Stable operation of mode-locked fiber lasers: similariton regime

Pierre-André Bélanger

doi:10.1364/OE.15.011033

1. Introduction

Generation of short and high power pulses has been realized in gas as well as in solid-state lasers for many years using active or passive mode-locking techniques [1]. Today, polarization additive pulse mode-locking technique allows fiber lasers to generate short pulses at high repetition rates so that fiber lasers are replacing solid-state lasers in several applications. The recent papers [2,3] reporting the operation of an all-fiber mode-locked laser enables one to expect a broader range of applications in several fields such as engineering, biology, medecine etc. The design of a fiber laser first requires a good numerical analysis before choosing the type of fiber, the dispersive component, the amplifying fiber and the mode-locking mechanism. However, before turning to the numerical analysis, I think that studying carefully the solitary pulse solution of the master equation of Ref. [1], where all the intracavity components are distributed along the propagation axis, is necessary and helpful. The solitary pulse solution of the master equation only gives insight on the average pulse profile and spectrum and not on the dynamics of the pulse propagation [4]. However, I shall show, in this paper, that a careful analysis of the solitary pulse parameters can give a realistic indication on how to optimize the energy output after properly selecting the laser components.

2. The distributed model

Following Haus [1], such a short pulse laser can be modeled as a mode-locked laser system using a fast saturable absorber. Here, I write the master equation of the distributed model in the form of a Ginzburg-Landau (GL) equation:

i V_{x} + \frac{(\overset{̅}{β} - i ε)}{2} V_{τ τ} - i (g - l) V - γ_{0} (1 + i ε_{0}) {∣ V ∣}^{2} V = 0

The laser components are distributed along the normalized propagation distance $x = \frac{z}{L}$ where L is the length of the laser. β̄ is the second-order dispersion (SOD) and ε is the filtering coefficient of the gain medium. The mode-locked gain is g while l is the distributed loss. γ ₀ stands for the Kerr coefficient whereas γ ₀ ε ₀ is the fast saturable absorber coefficient that models the action of the nonlinear pulse evolution (NPE) mechanism. As a result of this normalization, all the laser parameters are total ones, for exemple, β̄ is the total dispersion of the system in ps ². The average behaviour of the short pulse profile generated by the mode-locked pulse laser can be extracted from the solitary-wave solution of the differential equation (1) namely:

V (τ, x) = V_{0} sech (α τ) \exp {- i [β \ln (sech (α τ)) + Γ x]}

where the phase-shift Γ and the peak amplitude of the pulse V ₀ are related to the pulse parameters by:

Γ = β α^{2} [ε + \frac{\overset{̅}{β}}{2 β} (β^{2} - 1)]

2 γ_{0} V_{0}^{2} = α^{2} [(β^{2} - 2) \overset{̅}{β} + 3 β ε]

The energy in each pulse is calculated as:

γ_{0} P_{0} = \frac{2 γ_{0} V_{0}^{2}}{α}

In order to satisfy the differential equation (1) (see Ref. [5]), the gain and loss of the system must be related to the pulse width α and chirp β such as:

(g - 1) = - β α^{2} [\overset{̅}{β} - \frac{ε}{2 β} (β^{2} - 1)]

and the NPE action parameter ε ₀ must be given by:

ε_{0} = \frac{[3 β \overset{̅}{β} - (β^{2} - 2) ε]}{[(β^{2} - 2) \overset{̅}{β} + 3 β ε]}

The gain (g) and loss (l) parameters as well as the NPE parameter ε ₀ are external parameters that can be changed by the laser operators as opposed to the pulse width α and chirp β which are internal parameters which are fixed for a particular value of the gain and NPE whereas β̄,ε and γ ₀ are known as they are inherent of the laser components.

3. An optimal operating point

In such a laser, modeled appropriately by the GL differential equation, the pulse or the solitary-wave solution is in competition with the CW operation where the gain just compensates the loss (g-l=0). In order to generate a good and stable mode-locked pulse from the laser [1], we have to require that (g-l)<0 in Eq. (6). Of course, a more perturbative stable mode-locked operation will be obtained for the largest negative (g-l) parameter. Here, for finding an optimal operation point, I shall require that $\frac{\partial}{\partial β} (g - l) = 0$ for a fixed pulse width α. This extremum condition, when applied to Eqn. (6), requires that:

β = \frac{\overset{̅}{β}}{ε}

and then:

g - l = \frac{- α^{2} ε}{2} (β^{2} + 1)

and the NPE parameter ε₀ of Eq. (7) is now given by:

ε_{0} = \frac{2}{β}

At this operation point, the chirp parameter β must be positive (Eq. (10)) and therefore the SOD β̄ must be positive whereas the mode-locked gain term (g-l) must, as imposed, be negative.

For these optimal parameters, the phase shift Γ is given by:

Γ = \frac{α^{2} \overset{̅}{β}}{2} (β^{2} + 1)

while the amplitude V ₀ is obtained by:

γ_{0} V_{0}^{2} = Γ

and the pulse energy will then be:

γ_{0} P_{0} = α \overset{̅}{β} (β^{2} + 1)

For a parabolic gain medium, the filtering action of the gain can be approximated by:

ε = \frac{2}{ω_{g}^{2}} = \frac{1}{2 π^{2} v_{g}^{2}} \approx \frac{0.05}{v_{g}^{2}}

where ν _g is the spectral width of the gain parameter. The chirp β, given by Eq. (8), can now be calculated as:

β = 20 \overset{̅}{β} v_{g}^{2}

For this optimal operation point, the NPE device (ε ₀) must be adjusted to satisfy Eqs. (15) and (10) and the pulse width α can be varied according to the pump power that controls the mode-locked gain (Eq. (9)). The full-width-half-maximum(FWHM) τ ₀(ατ ₀=1.7627) can be related to the spectral FWHM ν _f of the pulse (Ref. [5]) by:

τ_{0} v_{f} = 0.3572 {sinh}^{- 1} [\cosh (\frac{π}{2} β)]

The objective of mode-locking a laser is to generate a pulse having a spectral width as close as possible to the gain spectral width. Here, I shall impose that the pulse spectrum respects the following condition: ν _f=ν _g. This restriction, applied to Eqs. (15) and (16), yields an average pulse width τ ₀ of the laser system. However, it is unsure whether or not the mode-locking mecanisms (Kerr and NPE) can be adjusted in a practical system so that this condition is satisfied.

4. Similariton operation at the optimal point

In a recent paper [5], I have shown that the so-called similariton laser can be modelled by a GL distributed equation when the chirp parameter β is very large. In an experimental laser, Ruehl et al [6] have indeed shown that the pulse profile is clearly close to this sech profile and in a more recent paper [7], the measured phase profile clearly shows the linear asymptotic behavior of the thoeretical phase. When β≫1, the spectral width is related to the pulse width through πν _f=αβ and at the optimal operation point, we have the following system of equations:

(g - l) = - 0.25 {(\frac{v_{f}}{v_{g}})}^{2}

τ_{0} = 0.5611 \frac{β}{v_{f}}

Γ = \frac{π^{2}}{2} \overset{̅}{β} v_{f}^{2}

γ_{0} P_{0} = \frac{2 Γ}{α} = 1.1346 Γ τ_{0}

In Ref. [8], the Cornell group have reported the operation of a mode-locked ytterbium (Yb) fiber laser working completely in normal dispersion. In their numerical simulations, they also show how the pulse duration along the propagation distance varies for a laser having a total SOD β̄=0.1ps ² and for a pulse having exactly the same spectral width as the gain bandwidth namely ν _g=ν _f=2.82ps ^-1. For these parameters, Eqs. (17) predict an average pulse width τ ₀=3.16ps, a total phase shift Γ=1.25π and the energy per pulse should be $P_{0} = \frac{14.1}{γ_{0}}$ .

According to their numerical simulations, the pulse width varies from 2.5ps to 4ps for an average value of τ ₀=3.25ps which is very close to the one predicted here. The length of their laser was 4 meters, the assumed Kerr parameter was γ ₀=3.4W ^-1 km ^-1 (personal communication with Prof. Wise), and the predicted energy here should be P ₀=1nJ. In their simulations, they claimed a pulse energy of 2nJ. In the present model, the losses have been distributed along the laser whereas in a practical laser, most of the losses occur where the coupling with the NPE device occurs. Neglecting the internal loss, the loss coefficient parameter l can be related to the transmission T by:

e^{- 2 l} = T = 1 - R

where R is the reflection coefficient. Now the threshold condition for the mode-locked gain becomes

R e^{2 g} = 1

The last two equations, when used for the gain/loss term (g-l) of Eq. (17) result in:

{(\frac{v_{f}}{v_{g}})}^{2} = 2 \ln (\frac{R}{(1 - R)})

In order to build a system where ν _f=ν _g, the reflection coefficient should be R=0.6225. For a reflection coefficient R=0.6, Eq. (20) predicts a spectral width ν _f=0.9ν _g. In such a laser, the pulse energy varies from a minimum to a maximum according to P _max=e ^2g P _min after the passage through the amplifier section. In a similariton laser, different numerical simulations [6,9] have shown that the spectral width varies only slightly during the propagation. Assuming also that the pulse profile maintains its shape along the propagation, Eqs. (17c) and (17b) allow one to deduce that the pulse width should evolve as ((τ ₀)_max=(τ ₀)_min e ^2g). In terms of the average pulse width given by Eq. (17b), the pulse breathing can be written as:

{(τ_{0})}_{max} = \frac{2 τ_{0}}{(1 + R)}

{(τ_{0})}_{min} = {R (τ_{0})}_{max}

For the ideal numerical simulation of Ref. [7] with R=0.6225, I find that (τ ₀)_max=3.9ps and (τ ₀)_min=2.42_ps which are very close to their results.

For this exemple, the chirp parameter β=16 and the NPE coefficient is ε ₀=0.125. It can therefore be observed that while in similariton operation, the contribution of the NPE mecanism to the mode-locking of the laser is much smaller than the contribution of the Kerr mecanism. However, the contribution of the NPE is small but necessary. For example, if one tries to exploit the whole gain bandwidth of this Yb laser (ν _g=4×2.82ps ^-1) and operate at the optimal point with ν _f=ν _g, the value of the chirp turns out to be β=256 and the NPE coefficient ε ₀=0.004. I presume that this value of ε ₀ is too small to start the mode-locking mecanism and I also presume that this is the reason why the Cornell group [8] have introduced a spectral filter.

5. Frequency filter along with NPE

Adding a frequency filter in front of the gain medium will only reduce the gain bandwidth and might, if properly adapted, enable operation of the laser at the optimal point as described above. The Cornell group have introduced in their system a frequency filter (ν ₀=2.82ps ^-1) and were able to generate pulses having ν _f=5.64ps ^-1 because they have put their filter at a critical place namely combining it with their NPE device.

The combined action of the filter and the NPE can be modelled by first observing the local change that the NPE makes on the pulse. In appendix B of Ref.[5], I have shown, working in the frequency domain, that the saturable absorber operation produced by the NPE device is equivalent to changing the spectrum according to :

{\hat{V}}_{NPE} = {\hat{V}}_{i} \exp (\frac{- ε_{0} γ_{0} V_{0}^{2} ω^{2}}{α^{2} β^{2}}) for β > > 1

For the optimal operation point in the similariton regime, $γ_{0} V_{0}^{2} = \frac{\bar{β} α^{2} β^{2}}{2}$ and the operation of the NPE device becomes:

{\hat{V}}_{NPE} = {\hat{V}}_{i} \exp (\frac{- ε_{0} \overset{̅}{β} ω^{2}}{2})

Next, combining a frequency filter of width ω _0F (ν _0F) with the NPE device changes the NPE operation to:

{\hat{V}}_{NPE + FILTER} = {\hat{V}}_{i} \exp [- ω^{2} (\frac{ε_{0} \overset{̅}{β}}{2} + \frac{1.3863}{ω_{0 F}^{2}})]

This combined operation is equivalent to modifying the NPE coefficient by:

ε_{0} \to ε_{0} + \frac{0.07}{β v_{0 F}^{2}}

Notice here that this result does not mean that you can get rid of the NPE device and replace it by a passive frequency filter. In fact the NPE device is a dynamic device that starts the mode-locking operation and in the frequency domain the associated frequency filter amplifies its action.

The operation (Eq. (25)) can also be used to calculate the chirp parameter β of the new laser system according to:

\frac{\overset{̅}{β}}{β} = \frac{0.05}{v_{g}^{2}} + \frac{0.0351}{v_{0 F}^{2}}

For the experimental results of Ref. [8] with ν _0F=2.82ps ^-1, ν _g=11.28ps ^-1 and β̄=0.1ps ², an estimated chirp parameter β=20.8 can be calculated. The effective NPE parameter ε ₀=0.0962 is small now but not negligeble as the one calculated without the filter. Using the reported observed spectral width ν _F=5.64ps ^-1, Eq. (17) predicts an average pulse width τ ₀=2.07ps, a phase shift Γ=5π and an energy per pulse of P ₀=2.71nJ. Finally, Eq. (20) yields a reflection factor R=0.5312 and Eq. (21) gives a minimum pulse (τ ₀)min=1.436ps which is very close to the experimentally observed value (τ ₀=1.4ps). The width of the chirp-free pulse was measured to be 170 fs while the estimated one using τ _f ν _f=0.9309 of Ref. [5] is 165 fs. P ₀ is the pulse energy necessary to support the solitary solution of the system. Before being ejected (1-R) from the cavity, the pulse is amplified (e ^2g) by the local gain medium. Thus, I can write P _out=(1-R)e ^2g P ₀ and combining this relation to the threshold condition R=e ^-2g, gives the following relation:

P_{out} = \frac{(1 - R)}{R} P_{0}

For the experiment discussed here, the energy of the coupled exiting pulse should be P _out=2.4nJ which is very close to the measured one.

6. Third-order dispersion (TOD) and optimal operation point

For some similariton lasers, TOD can be high enough (due to the presence of a grating) to perturb significantly the solitary pulse solution (Eq. 2). However, adding the self-steepening (SS) and intrapulse Raman scattering (IRS) into the master equation model enables the same form of solitary pulse to be derived (see Ref. [5]) namely:

V (τ, x) = V_{0} {sech [α (τ + b x)]}^{1 - i β} \exp [i (a τ - Γ x)]

The main difference is now that the pulse is frequency shifted by a factor a=2πΔν relative to the gain spectrum (In Eq. (2) of Ref. [5], the frequency shift factor is written as α which is to be corrected). Using the exact Eqs. (A2) and (A3) of this reference, it is also possible to fix an optimum condition for the gain $(\frac{\partial}{\partial β} (g - l) = 0)$ . The resulting condition can be expressd as:

\overset{̅}{β} = ε β + \frac{5}{3} β_{3} a

β_{3} a = \frac{- 3 ε}{β} \frac{a^{2}}{α^{2}}

For the similariton condition β≫ 1,Eqs. (29) can be written as:

\overset{̅}{β} = ε β [1 - 20 {(\frac{Δ v}{v_{F}})}^{2}]

and the optimum gain becomes:

{(g - l)}_{op} = - \frac{ε}{2} π^{2} v_{F}^{2} [1 - 20 {(\frac{Δ v}{v_{F}})}^{2}]

The mode-locked gain will be negative only if the frequency shift Δν<0.234ν _F. In their experimental setup [6], the similariton starts mode-locking when Δν=1.9THz with a pulse spectral width ν _F=8.4ps ^-1 which corresponds almost exactly to Δν=0.234ν _F. At this operation point, the calculated TOD, according to Eq. (29b), is β ₃=0.000576ps ³ which is close to their measured value of β ₃=0.00066ps ³ (these experimental parameters are from a personal communication with A. Rühl).

7. Conclusion

The optimal operation point, derived in this communication, seems to be a natural and stable operation point for the highly-chirped regime (β≫1) of a fiber mode-locked laser. Indeed, the pulse parameters associated to this optimization appears to be close to the full numerical ones and also to some measured ones. The pulse breathing, introduced (Eq. 21) in the model, could be a useful parameter to check, in the laboratory, how close a design is from the ideal one. This simple GL model has confirmed the positive contribution of the combined action of the NPE and the frequency filter. The extended GL model has shown that TOD is a limiting factor to the optimal operation point and imposes a threshold condition to the frequency shift.

In appendix A, I have shown that, for a given gain (g-l) and gain bandwidth ε, the energy of the mode-locked pulse is maximum for a certain NPE parameter ε ₀ (Eq. A2). For a highly-chirped pulse (β≫1), the optimal parameter ε ₀ for the energy and for the gain (g-l) is the same, i.e $ε_{0} = \frac{2}{β}$ . According to this GL distributed model, I think that these two conditions can serve as a clear definition of the so-called similariton regime of a mode-locked laser. This regime has also been called gain-guided soliton [12] in order to emphasize the existence of a soliton-type pulse in a normal dispersion fiber in presence of gain. In a laser system, the mode-locked pulse is generated and formed by gain bandwidth, the nonlinear Kerr effect and the mode-locking mechanism. In the present GL distributed model, the gain bandwidth can be totally eliminated by setting ε=0 in Eq. (7) and the NPE parameter $ε_{0} = \frac{3}{β}$ for β≫1. This result shows that a similariton pulse can be obtained for a uniform gain that compensates the loss but the NPE device must be 1.5 times effective than in the presence of a gain bandwidth. Of course, in such a situation, the laser system will be only be marginally stable (g-l=0).

A. Appendix A

The optimal operation point was derived after optimizing the mode-locked gain in order to ensure stable operation in the mode-locked regime. However, one may prefer trying to find an operation point that maximizes the energy of the pulse. In this appendix, I shall show that in the similariton regime, the optimal point for energy is the same as the one derived for the gain. Using Eqs. (14)–(17), it is possible to write the energy as:

{(γ_{0} P_{0})}^{2} = \frac{{2 (g - l) ε (β^{2} + 1) (β^{2} + 4)}^{2}}{[ε_{0} (β^{2} - 2) - 3 β] [ε_{0} (β^{2} + 2) - β]}

In the laser system, ε is fixed and (g-l) is adjusted in order to reach mode-locking operation and finally the NPE device is tuned to achieve the best pulse output. In this model, tuning the NPE device changes the ε ₀ parameter. Therefore it is possible to find the best operation point after requiring that $\frac{\partial}{\partial ε_{0}} P_{0} = 0$ . The result is:

ε_{0} = \frac{2 β (β^{2} + 1)}{(β^{4} - 4)}

ε ₀ must be positive and we have to conclude that a positive solution exists if β ₂>2. For this particular of ε ₀, the SOD is:

\overset{̅}{β} = \frac{ε (β^{4} + 2)}{β (β^{2} - 2)}

Again, this optimal point will also be in the normal dispersion regime (β̄>0) with the gain negative (g-l<0).

For the similariton regime, i.e β≫1, Eq. (A2) becomes:

ε_{0} = \frac{2}{β}

which is exactly the same expression derived for the optimum gain.

B. Appendix B-The chirp parameter β and the root-mean-square (RMS) chirp

The RMS chirp of the solitary pulse given by Eq. (2) is related to the second-order moment (SOM) [10] σ² by:

C = \frac{β}{2 σ^{2}}

where $σ^{2} = \frac{π^{2}}{12 α^{2}}$ for a sech profile.

In the similariton regime, i.e β≫1, the RMS chirp can be writen as:

C = 3.3665 \frac{v_{F}}{τ_{0}}

The minimum and maximum RMS chirp can be estimated from this equation after using the appropriate value for (τ ₀)_min and (τ ₀)_max of Eqs. (21). The chirp of the pulse can also be characterized by evaluating the amount of dispersion D (ps ²) needed to un-chirp the pulse. According to SOM pulse propagation, it can be shown that:

D = \frac{C σ^{2}}{\hat{σ^{2}}}

where ${\hat{σ}}^{2} = \frac{α^{2} (β^{2} + 1)}{3}$ for the solitary pulse given by Eq. (2). For the similariton regime, the un-chirped dispersion D will be given by:

D = \frac{- 0.271 τ_{0}}{v_{f}}

The sign of D is of course negative here because β̄ is positive in the laser system. The dispersion D will be minimum if the output coupling is at (τ ₀)_min. This evaluation is valid for a linear unchirping of the pulse. Ortaç [11] et al have already used this parameter to characterize the chirp of their similariton laser. From their experimental data of τ ₀=5.6ps and spectral bandwidth of ν _F=7ps ^-1. Equation (B4) than yields D=-0.217ps ² which is very close to their measured value D=-0.2ps ².

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