_{out}(t)=\widehat{T}_{SOA} A_{in}(t)= \exp{\left[\frac{1-i \;\alpha_{H}}{2} \; h(t\right]} \times A_{in}(t),

where

\frac{d h}{dt} ={-} \frac{h - h_0}{T_{SOA}} - \frac{\exp{(h)} -1}{E_{sat}} \times |A_{in}(t)|^2,

where $A_{in}(t)$ and $A_{out}(t)$ are respectively the input and output optical fields, $h_0$ is related to the small signal gain $G_0=\exp (h_0)$, $T_{SOA}$ the SOA gain recovery time, $E_{sat}$ is a characteristic saturation energy that defines the SOA saturation power $P_{sat}=E_{sat}/T_{SOA}$, and $\alpha _H$ is the so-called Henry linewidth enhancement factor. In the limit of short enough pulses (with width much smaller than $T_{SOA}$), one can use a well-known approximation [17] to simplify the description of an SOA-induced transformation of the input signal $A_{in}(t)$:

(1)$$ h(t) ={-} \ln\left[1-\left(1-\frac{1}{G_0}\right) \exp\left(-\frac{\int_{-\infty}^{t}{P_{in}(t') dt'}}{E_{sat}}\right) \right].$$

The instantaneous frequency ($2 \pi \nu = - d \arg (A)/dt$) of a pulse after every passing of the SOA acquires a spectral shift $\Delta \nu$ [17]:

(2)$$ \Delta \nu = \frac{\alpha_H}{4 \pi} \frac{d h}{d t} ={-} \frac{\alpha_H (G_0 - 1)}{4 \pi G_0 E_{sat}} \; \exp\left[-\frac{\int_{-\infty}^{t}{P_{in}(t') dt'}}{E_{sat}} \right]\times |A_{out}(t)|^2.$$

This means, in particular, that without compensation of this redshift in the cavity, the pulse central frequency will be continuously moving until it will reach the end of the spectral gain curve. This continuous shifting of the pulse spectrum with each round trip can be stabilized by a spectral filtering. In this work, we consider using the soliton perturbation theory effect of the optical filter $H(\omega )$ off-set by $\omega _{f}$ from the central frequency of SOA $\omega _c$:

H(\omega) = H(0) \exp\left[-\frac{(\omega -\omega_c - \omega_{f})^2}{2 B^2}\right].

The direct numerical modeling of the consecutive transformations of light by each in-cavity element allows accurate description of the pulse evolution in a resonator though, due to the nonlinear nature of the dynamics of radiation, this approach is time consuming, especially in the multi-parametric design simulations, and it often does not provide intuition about the underlying physics. The classical mode-locking theory developed by H. Haus et al. [18,33–35] is effectively a path-average model. Haus’ theory of mode-locking lasers is based on the distributed equation averaged over a round trip that assumes small fractional changes of optical pulse characteristics during the propagation along the cavity and, effectively, additive summation of those variations over one transit of the resonator (see for details, e.g., [18,21,33–35] and references therein). Though path-average models are not well developed for the case of large in-cavity variations of signal parameters [21,36] and, thus, they cannot be applied to describe some important lasing regimes, they are still attractive for qualitative understanding of pulse dynamics within the limits of their validity.

Following [37–39], we introduce the master model describing the evolution of an optical pulse in a fiber cavity with an SOA (with the central frequency at $\omega _c$) path averaged over the one round trip $L$ as

(3)$$\begin{split} \frac{ \partial A}{\partial z} = \Delta g A - i\frac{\beta_2}{2} \frac{\partial^2 A}{\partial t^2} + i \gamma r |A|^2 A + \tilde{\eta}\; \left(\frac{\partial }{\partial t} +i \,\omega_f\right)^2\, &\\ \times A- \frac{1-i \alpha_H}{2 L E_{sat}} \left(1- \frac{1}{G_0}\right) \;A \int_{-\infty}^{t}{|A(t')|^2\,dt'}\end{split}$$

here, coefficient $r=[1-\exp (-\alpha L)]/(\alpha L)$ is a standard path-average coefficient [20] ($\alpha$ is an effective distributed cavity loss, defined as $\alpha = (\sum _ n \alpha _{n} L_{n} + \sum _k \alpha _{k})/L$), where sum over $n$ corresponds to all distributed losses (e.g., spans of SMF, DCF, and so on) and sum over $k$ is for all discrete losses (out-coupler, insertion losses of any discrete cavity elements); $\beta _2$ is, in the similar manner, an effective distributed group velocity dispersion parameter that is an average accumulated dispersion of the cavity, $\tilde {\eta }=1/(2 B^2 L)$ is the path-average parameter characterizing filtering in the resonator; $G_0 = \exp [\alpha L]$ is the linear SOA gain chosen to compensate for the effective distributed cavity loss, thus, $r=\sqrt {\;(G_0-1)/(G_0 \ln [G_0])}$. This path-average equation holds within a first-order expansion of the gain saturation of the SOA. The validity of this approximation of the gain saturation of the amplifier over one round trip is defined by the condition that power dependent modifications of the gain are small: $\int _{-\infty }^{t}{|A(t')|^2 dt'}/E_{sat} \ll 1$. This condition means that the average signal power is below the saturation power of the amplifier and the energy of the single pulse ($E=\int {|A|^2 dt}$) is below the amplifier saturation energy $E_{sat}$. Here, $\Delta g$ is the amplifier excess gain (averaged over $L$) introduced to balance losses due to the filtering and the nonlinear SOA gain saturation:

2\Delta g E= 2 \tilde{\eta} \int{\left|\frac{\partial A}{\partial t} +i \,\omega_f A\right|^2 dt}+(G_0-1) E^2/(2 G_0 L E_{sat}).

Equation (3) can be normalized in a standard manner, using the soliton units [20]: $A(t,z)$ characteristics are found from the normalized variables as follows (index $norm$ is for normalization): $A(t,z) = \sqrt {P_{norm}}\;u(\tau,Z)$, $\tau =t/T_{norm}$, $Z=z/L_D$, with $L_D=T_{norm}^{2}/|\beta _2|$, $P_{norm}=|\beta _2|/( r \gamma T_{norm}^{2})$. In the normalized soliton units, the master equation reads:

(4)$$\begin{split} \frac{ \partial u}{\partial Z} - \sigma_{b} \frac{i }{2} \frac{\partial^2 u}{\partial \tau^2} -i |u|^2 u = \delta u \;+ \eta\; \left(\frac{\partial }{\partial \tau}+ i\Omega_f\right)^2 &\\ \times u- \rho (1- i\alpha_H) \;u \int_{-\infty}^{\tau}{|u(\tau')|^2\,d\tau'};\end{split}$$

here, $\sigma _b =-\textrm{sgn}(\beta _2)=\pm 1$. In what follows, we will consider anomalous dispersion ($\beta _2<0$), for which $\sigma _b = 1$. Parameter $\delta = \Delta g T_{norm}^2/|\beta _2|$ corresponds to the normalized average cavity excess gain; $\eta = \tilde {\eta }/|\beta _2|=1/(2 B^2 L |\beta _2|)$ is the normalized distributed filtering strength (curvature of the filter), $\Omega _f =\omega _{f} T_{norm}$ is the normalized shift of the filter relative to the SOA gain maximum, and the normalized gain saturation coefficient $\rho = \ln [G_0] T_{norm}/(2 L E_{sat} \gamma )$.

Let us start from the analysis of the stabilization of a fundamental transform limited optical soliton by the filtering. Consider a standard perturbation theory for a single soliton [30,31,38,40]:

u(Z,\tau) = A(Z)\, \text{sech}[A(Z) (\tau - T)] \times \exp[- i \Omega (\tau - T) +i \phi],

with $A$ (assumed to be positive), $T$, $\Omega$, and $\phi$ being functions of $z$. Applying the standard soliton perturbation theory, we can describe the adiabatic changes of the fundamental soliton parameters under the combined action of the filtering and amplification by the SOA. Evolution of the parameters $A$, $\Omega$, and $T$ (dynamics of $\phi$ can be separated) is given by the following set of the ordinary differential equations:

(5)$$\frac{d \Omega}{d Z} = \frac{4}{3}\, \eta\, \Omega_{f}\, A^2 -\frac{4}{3}\, \eta\, \Omega\, A^2-\frac{2}{3}\, \alpha_H \, \rho\, A^2,$$

(6)$$\frac{d A}{d Z} = 2 A \left[\delta - \eta\, (\Omega-\Omega_{f})^2- \frac{A^2}{3}\, \eta -\rho A\right],$$

(7)$$\frac{d T}{d Z} ={-} \Omega -\rho.$$

The nonlinear effects in the SOA (the term proportional to $\rho$) make soliton dynamics in such lasers different from the conventional soliton lasers and traditional dissipative solitons in fiber lasers. It is seen that without a filter ($\eta =0$), while the change of the soliton amplitude with round trips (in $Z$) can be controlled for $\tilde {A}=\delta /\rho$, overall, there are no stationary solutions and the soliton is continuously shifting both its frequency $\tilde {\Omega }(Z)=\tilde {\Omega }(0)- 2 \alpha _H \delta ^2 Z/(3 \rho )$ and temporal position (point of the maximum of power distribution—peak power) $\tilde {T}(Z)=\tilde {T}(0) - (\rho +\tilde {\Omega }(0)) Z +\alpha _H \delta ^2 Z^2/(3 \rho )$. An important challenge here is to stabilize these continuous shifts in the soliton temporal position and frequency created by the SOA. In the case of the filter centered on the maximum of the gain curve ($\Omega _f=0$), the frequency of the steady-state solution is fixed as $\Omega _{c}= - \rho \, \alpha _H/(2 \eta )$ and cannot be tuned. The steady-state (with $\Omega$ and $A$ not changing with rounds trips) solutions of Eqs. (5)–(7) are

(8)$$\Omega_s = \Omega_{f} - \frac{\rho\,\alpha_H}{2\, \eta},$$

(9)$$A_s = \frac{-3 \rho + \sqrt{9 \rho^2+3 [4 \delta \eta- \rho^2 \alpha_H^2]}}{2 \eta}.$$

It is straightforward to check that this solution is stable against small perturbations of the soliton parameters. Assumption of the positive amplitude of the soliton leads to the condition on the access gain parameter: $\delta > \rho ^2 \alpha _H^2/(4 \eta )$. Note that use of the off-set filter allows to control the soliton frequency $\Omega _s$ that is not possible in the case of a filter centered at the SOA gain maximum. It can be pointed out that the frequency of the soliton is always shifted from the filter central frequency, opening a possibility for dynamic spectral tunability of the SOA-based laser [15,16]. It is also possible to achieve a regime with $d \Omega /dZ = dT/dZ =0$, for $\Omega _f = \rho \; [\alpha _H/(2 \eta ) - 1]$ and $\Omega _s = - \rho$.

Note that the fundamental soliton is not necessarily an attractor for the solution formed from the initial noise. It is just a particular branch of the more general family of steady state waveforms that can be generated in SOA-amplified fiber lasers. As the SOA transfers amplitude modulation into the phase modulation, we can expect in these lasers coherent structures with time-dependent phase—chirp pulses. Indeed, there are exact analytical solutions of Eq. (4) in the form of chirped solitons derived for similar models in [41] and [37] in different contexts:

(10)$$u_{chs}(Z,\tau) = \frac{A_0}{\cosh^{1+i q}(\frac{\tau- \sigma Z}{T_0})} \times \exp[{-}i \omega_0 \tau + i k Z],$$

here parameters of the solution: $A_0, q, \sigma, \omega _0, T_0, k$ are expressed through the parameters of the equation ($\eta$, $\Omega _f$, $\delta$, $\rho$, $\alpha _H$) as follows:

(11)$$q ={-} \frac{3}{4 \eta} + \sqrt{\frac{9}{16 \eta^2}+2}, \,\;\;\; T_0^2 A_0^2 = 3 q \eta\left(1+\frac{1}{4 \eta^2}\right)=S(\eta),$$

(12)$$\omega_0 = \Omega_f - \frac{3 \,\rho\, q}{2} \times \frac{\alpha_H + q}{1+ q^2} \times\left(1+ \frac{1}{4 \eta^2}\right) = \Omega_f- R, \;$$

(13)$$\sigma={-} \Omega_f +\left[1- 2 \eta \frac{ 1- \alpha_H q}{\alpha_H + q}\right] \times R(\eta,\rho,\alpha_H),$$

(14)$$\delta = \eta \left[(\omega_0 - \Omega_f)^2 + \frac{1}{T_0^2} \right] + \rho A_0^2 T_0- \frac{q}{2 T_0^2}.$$

Equation (15) determines the pulse width parameter $T_0$ as a function of $\eta, \rho$, and $\alpha _h$:

(15)$$T_0=\frac{\rho S+\sqrt{\rho^2 \,S^2+2(\delta - \eta R^2)\;(2 \eta-q)}}{2(\delta - \eta R^2)}.$$

As $2 \eta - q(\eta ) >0$, the requirement $T_0>0$ imposes a condition on the excess gain parameter $\delta$: $\delta >\delta _{th}= \eta R^2(\eta,\rho,\alpha _H)$. Note that at small $\eta \to 0$, this coincides with the criteria derived above using the soliton perturbation theory: $\delta _{th} \to \rho ^2 \alpha _H^2/(4 \eta )$ (see Fig. 1).

Fig. 1. An access gain threshold $\delta _{th}= \eta R^2(\eta )$ versus $\eta$. Here, $\rho =0.2, \, \alpha _H=4.$

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Energy $E$ of the chirped soliton is $E=\int {|u_{chs}|^2 d\tau }= 2A_0^2 T_0$ and the full pulse width at half maximum is the same as for the fundamental soliton $T_{FWHM}=1.763 T_0$. The spectral power density for chirped soliton reads:

|u_{chs}(Z,\omega)|^2= \frac{ 2 \pi A_0^2 T_0^2}{q}\times \frac{\text{tanh}(\pi q)}{1+\text{sech}[\pi q] \text{cosh}[\pi (\omega-\omega_0) T_0]}.

Therefore, the full width at half maximum bandwidth of the chirped soliton can be found as

B_{FWHM} = \frac{2 \ln[\text{cosh}(\pi q/2) +\sqrt{1+ \text{cosh}^2(\pi q/2)}]}{\pi^2 T_0}.

Figure 2(a) depicts how the time-bandwidth product $T_{FWHM} \times B_{FHWM}$ increases from the fundamental soliton value of 0.315 with growing $\eta$. Figure 2(b) shows that the chirped soliton $E$ is inversely proportional to the pulse width $T_{FWHM}$ with a scaling parameter increasing with the filter strength parameter $\eta$. Note that at $\eta \to 0$, $q \to 4 \eta /3 \to 0$, and $E \times T_{FWHM} \to 3.526$.

Fig. 2. (a) Normalized time-bandwidth product and (b) energy $E$ product with $T_{FWHM}$ versus $\eta$; $\rho =0.2, \,\alpha _H=4.$

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These analytical solutions can be used to determine the conditions of existence of the stationary chirped soliton pulses that are not moving ($\sigma =0$) in the considered frame. This can be achieved by adjusting the off-set filter as

(16)$$ \Omega_{f1}= [1- 2 \eta \frac{ 1- \alpha_H q}{\alpha_H + q}] \times R(\eta,\rho,\alpha_H).$$

In this case, the soliton frequency $\omega _{01}$ is found as

(17)$$ \omega_{01} ={-} 2 \eta \frac{ 1- \alpha_H q}{\alpha_H + q} \times R(\eta,\rho,\alpha_H).$$

It is seen that by varying the filter strength $\eta$ around the point $\eta _1 = 0.194$ [that corresponds to $q(\eta _1)=1/\alpha _H$], the soliton frequency $\omega _{01}$ changes sign making possible both a red and blue off-set of the pulse from the SOA central frequency.

While the path-average model analysis is limited to small changes of the pulse power and phase over a single round trip (see, though, recent work [42]), it provides a useful analytical description of the soliton solutions. In particular, it shows that off-set spectral filtering can be used to manipulate the central frequency of the soliton and its velocity. It has been shown recently in [15,16] that nonlinear dynamics in SOA-based fiber laser can be used to provide for a spectral tunability of the generated pulses. The analytical description above [i.e., Eq. (17)] quantifies this possibility within the limit of validity of the path-average model.

Note that the access gain $\delta >0$ gives rise to the instability of the CW waves outside of the soliton solution. However, this instability is suppressed in the laser by the saturable absorber. The saturable absorber effect is assumed here to be responsible for the self-starting of the pulse generation from noise, but not affecting soliton shaping at higher powers. Contribution of the saturable absorber can be accounted for both in the soliton perturbation analysis and in the description of the chirped solitons (see for details [39]) leading to more cumbersome analytical results. The impact of the saturable absorber on soliton properties in SOA-based fiber lasers will be presented elsewhere.

Funding

Engineering and Physical Sciences Research Council (EP/W002868/1).

Acknowledgments

I would like to thank Daria Khudozhitkova, Anastasia Bednyakova, Natalia Manuylovich, and Elena Turitsyna for help with the preparation of the manuscript.

Disclosures

The author declares no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Soliton control in fiber lasers with a semiconductor optical amplifier by off-set filtering

Abstract

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Data availability

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Optics Letters