5.5: 5.5 Active Mode Locking with Soliton Formation

Experimental results with fiber lasers [8, 9, 11] and solid state lasers [10] indicated that soliton shaping in the negative GDD regime leads to pulse stabilization and considerable pulse shorting. With sufficient negative dispersion and self-phase modulation in the system and picosecond or even femtosecond pulses, it is possible that the pulse shaping due to GDD and SPM is much stronger than due to modulation and gain filtering, see Figure 5.8. The resulting master equation for this case is

\[T_R \dfrac{\partial A}{\partial T} = \left [g + (D_g - j |D|) \dfrac{\partial ^2}{\partial t^2} - l - M (1 - \cos (\omega_M t)) - j \delta |A|^2 \right ] A.\label{eq5.5.1} \]

For the case, that soliton formation takes over, the steady state solution a soliton plus a continuum contribution

\[A(T, t) = (a(x) e^{jpt} + a_c (T, t)) e^{-j \theta}\label{eq5.5.2} \]

with

\[a(x) = A \text{sech} (x), \text{ and } x = \dfrac{1}{\tau} (t + 2 D \int_0^T p (T') dT' - t_0) \nonumber \]

where \(a_c\) is the continuum contribution. The phase is determined by

\[\theta (T) = \theta_0 (T) - \dfrac{D}{T_R} \int_0^T \left (\dfrac{1}{\tau (T')^2} - p(T')^2 \right ) dT', \nonumber \]

whereby we always assume that the relation between the soliton energy and soliton width is maintained (3.3.7)

\[\dfrac{|D|}{\tau (T)^2} = \dfrac{\delta A(T)^2}{2} \nonumber \]

We also allow for a continuous change in the soliton amplitude \(A\) or energy \(W = 2A^2 \tau\) and the soliton variables phase \(\theta_0\), carrier frequency \(p\) and timing \(t_0\). \(\phi_0\) is the soliton phase shift per roundtrip

\[\phi_0 = \dfrac{|D|}{\tau^2}. \nonumber \]

However, we assume that the changes in carrier frequency, timing and phase stay small. Introducing (\(\ref{eq5.5.2}\)) into (\(\ref{eq5.5.1}\)) we obtain according to the soliton perturbation theory developed in chapter 3.5

\[\begin{array} {ll} \ & {T_R \left [\dfrac{\partial a_c}{\partial T} + \dfrac{\partial W}{\partial T} \text{f}_{\omega} + \dfrac{\partial \Delta \theta}{\partial T} \text{f}_{\theta} + \dfrac{\partial \Delta p}{\partial T} \text{f}_{p} + \dfrac{\partial \Delta t}{\partial T} \text{f}_{t} \right ]} \\ = & {\phi_0 L(a_c + \Delta p \text{f}_p) + R(a + \Delta p \text{f}_p + ac) - M \omega_M \sin (\omega_M \tau x) \Delta t a(x)} \end{array}\label{eq5.5.7} \]

The last term arises because the active modelocker breaks the time invariance of the system and leads to a restoring force pushing the soliton back to its equilibrium position. \(L, R\) are the operators of the linearized NSE and of the active mode locking scheme, respectively

\[R = g\left ( 1 + \dfrac{1}{\Omega_g^2 \tau^2} \dfrac{\partial ^2}{\partial x^2} \right ) - l - M (1 - \cos (\omega_M \tau x)), \nonumber \]

The vectors \(\text{f}_{\omega}\), \(\text{f}_{\theta}\), \(\text{f}_p\) and \(\text{f}_t\) describe the change in the soliton when the soliton energy, phase, carrier frequency and timing varies.

Stability Condition

We want to show, that a stable soliton can exist in the presence of the modelocker and gain dispersion if the ratio between the negative GDD and gain dispersion is sufficiently large. From (\(\ref{eq5.5.7}\)) we obtain the equations of motion for the soliton parameters and the continuum by carrying out the scalar product with the corresponding adjoint functions. Specifically, for the soliton energy we get

\[T_R \dfrac{\partial W}{\partial T} = 2 \left (g - l - \dfrac{g}{3\Omega_g^2 \tau^2} - \dfrac{\pi^2}{24} M\omega_M^2 \tau^2 \right ) W + < \text{f}_{\omega}^{(+)}| Ra_c>. \nonumber \]

We see that gain saturation does not lead to a coupling between the soliton and the continuum to first order in the perturbation, because they are or- thogonal to each other in the sense of the scalar product (3.5.23). This also means that to first order the total field energy is contained in the soliton.

Thus to zero order the stationary soliton energy \(W_0 = 2A_0^2 \tau\) is determined by the condition that the saturated gain is equal to the total loss due to the linear loss \(l\), gain filtering and modulator loss

\[g - l = \dfrac{\pi^2}{24} M \omega_M^2 \tau^2 + \dfrac{g}{3 \Omega_g^2 \tau^2}\label{eq5.5.10} \]

with the saturated gain

\[g = \dfrac{g_0}{1 + W_0/E_L}. \nonumber \]

Linearization around this stationary value gives for the soliton perturbations

\[T_R \dfrac{\partial \Delta W}{\partial T} = 2 \left (-\dfrac{g}{(1 + W_0/E_L)} \left (\dfrac{W_0}{E_L} + \dfrac{1}{3\Omega_g^2 \tau^2} \right ) + \dfrac{\pi^2}{12} M \omega_M^2 \tau^2 \right ) \Delta W + < \text{f}_{\omega}^{(+)} | Ra_c>\label{eq5.5.12} \]

\[T_R \dfrac{\partial \Delta \theta}{\partial T} = <\text{f}_{\theta}^{(+)} | Ra_c>\label{eq5.5.13} \]

\[T_R \dfrac{\partial \Delta p}{\partial T} = -\dfrac{4g}{3\Omega_g^2 \tau^2} \Delta p + < \text{f}_{p}^{(+)} | Ra_c> \nonumber \]

\[T_R \dfrac{\partial \Delta t}{\partial T} = -\dfrac{\pi^2}{6} M \omega_M^2 \tau^2 \Delta t + 2|D| \Delta p + <\text{f}_{t}^{(+)} | Ra_c>\label{eq5.5.15} \]

and for the continuum we obtain

\[\begin{array} {rcl} {T_R \dfrac{\partial g(k)}{\partial T}} & = & {j \Phi_0 (k^2 + 1) g(k) + <\text{f}_k^{(+)} | Ra_c >} \\ {} & + & {< \text{f}_k^{(+)} | R(a_0(x) + \Delta \omega \text{f}_{\omega} + \Delta p f_p) >} \\ {} & - & {< \text{f}_k^{(+)} | M \omega_M \sin (\omega_M \tau x) a_0 (x)> \Delta t} \end{array}\label{eq5.5.16} \]

Thus the action of the active modelocker and gain dispersion has several effects. First, the modelocker leads to a restoring force in the timing of the soliton (\(\ref{eq5.5.15}\)). Second, the gain dispersion and the active modelocker lead to coupling between the perturbed soliton and the continuum which results in a steady excitation of the continuum.

However, as we will see later, the pulse width of the soliton, which can be stabilized by the modelocker, is not too far from the Gaussian pulse width by only active mode locking. Then relation

\[\omega_M \tau \ll 1 \ll \Omega_g \tau \nonumber \]

is fulfilled. The weak gain dispersion and the weak active modelocker only couples the soliton to the continuum, but to first order the continuum does not couple back to the soliton. Neglecting higher order terms in the matrix elements of eq.(\(\ref{eq5.5.16}\)) [6] results in a decoupling of the soliton perturbations from the continuum in (\(\ref{eq5.5.12}\)) to (\(\ref{eq5.5.16}\)). For a laser far above threshold, i.e. \(W_0/E_L \gg 1\), gain saturation always stabilizes the amplitude perturbation and eqs.(\(\ref{eq5.5.13}\)) to (\(\ref{eq5.5.15}\)) indicate for phase, frequency and timing fluctuations. This is in contrast to the situation in a soliton storage ring where the laser amplifier compensating for the loss in the ring is below threshold [14].

By inverse Fourier transformation of (\(\ref{eq5.5.16}\)) and weak coupling, we obtain for the associated function of the continuum

\[T_R \dfrac{\partial G}{\partial T} = \left [ g - l + j \Phi_0 + \dfrac{g}{\Omega_g^2} (1 - jD_n) \dfrac{\partial^2}{\partial t^2} - M(1 - \cos (\omega_M t)) \right ] G + \mathcal{F}^{-1} \left \{<\text{f}_k^{(+)} | Ra_0 (x)> - <\text{f}_k^{(+)} |M \omega_M \sin (\omega_M \tau x) a_0 (x) > \Delta t\right \}\label{eq5.5.18} \]

where \(D_n\) is the dispersion normalized to the gain dispersion

Note, that the homogeneous part of the equation of motion for the continuum, which governs the decay of the continuum, is the same as the homogeneous part of the equation for the noise in a soliton storage ring at the position where no soliton or bit is present [14]. Thus the decay of the continuum is not affected by the nonlinearity, but there is a continuous excitation of the continuum by the soliton when the perturbing elements are passed by the soliton. Thus under the above approximations the question of stability of the soliton solution is completely governed by the stability of the continuum (\(\ref{eq5.5.18}\)). As we can see from (\(\ref{eq5.5.18}\)) the evolution of the continuum obeys the active mode locking equation with GVD but with a value for the gain determined by (\(\ref{eq5.5.10}\)). In the parabolic approximation of the cosine, we obtain again the Hermite Gaussians as the eigensolutions for the evolution operator but the width of these eigensolutions is now given by

\[\tau_c = \tau_a \sqrt[4] {(1 - j D_n)}. \nonumber \]

and the associated eigenvalues are

\[\lambda_m = j\Phi_0 + g - l - M \omega_M^2 \tau_a^2 \sqrt{(1 - j D_n)} (m + \dfrac{1}{2}).\label{eq5.5.20} \]

The gain is clamped to the steady state value given by condition (\(\ref{eq5.5.10}\)) and we obtain

\[\lambda_m = +j \Phi_0 + \dfrac{1}{3} \sqrt{D_g M_s} \left [\left (\dfrac{\tau_a}{\tau} \right)^2 + \dfrac{\pi^2}{4} \left (\dfrac{\tau_a}{\tau} \right)^{-2} - 6\sqrt{(1 - j D_n)} (m + \dfrac{1}{2}) \right ].\label{eq5.5.21} \]

Stability is achieved when all continuum modes see a net loss per roundtrip, \(\text{Re} \{ \lambda_m \} < 0\) for \(m \ge 0\), i.e. we get from (\(\ref{eq5.5.21}\))

\[\left (\dfrac{\tau_a}{\tau} \right)^2 + \dfrac{\pi^2}{4} \left (\dfrac{\tau}{\tau_a} \right)^2 < 3 \text{Re} \{\sqrt{(1 - j D_n)} \}.\label{eq5.5.22} \]

Relation (\(\ref{eq5.5.22}\)) establishes a quadratic inequality for the pulse width reduction ratio \(\xi = (\tau_a/\tau)^2\), which is a measure for the pulse width reduction due to soliton formation

\[\xi^2 - 3 \text{Re} \{ \sqrt{(1 - j D_n)} \} \xi + \dfrac{\pi^2}{4} < 0.\label{eq5.5.23} \]

As has to be expected, this inequality can only be satisfied if we have a minimum amount of negative normalized dispersion so that a soliton can be formed at all

\[D_{n, crit} = 0.652. \nonumber \]

Therefore our perturbation ansatz gives only meaningful results beyond this critical amount of negative dispersion. Since \(\xi\) compares the width of a Gaussian with that of a secant hyperbolic it is more relevant to compare the full width half maximum of the intensity profiles [?] of the corresponding pulses which is given by

\[R = \dfrac{1.66}{1.76} \sqrt{\xi}. \nonumber \]

Image removed due to copyright restrictions.

Please see:
Kaertner, F., D. Kopf, and U. Keller. "Solitary-pulse stabilization and shortening in actively mode-locked lasers." Journal of the Optical Society of America B 12, no. 3 (March 1995): 486.

Figure 5.9: Pulsewidth reduction as a function of normalized dispersion. Below \(D_{n, crit} = 0.652\) no stable soliton can be formed.

Figure 5.9 shows the maximum pulse width reduction \(R\) allowed by the stability criterion (\(\ref{eq5.5.23}\)) as a function of the normalized dispersion. The critical value for the pulse width reduction is \(R_{crit} \approx 1.2\). For large normalized dispersion Figure 1 shows that the soliton can be kept stable at a pulse width reduced by up to a factor of 5 when the normalized dispersion can reach a value of 200. Even at a moderate negative dispersion of \(D_n = 5\), we can achieve a pulsewidth reduction by a factor of 2. For large normalized dispersion the stability criterion (\(\ref{eq5.5.23}\)) approaches asymptotically the behavior

\[\xi , \sqrt{\dfrac{9D_n}{2}} \text{ or } R< \dfrac{1.66}{1.76} \sqrt[4]{\dfrac{9D_n}{2}}.\label{eq5.5.26} \]

Thus, the possible pulse-width reduction scales with the fourth root of the normalized dispersion indicating the need of an excessive amount of dispersion necessary to maintain a stable soliton while suppressing the continuum. The physical reason for this is that gain filtering and the active modelocker continuously shed energy from the soliton into the continuum. For the soliton the action of GVD and SPM is always in balance and maintains the pulse shape. However, as can be seen from (\(\ref{eq5.5.18}\)), the continuum, which can be viewed as a weak background pulse, does not experience SPM once it is generated and therefore gets spread by GVD. This is also the reason why the eigenstates of the continuum consist of long chirped pulses that scale also with the fourth root of the dispersion (\(\ref{eq5.5.20}\)). Then, the long continuum pulses suffer a much higher loss in the active modulator in contrast to the short soliton which suffers reduced gain when passing the gain medium due to its broader spectrum. The soliton is stable as long as the continuum sees less roundtrip gain than the soliton.

In principle by introducing a large amount of negative dispersion the theory would predict arbitrarily short pulses. However, the master equation (\(\ref{eq5.5.1}\)) only describes the laser system properly when the nonlinear changes of the pulse per pass are small. This gives an upper limit to the nonlinear phase shift \(\Phi_0\) that the soliton can undergo during one roundtrip. A conservative estimation of this upper limit is given with \(\Phi_0 = 0.1\). Then the action of the individual operators in (\(\ref{eq5.5.1}\)) can still be considered as continuous. Even if one considers larger values for the maximum phase shift allowed, since in fiber lasers the action of GVD and SPM occurs simultaneously and therefore eq.(\(\ref{eq5.5.1}\)) may describe the laser properly even for large nonlinear phase shifts per roundtrip, one will run into intrinsic soliton and sideband instabilities for \(\Phi_0\) approaching \(2\pi\) [30, 31]. Under the condition of a limited phase shift per roundtrip we obtain

\[\tau^2 = \dfrac{|D|}{\Phi_0}.\label{eq5.5.27} \]

Thus from (5.2.11), the definition of \(\xi\), (\(\ref{eq5.5.26}\)) and (\(\ref{eq5.5.27}\)) we obtain for the maximum possible reduction in pulsewidth

\[R_{\max} = \dfrac{1.66}{1.76} \sqrt[12]{\dfrac{(9\Phi_0 /2)^2}{D_gM_s}}\label{eq5.5.28} \]

and therefore for the minimum pulsewidth

\[\tau_{\min} = \sqrt[6] {\dfrac{2D_g^2}{9\Phi_0 M_s}} \nonumber \]

The necessary amount of normalized negative GVD is then given by

\[D_n = \dfrac{2}{9} \sqrt[3] {\dfrac{(9\Phi_0/2)^2}{D_gM_s}}\label{eq5.5.30} \]

Image removed due to copyright restrictions.

Please see:
Kaertner, F., D. Kopf, and U. Keller. "Solitary-pulse stabilization and shortening in actively mode-locked lasers."

Journal of the Optical Society of America B 12, no. 3 (March 1995): 486.

Table 5.1: Maximum pulsewidth reduction and necessary normalized GVD for different laser systems. In all cases we used for the saturated gain \(g = 0.1\) and the soliton phase shift per roundtrip \(\Phi_0 = 0.1\). For the broadband gain materials the last column indicates rather long transient times which calls for regenerative mode locking.

Eqs.(\(\ref{eq5.5.28}\)) to (\(\ref{eq5.5.30}\)) constitute the main results of this paper, because they allow us to compute the possible pulse width reduction and the necessary negative GVD for a given laser system. Table (5.1) shows the evaluation of these formulas for several gain media and typical laser parameters.

Table 5.1 shows that soliton formation in actively mode-locked lasers may lead to considerable pulse shortening, up to a factor of 10 in Ti:sapphire. Due to the 12th root in (\(\ref{eq5.5.28}\)) the shortening depends mostly on the bandwidth of the gain material which can change by several orders of magnitude for the different laser materials. The amount of negative dispersion for achieving this additional pulse shortening is in a range which can be achieved by gratings, Gires-Tournois interferometers, or prisms.

Of course, in the experiment one has to stay away from these limits to suppress the continuum sufficiently. However, as numerical simulations show, the transition from stable to instable behaviour is remarkably sharp. The reason for this can be understood from the structure of the eigenvalues for the continuum (\(\ref{eq5.5.20}\)). The time scale for the decay of transients is given by the inverse of the real part of the fundamental continuum mode which diverges at the transition to instability. Nevertheless, a good estimate for this transient time is given by the leading term of the real part of (\(\ref{eq5.5.20}\))

\[\dfrac{\tau_{trans}}{T_R} = \dfrac{1}{\text{Re} \{ \lambda_0 \}} \approx \dfrac{3}{\sqrt{D_g M_s} R^2}\label{eq5.5.31} \]

This transient time is also shown in Table (5.1) for different laser systems. Thus these transients decay, if not too close to the instability border, on time scales from approximately 1,000 up to some 100,000 roundtrips, depending strongly on the gain bandwidth and modulation strength. Consequently, to first order the eigenvalues of the continuum modes, which are excited by the right hand side of (\(\ref{eq5.5.16}\)), are purely imaginary and independent of the mode number, i.e. \(\lambda_n \approx j \Phi_0\). Therefore, as long as the continuum is stable, the solution to (\(\ref{eq5.5.16}\)) is given by

\[G(x) = \dfrac{-j}{\Phi_0} \mathcal{F}^{-1} \left \{ < \text{f}_k^{(+)}|Ra_0 (x) > - M_s \tau^2 < \text{f}_k^{(+)}|x a_0 (x) > \dfrac{\Delta t}{\tau} \right \}. \nonumber \]

Thus, in steady state the continuum is on the order of

\[|G(x)| \approx \dfrac{A_0}{\Phi_0} \dfrac{D_g}{\tau^2} = \dfrac{A_0}{D_n}.\label{eq5.5.33} \]

which demonstrates again the spreading of the continuum by the dispersion. Equation (\(\ref{eq5.5.33}\)) shows that the nonlinear phase shift of the solitary pulse per round trip has to be chosen as large as possible. This also maximizes the normalized dispersion, so that the radiation shed from the soliton into the continuum changes the phase rapidly enough such that the continuum in steady state stays small. Note that the size of the generated continuum according to (\(\ref{eq5.5.33}\)) is rather independent of the real part of the lowest eigenvalue of the continuum mode. Therefore, the border to instability is very sharply defined. However, the time scale of the transients at the transition to instability can become arbitrarily long. Therefore, numerical simulations are only trustworthy if the time scales for transients in the system are known from theoretical considerations as those derived above in (\(\ref{eq5.5.31}\)). The simulation time for a given laser should be at least of the order of 10 times \(\tau_{trans}\) or even longer, if operated close to the instability point, as we will see in the next section.

Numerical simulations

Table 5.1 shows that soliton formation in actively mode-locked lasers may lead to considerable pulse shortening, up to a factor of 10 in Ti:sapphire. We want to illustrate that at the example of a \(\ce{Nd:YAG}\) laser, which is chosen due to its moderate gain bandwidth, and therefore, its large gain dispersion. This will limit the pulsewidth reduction possible to about 3, but the decay time of the continuum (\(\ref{eq5.5.31}\)) (see also Table 5.1) is then in a range of 700 roundtrips so that the steady state of the mode-locked laser can be reached with moderate computer time, while the approximations involved are still satisfied. The system parameters used for the simulation are shown in table 5.2. For the simulation of eq.(\(\ref{eq5.5.1}\)) we use the standard split-step Fourier transform method. Here the discrete action of SPM and GDD per roundtrip is included by choosing the integration step size for the \(T\) integration to be the roundtrip time \(T_R\). We used a discretisation of 1024 points over the bandwidth of \(1THz\), which corresponds to a resolution in the time domain of \(1ps\). The following figures, show only one tenth of the simulated window in time and frequency.

Table 5.2: Parameters used for numerical simulations

parameter	value
\(l\)	0.1
\(g_0\)	1
\(P_L\)	1\(W\)
\(\Omega_g\)	\(2\pi \cdot 60 GHz\)
\(\omega_M\)	\(2\pi \cdot 0.25 GHz\)
\(T_R\)	\(4ns\)
\(M\)	0.2
\(\delta\)	\(1.4 \cdot 10^{-4} W^{-1}\)
\(D\)	\(-17 ps^2 / -10 ps^2\)

Image removed due to copyright restrictions.

Please see:
Kaertner, F., D. Kopf, and U. Keller. "Solitary-pulse stabilization and shortening in actively mode-locked lasers." Journal of the Optical Society of America B 12, no. 3 (March 1995): 486.

Figure 5.10: Time evolution of the pulse intensity in a \(\ce{Nd:YAG}\) laser for the parameters in Table 5.2, \(D = −17ps^2\), for the first 1,000 roundtrips in the laser cavity, starting with a 68ps long Gaussian pulse.

Figure 5.10 shows the result of the simulation starting with a 68-ps-long Gaussian pulse with a pulse energy of \(W = 40\) nJ for \(D_n = 24\), i.e. \(D = -17 \text{ps}^2\). For the given SPM coefficient this should lead to stable pulse shortening by a factor of \(R = 2.8\). Thus after at least a few thousand roundtrips the laser should be in steady state again with a FWHM pulsewidth of 24 ps. Figure 5.10 shows the pulse evolution over the first thousand round-trips, i.e. 4μs real time. The long Gaussian pulse at the start contains an appreciable amount of continuum. The continuum part of the solution does not experience the nonlinear phase shift due to SPM in contrast to the soliton. Thus the soliton interferes with the continuum periodically with the soliton period of \(T_{soliton}/T_R = 2\pi /\phi_0 = 20\pi\). This is the reason for the oscillations of the pulse amplitude seen in Figure 5.10 which vanish with the decay of the continuum. Note also that the solitary pulse is rapidly formed, due to the large nonlinear phase shift per roundtrip. Figure 5.11 shows the simulation in time and frequency domain over 10,000 roundtrips. The laser reaches steady state after about 4,000 roundtrips which corresponds to \(6 \times \tau_{trans}\) and the final pulsewidth is 24 ps in exact agreement with the predictions of the analytic formulas derived above.

Lower normalized dispersion of \(D_n = 15\) or \(D = -10 \text{ps}^2\) only allows for a reduction in pulsewidth by \(R = 2.68\). However, using the same amount of SPM as before we leave the range of stable soliton generation.

Image removed due to copyright restrictions.

Please see:
Kaertner, F., D. Kopf, and U. Keller. "Solitary-pulse stabilization and shortening in actively mode-locked lasers." Journal of the Optical Society of America B 12, no. 3 (March 1995): 486.

Figure 5.11: Time evolution of the intensity (a) and spectrum (b) for the same parameters as Figure 2 over 10,000 roundtrips. The laser reaches steady state after about 4,000 rountrips.

Image removed due to copyright restrictions.

Please see:
Kaertner, F., D. Kopf, and U. Keller. "Solitary-pulse stabilization and shortening in actively mode-locked lasers." Journal of the Optical Society of America B 12, no. 3 (March 1995): 486.

Figure 5.12: (a) Time evolution of the intensity in a \(\ce{Nd:YAG}\) laser for the parameters in Table 5.2 over the first 1,000 round-trips. The amount of negative dispersion is reduced to \(D = −10ps^2\), starting again from a 68ps long pulse. The continuum in this case does not decay as in Figure 5.2 and 5.3 due to the insufficient dispersion. (b) Same simulation over 50,000 round-trips.

Figure 5.12(a) shows similar to Figure 5.10 the first 1,000 roundtrips in that case. Again the solitary pulse is rapidly formed out of the long Gaussian initial pulse. But in contrast to the situation in Figure 5.10, the continuum does not any longer decay on this time scale. The dispersion is too low to spread the continuum rapidly enough. The continuum then accumulates over many roundtrips as can be seen from Figure 5.12(b). After about 10,000 roundtrips the continuum has grown so much that it extracts an appreciable amount of energy from the soliton. But surprisingly the continuum modes stop growing after about 30,000 roundtrips and a new quasi stationary state is reached.

Experimental Verification

The theory above explains very well the ps Ti:saphire experiments [10] in the regime where the pulses are stabilized by the active modelocker alone. Gires-Tournois interferometers were used to obtain large amounts of negative GDD to operate the laser in the stable soliton regime derived above. Here we want to discuss in more detail the experimental results obtained recently with a regeneratively, actively mode-locked Nd:glass laser [7], resulting in 310 fs. If SPM and GVD could be neglected, the weak modelocker would produce Gaussian pulses with a FWHM of \(\tau_{a,FWHM} = 10\) ps. However, the strong SPM prevents stable pulse formation. The negative dispersion available in the experiment is too low to achieve stable soliton formation, because the pulse width of the soliton at this power level is given by \(\tau = 4|D|/(\delta W) = 464\) fs, for the example discussed. The normalized dispersion is not large enough to allow for such a large pulse width reduction. Providing enough negative dispersion results in a 310 fs perfectly sech-shaped soliton-like pulse as shown in Figure 5.13. A numerical simulation of this case would need millions of roundtrips through the cavity until a stationary state is reached. That means milliseconds of real time, but would necessitate days of computer time. Also the transition to instable behaviour has been observed, which is the characteristic occurence of a short solitary fs-pulse together with a long ps-pulse due to the instable continuum as we have found in the numerical simulation for the case of a \(\ce{Nd:YAG}\) laser (see Figure 5.12(b)). Figure 5.14 shows the signal of a fast detector diode on the sampling oscilloscope. The detector has an overall bandwidth of \(25GHz\) and therefore can not resolve the fs-pulse, but can resolve the width of the following roughly 100ps long pulse.

Image removed due to copyright restrictions.

Please see:
Kaertner, F., D. Kopf, and U. Keller. "Solitary-pulse stabilization and shortening in actively mode-locked lasers." Journal of the Optical Society of America B 12, no. 3 (March 1995): 486.

Figure 5.13: Autocorrelation of the actively mode-locked pulse (solid line) and corresponding \(sech^2\) fit (dashed line) with additional soliton formation.

Image removed due to copyright restrictions.

Please see:
Kaertner, F., D. Kopf, and U. Keller. "Solitary-pulse stabilization and shortening in actively mode-locked lasers."

Journal of the Optical Society of America B 12, no. 3 (March 1995): 486.

Figure 5.14: Sampling signal of fast detector when the mode-locked laser operates at the transition to instability. The short fs pulse can not be resolved by the detector and therefore results in a sharp spike corresponding to the detector response time. In advance of the fs-pulse travels a roughly \(100ps\) long pulse.