**2. Simulation of passively mode‐locked ytterbium‐doped fiber laser**

Ultrashort‐pulse propagation in optical fiber can be accurately modeled by one or more coupled partial differential equations. Various of simulation methods with different theoreti‐ cal models have been introduced to study laser pulse phenomenon and dynamic processes in the cavity with the parameters of dispersion, nonlinearity, gain, loss, etc. The well‐known master mode‐locking equation first proposed by H. A. Haus [22], which being a perturba‐ tion from the nonlinear Schrödinger equation, has the capacity of describing both the energy saturation and the pulse stabilization process. To prove the possibility of stable pulse gener‐

ation in the presented cavity, numerical simulation of the generalized nonlinear Schrödinger equation is performed here, which provides an insights into the mode‐locking dynamics of ytterbium‐doped fiber lasers and directs to the performance optimization for the pulses.

The whole cavity is schematically shown as an analytical model in **Figure 3**. The main parameters of each fiber can be found in **Table 1**. Firstly, we have numerically simulated the pulse propagation and formation in the cavity governed by the following equation [23]:

$$\frac{\partial \mathbf{A}}{\partial \mathbf{z}} = \frac{\mathbf{g} - \alpha}{2} \mathbf{A} + \frac{\mathbf{g}}{2 \Omega\_{\text{g}}^{2}} \frac{\partial^{2} \mathbf{A}}{\partial \tau^{2}} + \text{i}\gamma \left| \mathbf{A} \right|^{2} \mathbf{A} - \text{i}\frac{\mathcal{B}\_{2}}{2} \frac{\partial^{2} \mathbf{A}}{\partial \tau^{2}} + \text{i}\frac{\mathcal{B}\_{2}}{2} \frac{\partial^{3} \mathbf{A}}{\partial \tau^{3}} \tag{1}$$

where A is the slowly varying envelope of the optical field, z is the axial distance, *τ* is the local time, α accounts for the loss, γ is the nonlinear coefficient of fiber, which accounts for the self‐ phase modulation effect, and *β*2 is the second‐order derivative of the propagation constant. During each cavity round‐trip time, the pulse goes through different cavity components, and the output from one component is used as the input to the other, as described in **Figure 3**. Also, the pulse goes through the saturable absorber with a nonlinear loss and through the coupler with a fraction of the pulse energy outputted. The fiber parameters were chosen to match the measured or specified parameters of different elements used in the experiment.

**Figure 3.** Proposed model of passively mode‐locked fiber laser. YDF, ytterbium‐doped fiber; SMF, single mode fiber; OC, output coupler; SA, saturable absorber.


**Table 1.** Summary of the fiber parameters used in the simulation.

The parameter g denotes the gain coefficient of ytterbium‐doped fiber, which can be descri‐ bed as the gain function and approximately expressed by:

ation in the presented cavity, numerical simulation of the generalized nonlinear Schrödinger equation is performed here, which provides an insights into the mode‐locking dynamics of ytterbium‐doped fiber lasers and directs to the performance optimization for the pulses.

The whole cavity is schematically shown as an analytical model in **Figure 3**. The main parameters of each fiber can be found in **Table 1**. Firstly, we have numerically simulated the pulse propagation and formation in the cavity governed by the following equation [23]:

where A is the slowly varying envelope of the optical field, z is the axial distance, *τ* is the local time, α accounts for the loss, γ is the nonlinear coefficient of fiber, which accounts for the self‐ phase modulation effect, and *β*2 is the second‐order derivative of the propagation constant. During each cavity round‐trip time, the pulse goes through different cavity components, and the output from one component is used as the input to the other, as described in **Figure 3**. Also, the pulse goes through the saturable absorber with a nonlinear loss and through the coupler with a fraction of the pulse energy outputted. The fiber parameters were chosen to match the

**Figure 3.** Proposed model of passively mode‐locked fiber laser. YDF, ytterbium‐doped fiber; SMF, single mode fiber;

**m-1)** *β3* **(fs3**

Ytterbium‐doped fiber 0.021 0.0254 0.0048 10 Other fiber elements 0.022 0.0254 0.0044 5

Hi1060 fiber 0.022 0.0254 0.0044 20 or Var.

Ag g A A A <sup>A</sup> i AA i i z2 2 2 2

g

¶- ¶ ¶ ¶ =+ + - +

measured or specified parameters of different elements used in the experiment.

g

t

a

284 286High Energy and Short Pulse Lasers

OC, output coupler; SA, saturable absorber.

**Fiber type** *β2* **(ps2**

**Table 1.** Summary of the fiber parameters used in the simulation.

<sup>2</sup> 2 3 <sup>2</sup> 2 2 2 2 2 3

b

¶ W ¶ ¶ ¶ (1)

 b

tt

**m-1)** *γ* **(W-1 m-1) Length (m)**

$$\mathbf{g}\_{i} = \frac{\mathbf{g}\_{0}}{\mathbf{l} + \mathbf{E}\_{\text{pulse}}} \overbrace{\mathbf{E}\_{\text{sat}}}^{\text{s}} \tag{2}$$

where Esat is the saturation energy due to the limited pump power, which is defined as Esat=(hv / σ)Aeff with the dependence of pump power. The pulse energy Epulse is given by Epulse= *∫*TR/2 −TR/2 |A(z, τ)| <sup>2</sup> dτ , where TR is the cavity round‐trip time. The same small signal gain g0, depending on the doping concentration, can be assumed to be constant if only a small fraction of the pump light is absorbed provided an approximation of uniform pumping. Ytterbium‐doped fiber is modeled with a total unsaturated gain of 30 dB, corresponding to these parameters: g0 =6.9m−<sup>1</sup> and Esat=1nJ. *Ωg* is the gain bandwidth of ytterbium‐doped fiber, which is related to the bandwidth Δλ through <sup>Ω</sup><sup>g</sup> <sup>=</sup> |2*π<sup>c</sup>* / *<sup>λ</sup>* <sup>2</sup> <sup>|</sup> <sup>∆</sup>*λ*, where ∆λ is chosen to be 55nm bandwidth.

The parameter β<sup>2</sup> represents dispersion of the group velocity contributing to time‐domain broadening of laser pulse, that is, the so‐called group velocity dispersion, which is common‐ ly used by physicists in units of ps2∆m−<sup>1</sup> . For optical fibers, the group velocity dispersion usually refers to the chromatic dispersion parameter D that defined as a derivative dβ<sup>1</sup> / dλ, which is also used in practice with the relation of β2 and n as: β<sup>2</sup> <sup>=</sup> <sup>−</sup>(λ<sup>2</sup> / <sup>2</sup>πc)∆D, where λ is the operating wavelength. The higher order dispersion and higher order nonlinear effects were ignored in simulations. Fiber nonlinear parameter γ relates the wavelength λ and effective area Aeff to the nonlinear index n2 with an expression as: γ=2πn<sup>2</sup> / λ0Aeff, when the radial field distribution is known.

Nonlineartransmission of the saturable absorber can be described by T=exp −(α<sup>l</sup> + αnl) , where α<sup>l</sup> is nonsaturable absorption loss, and αnl denotes power‐dependent nonlinear absorption loss, which is given by αnl=α<sup>0</sup> /(1 + P(τ)/ Psat), where α0 is the saturable loss due to the absorption, that is, the modulation depth. P(τ) is the instantaneous pulse power and Psat is the saturation power of the saturable absorber. The saturable loss, which acts as an equivalent‐filtering effect, has a significant impact on the pulse duration and bandwidth of laser pulse. It is expected, for high power/energy pulse, that the saturable absorber with a larger modulation depth can be used in a fiberlaser with large normal dispersion and strong nonlinearity. The furtherincrease of the modulation depth should be carefully designed due to the limitation of the nonsatura‐ ble loss. One probable way for the increase of modulation depth is to reduce the evanescent field leaking of D‐shaped fiber and enlarge the evanescent field interaction with the satura‐ ble absorber by lengthening the fiber D‐shaped domain. Here, the parameters of the satura‐ ble absorber in the simulation model are as follows: αl=45%, α<sup>0</sup> =27%, and Psat=1000W.

#### **2.1. Numerical simulation and results**

All optical fibers in the model above have normal dispersion within the laser spectral range. A 20 m length of Hi1060 fiber was inserted into the cavity aiming to increase the cavity length. The total dispersion is calculated to be about 0.75 ps<sup>2</sup> . Eq. (1) has been solved with the standard split‐step Fourier method. The simulation field is represented on a temporal grid (and via the Fourier transform on a frequency grid) consisting of 2<sup>11</sup> points with a width of 0.2 ns. One initialized weak signal was introduced into the round‐trip propagation in the cavity, and this pulse consecutively experiences each action of cavity components along the routes.

**Figure 4.** Transient evolution of the pulse width (a) and peak power (b).

**Figure 5.** Pulse profile and chirp (a), and spectrum (b) of mode‐locked laser. The inset is the spectrum in decibel scale.

After a finite number ofround trips, the pulse started to converge into stable dissipative soliton solutions. The simulation results indicate that stable solutions do exist in such a laser, which can be confirmed from the peak power and pulse duration evolutions as shown in **Figure 4**. The pulse duration is ∼40.6 ps and the spectral edge‐to‐edge bandwidth is 11.2 nm as shown in **Figure 5**, which both evidently indicate that the pulses are highly chirped. As shown in **Figure 5(b)**, the spectrum on a linear scale and a logarithmic scale is characterized by their steep edges, that is, the so‐called M‐shaped optical spectra. The higher peak power of the pulse after amplification by gain fiber induces a substantial nonlinear phase shift in the single‐mode fiber, which results in sharp peaks on the spectrum edges. The gain and loss coexist in the dissipative system and play an essential role in the formation of dissipative solitons. Thus, dissipative soliton must be self‐organized and its dynamics differ from that of the conven‐ tional soliton. It is noted that the saturable absorber has a nonlinear transmission depending on the light intensity, and the nonlinear phase shift is gradually varied in the process of gain saturation when an initial pulse is circulating in the cavity [24].

**Figure 6.** Evolution of the pulse width and spectral bandwidth in one cavity round trip.

**2.1. Numerical simulation and results**

286 288High Energy and Short Pulse Lasers

The total dispersion is calculated to be about 0.75 ps<sup>2</sup>

**Figure 4.** Transient evolution of the pulse width (a) and peak power (b).

All optical fibers in the model above have normal dispersion within the laser spectral range. A 20 m length of Hi1060 fiber was inserted into the cavity aiming to increase the cavity length.

split‐step Fourier method. The simulation field is represented on a temporal grid (and via the Fourier transform on a frequency grid) consisting of 2<sup>11</sup> points with a width of 0.2 ns. One initialized weak signal was introduced into the round‐trip propagation in the cavity, and this

**Figure 5.** Pulse profile and chirp (a), and spectrum (b) of mode‐locked laser. The inset is the spectrum in decibel scale.

After a finite number ofround trips, the pulse started to converge into stable dissipative soliton solutions. The simulation results indicate that stable solutions do exist in such a laser, which can be confirmed from the peak power and pulse duration evolutions as shown in **Figure 4**. The pulse duration is ∼40.6 ps and the spectral edge‐to‐edge bandwidth is 11.2 nm as shown in **Figure 5**, which both evidently indicate that the pulses are highly chirped. As shown in **Figure 5(b)**, the spectrum on a linear scale and a logarithmic scale is characterized by their

pulse consecutively experiences each action of cavity components along the routes.

. Eq. (1) has been solved with the standard

The evolutions of the pulse duration and spectral edge‐to‐edge bandwidth in one cavity round trip are shown in **Figure 6**. It can be seen that the significant broadening of the duration and spectral width of the pulse after passing through the gain fiber. The large normal dispersion of a long segment of single‐mode fiber induces large and positive chirp. The nonlinear phase shift broadens the spectrum of the pulse, while strong chirping induced by normal cavity dispersion enlarges the pulse pedestals, which are the lower intensity red‐shifted and blue‐ shifted spectral weights located at both pulse edges in time‐domain. When the pulse is travelling throughout the saturable absorber, pulse duration and spectrum width could be compressed. Here, the saturable absorber plays a key role of providing an effective filtering function to stabilize the mode locking.

When the cavity length was increased to 60m, the total cavity dispersion is up to ∼1.3 ps<sup>2</sup> . The laser works on a larger normal dispersion regime without dispersion compensation in this model as the same above. Simulation results showed that the stable pulse obtained with the pulse duration of 76.5 ps and the corresponding spectral width of 9.6 nm. With the larger positive dispersion, the stable pulses have been obtained again but highly chirp. Comparing the pulse characteristics of these two cavities, the broadened pulse induced mainly by self‐ phase modulation in the longer cavity has lower peak power, and both of the equivalent filter effect and lower self‐phase modulation result in the narrower spectrum.

For the deliverable high energy of the pulse, the stabilization of mode locking should be ensured in a normal and large cavity dispersion accompanied with a high nonlinearity. A spectral filter, which is added in such a laser cavity, can be considered as an effective absorb‐ er in the spectral domain to cut off the temporal wings of the pulse and stable the mode locking. It is assumed that the spectral filter has a Gaussian profile, so the spectral filter is numerical‐ ly implemented in the model by a function as: T(ω)=exp −(ω / Ω<sup>f</sup> )<sup>2</sup> , where ω is the angular frequency, and Ω<sup>f</sup> is the bandwidth of the spectral filter. The influence of the spectra filter‐ ing on pulse shaping could be investigated for the stabilization of the mode locking, as well as the performance optimization of the pulses [25].
