## Abstract

The operation of an intracavity frequency shifted feedback (FSF) laser exhibits a remarkable range of properties, some of which have been described previously. Here we report a more complete analysis of the dependence of the output power upon pump-laser power, based on simulations with an extended rate equation model and the use of phase space analysis. The effect of FSF is discussed in detail. The simulation of the operation of a titanium-sapphire laser with FSF reveals five separate regimes of operation, a superset of those observed in experiment. We predict the thresholds for each of these regimes for FSF-lasers with titanium-sapphire or neodymium doped crystals as gain medium.

©2003 Optical Society of America

## 1. Introduction

Frequency shifted feedback (FSF) lasers have, since the early work by Kowalski and coworkers [1, 2], been studied by several groups [3, 4, 5, 6, 7]. These lasers display a rich variety of operating characteristics, ranging from regular short pulsations to seemingly chaotic behavior, and have been designed with various layouts. Figure 1 illustrates a representative linear layout. The optical path within a laser cavity (comprising a gain medium and various frequency-dependent filters) is closed by means of the first-order diffraction from a traveling acoustic wave at frequency *v _{AOM}* in an acousto-optic modulator (AOM). Light circulating in the cavity is thereby shifted in frequency by Δ

*v*=2·

*v*each round trip. The diffraction efficiency into the first order is very high, perhaps exceeding 90%. The light left in the zeroth order leaves the cavity as output.

_{AOM}The output behavior of a FSF-laser depends upon the interplay of various controllable parameters. These are, in particular, the population inversion (as established by the pump-laser power *P _{in}*) and several parameters that define the FSF operation such as the round trip time

*τ*, the frequency shift per round trip Δ

_{RT}*v*, and the frequency dependent losses per roundtrip,

*γ*. When the parameters are suitably chosen, the output is broadband, exhibiting stationary mode structure [2, 3, 4, 8, 9]. By

_{cav}*stationary*we mean, as in stochastic theory, that output properties depend only on time differences; by

*steady*we mean constant cw output. The specific operation characteristics make the FSF laser a useful device for optical pumping [10], laser cooling [11] and other applications [12, 13, 14].

When seeded with monochromatic light, the FSF laser generates a frequency-comb [15, 16, 17]. Other operation conditions lead to a train of short (picosecond) pulses [2, 18, 19]. In fact a FSF laser allows the robust creation of a self-starting train of short pulses (as short as 2 ps) with fiber lasers [19]. Work by Nakamura and coworkers [20, 21, 22] has shown that, with appropriately chosen parameters, the output can be regarded as a comb of steadily shifting frequency components (i.e., a chirped comb). Using a titanium-sapphire (Ti:Sa) gain medium, Bonnet et al. [7] demonstrated the existence of a number of operating regimes, such as continuous-wave output, sustained spiking, and trains of short pulses with a duration of 50 ps.

In the present article we report on results from detailed simulation studies of the operation of a Ti:Sa FSF laser, extending the rate-equation model described earlier [7]. From these simulations and new experimental results we offer new insights into the operation of a FSF laser, particularly the influence of pump power upon the output characteristics. The simulation applies to the regime of broadband output at low pump power. The model is not adequate for short pulse-train operation because such operation requires consideration of nonlinear effects (e.g. four-wave-mixing within the laser crystal) and phase information. Such short pulses occur at sufficiently high pump power. Furthermore, we demonstrate the usefulness of a phase-space analysis as a tool for understanding the dynamics of a FSF laser, and to identify general ways in which the output characteristics (output power, pulse patterns, etc.) depend upon both the specific operating conditions (e.g. pump power) and on the laser parameters (e.g. frequency shift, cavity round trip time).We use three approaches to reveal the FSF output dynamics as the controllable pump power *P _{in}* is varied:

- Examination of the gain and the field spectrum in frequency and time
- Examination of the variation of time-averaged output power
*P̅*_{out} - Examination of a bifurcation diagram of the instantaneous output power
*P*versus_{out}*P*_{in}

The combination of these three diagnostic tools provides a concise way to identify the various output regimes, and to delineate the ranges of input power where each occurs. The bifurcation diagram provides a concise overview of the various output scenarios of an FSF laser.

## 2. Previous results

We summarize briefly the results from the earlier work of Bonnet et al. [7] who used a set of rate equations for the number of inverted atoms *N* and the photon number *M* to model the variation of the output power with time. A plot of the calculated averaged output power versus the input pump power reproduces the experimental data quite well, except for the short pulse regime, where rate equations are expected to fail (see Fig. 10 in [7]).

Bonnet et al. [7] introduced a criterion for a stationary spectrum. In their view, the envelope of the spectrum undergoes filtering on each round trip. The resulting spectral shift must be balanced by the frequency shift introduced by the AOM (see Eq. (6) in [7]). If this stationarity criterion is not fulfilled at a specified pump power, low-frequency (microsecond period) pulsations occur, in which laser operation begins and ceases. As an example, from the simulation results one finds that for a pump power less than 10W(see Figs. 11 and 12 in [7]) oscillation of the population inversion, and the lasing action damp out so rapidly that no field energy remains in the resonator at the end of a cycle. When the pump power exceeds 10 W some field energy remains in the resonator and interferes with the buildup of the next pulse.

The results of [7] raise questions regarding the reason for the different types of pulsations. Furthermore, the role of spontaneous emission and its influence on the repetition rate was not analyzed. Finally, it is of interest to learn how the previous results, obtained for a Ti:Sa laser, apply to other laser media, characterized by a different set of parameters. To answer these questions we supplement traditional studies in the time domain by an examination of the phase space of output power, defined by the coordinates (*P _{out}, Ṗ_{out}*). We find and characterize a distinct set of operating regimes and compare the numerical results of simulation with the data from experiments.

## 3. The numerical model

#### 3.1 The rate-equation model

The rate equations comprise two sets of coupled ordinary differential equations. One set describes the frequency components of the field, as affected by atomic population. The other describes the changes in population produced by the lasing field and the pump field.

Earlier rate-equation work [7] has modeled the output intensity of an FSF laser by distributing intensity amongst a set of discrete frequency bins, rather than dealing with a single frequency as is common in conventional laser models that treat only monochromatic light. We follow this same general approach, with some changes to be noted. We take spontaneous emission into account by the inclusion of a stochastic component.We introduce a numerically robust method for changing the frequency at each round trip, incorporate variable parameters, and consider a broader range of gain media than in the earlier work.

### 3.1.1 The equations for the laser field

We characterize the spectrum of the radiation by a spectral distribution of photon density *ρ*(*v, t*) (the number of photons per Hz), rather than by a dimensionless photon number *M*(*v, t*), as was done in [7]. The function *ρ*(*v, t*) is zero for frequencies outside the selected range. We discretize frequency and time by *v _{i}*=

*iδv*and

*t*=

_{j}*jδt*respectively. With this discretization the spectral photon density becomes a matrix of values

*ρ*≡

_{i,j}*ρ*(

*v*). We obtain the photon number by the relationship

_{i}, t_{j}*M*(

*v, t*)=

*ρ*(

*v, t*)

*δv*.

The rate equation for the photon spectral density reads

where the index *i* identifies a spectral component while the index *j* measures the progress in time. Here *N*(*t _{j}*) is the population inversion,

*B*≡

_{i}*B*(

*v*) is the stimulated emission coefficient and

_{i}*A*≡

_{i}*A*(

*v*) is the spontaneous emission coefficient (discussed in Section

_{i}*3.3*). The coefficient

*γ*≡

_{i}*γ*(

_{cav}*v*) is the photon loss from the cavity in one round trip, including the frequency dependent losses related to the transmission function of the etalon

_{i}*T*(

_{etal}*v*) and frequency independent losses incorporated into

*ε*≡(1-

*v*)·(1-

_{abs}*v*) (see table I)

_{out}We take the etalon transmission to be given by the Airy-function

where *R* is the reflectivity and *V* is the loss due to the etalon. The frequency is referenced to the maximum of the transmission function *T _{etal}*(

*v*) at

*v*=

*v*

_{0}. Without FSF the laser would operate at

*v*

_{0}(see Fig. 2(a)). In our experiment a birefringent filter is used to select a particular maximum of the periodic transmission function of the etalon. The free spectral range (FSR) of the intracavity etalon is the frequency range [

*v*] used in our modeling.

_{min},v_{max}### 3.1.2 The equation for the gain medium

We idealize the Ti:Sa gain medium as a four-level system. Because the relaxation rate of the lower laser level is fast, the population inversion *N*(*t*) is the number of atoms in the upper laser level. The rate equation for *N _{j}*≡

*N*(

*t*) expresses the change of the inversion due to pumping, spontaneous emission and stimulated emission. It reads

_{j}Here *η* is the absorption efficiency of pump photons, *λ _{P}* is the wavelength of the pump field,

*P*is the pump power (in Watts) and

_{in}*γ*is the spontaneous emission rate. The total number of photons in the field art time

_{sp}*t*is

_{j}*M*=Σ

_{j}*.*

_{i}ρ_{i, j}δv### 3.1.3 Solving the equations

Table I defines all the parameters of the model and lists numerical values. We numerically integrated the rate Eqs. (4) and (1) using a time step of *δ _{t}*=0.1

*τ*=1.07 ns with an Euler integrator [for which the error is

_{RT}*O*(

*dt*)]. The binned frequency range [

^{2}*v*]=200 GHz is the free spectral range of the intracavity etalon. In order to implement the AOM frequency shift in a numerically robust way, we chose the frequency increment

_{min},v_{max}*δv*to be

*δv*=2Δ

*v*=160 MHz, with a shift implemented in the calculation once per round trip. The frequency shift of the photon density distribution is implemented by storing the result

_{AOM}*ρ*

_{i, j+1}of each integration step

*dt*into

*ρ*

_{i+1, j+1}. This procedure works even for a noisy spectrum, as occurs when the spontaneous emission is modeled as a stochastic process.

### 3.1.4 Modeling a continuous shift device

We considered also the question [23, 24]: Would the behavior differ if the frequency shift occurred continuously (rather than as the result of an AOM as in our experiments)?

To predict the behavior of such a hypothetical “continuous shift device”we made several computations in which a frequency shift *δv*′ occurred with each time step. The shifts *δv*′ accumulate to *δv*=2Δ_{AOM} for a full round trip. We used various time steps, ranging from 10 steps per round trip to 100 steps per round trip. Our computation shows that such a device would behave in a manner identical with the FSF laser.

#### 3.2 Modeling spontaneous emission

The irregular low frequency pulsations in *P _{out}* observed by Bonnet

*et al.*[7] for

*P*≈10 W were attributed to the stochastic nature of spontaneous emission. To test this hypothesis we employed two models of spontaneous emission. In one model the emission occurs at a steady rate (Eq. (6)). In the other model the emission is a random process (Eq. (7)). When treating the spontaneous emission as a steady process we rely on results of standard laser theory (see [27], Sections 7.8 and 10.6), according to which the spontaneous emission rate per mode for a conventional laser cavity (lacking FSF) is

_{in}*A*=

_{con}*Bi/ρ*. When the cavity is closed through the frequency-shifting AOM, a stationary mode structure no longer holds. Nevertheless we assume that the number of photons emitted into a frequency interval Δ

*v*=1/

_{mode}*τ*does not change when the FSF mechanism is activated. With this assumption, the number of photons emitted into interval

_{RT}*i*of width

*δv*is (see Ref. [7])

For the Ti:Sa setup of Ref. [7] we obtain *A _{i}*=6×10

^{4}[cm

^{-2}Hz

^{-1}s

^{-1}]

*σ*. We assume the cross-section σ

_{i}_{i}≡σ(

*v*) for the gain medium Ti:Sa to be a constant σ

_{i}_{i}=σ) over the frequency range (

*v*-

_{max}*v*)=200 GHz. For Nd:YAG we take

_{min}*σ*(

*v*) to be a Gaussian function of

*v*, with the maximum value σ and width Δ

*v*(see table I).

_{gain}When treating spontaneous emission as a stochastic process we take the expected number of seed-photons *s _{i, j}* spontaneously emitted into the frequency interval

*δv*around frequency

*v*during the time interval

_{i}*δ*around time

_{t}*t*to be

_{j}Typically the value of *s _{i, j}* lies within the interval (0,1).

To analyze both the influence of the spontaneous emission and the reaction of the laser system to small perturbations, we substitute at any fixed *t _{j}* for every

*s*a non-negative integer random variable

_{i, j}*X*such that

_{i, j}*s*=〈

_{i, j}*X*〉. The number

_{i, j}*X*is generated from a binomial distribution of probabilities

_{i, j}where *X _{i, j}*=

*k*. Here

*S*=Σ

_{j}*is the total number of seed photons emitted into the*

_{i si, j}*n*spectral intervals of the frequency grid and

*p*=1/

*n*is the probability of choosing one out of

*n*frequency intervals. This stochastic model of spontaneous emission is an

*S*-step Bernoulli experiment in which

_{j}*S*photons are distributed randomly into

_{j}*n*bins.

#### 3.3 Diagnostic quantities

From the computed quantities *N _{j}* and

*ρ*(Eqs. (4) and (1)), which are not directly observable in an experiment, we evaluate observable quantities such as the output power at time step

_{i, j}*j*

We compute the time averaged power, averaged over *N* steps

by taking an interval of 100 *µ*s after the output has become stationary. We determine the frequency for which the spectral energy is maximum and the full width at half maximum (FWHM) of the spectrum around this frequency. The model provides quantities which, though not directly observable, give useful insight into the amplification processes within the FSF laser. In order to investigate the interplay between the inversion and the electromagnetic field we calculate the frequency distribution of the effective gain,

The frequency range where G>0 holds is the amplification bandwidth Δ*v _{gain}*. From Eq. (1) we find that in time-step

*t*the density

_{j}*ρ*at frequency

_{ij}*v*is amplified by the factor exp(

_{i}*G*). It is particularly useful to display a contour plot of the photon number

_{i, j}*M*=

_{i, j}*ρ*as a function of frequency and time. We refer to a plot of the photon density

_{i, j}δv*ρ*or the photon number

_{i, j}*M*versus frequency as the

_{i, j}*spectrum*.

#### 3.4 Phase space analysis of pulsations

We distinguish the different pulsation regimes by using several mapping techniques well known from nonlinear dynamics [28]. The first of these, within the two-dimensional phase space (*P _{out}, Ṗ_{out}*), is an examination of limit cycles (the limiting phase-space trajectory for

*t*→∞) for various values of pump power

*P*. We found that for

_{in}*t*>900

*µs*the trajectories converged in each case to the attractor, a point or a limit cycle. Figure 3 presents an example of a limit-cycle in the phase space for a pump power of 8 W.

The second technique of data reduction selects from the phase space trajectories those points for which *Ṗ _{out}*=0, thereby producing a

*Poincaré section*[28]. We repeat this sectioning for a set of pump power

*P*values, thereby obtaining a diagram with the output power on the ordinate and the input power as the abscissa. In our case the diagram contains all points (

_{in}*P*) for which

_{in},P_{out}*Ṗ*=0, including the local minima and maxima of

_{out}*P*.

_{out}For the FSF laser these plots are examples of a bifurcation diagram [28]. By viewing such a diagram one can distinguish three distinct types of pulsations for a fixed value of the input power *P _{in}*:

- continuous-wave (cw) output, comprising a single isolated point,
- period-
*n*pulsations, comprising 2n points, and - chaotic pulsations, comprising a dense set of points.

Figure 3(c) shows an example for a period-one pulsation.

## 4. Simulation results for Ti:Sa

We first present simulation results for the Ti:Sa system, for ready comparison with the earlier work [7]. To exhibit the consequences of the frequency shifting feedback loop, we devote the next section to a comparison of results with and without the AOM element.

We first display the frequency dependence of the effective gain *G _{i, j}* and the photon density and visualize their time evolution in a video. Following a discussion of the time dependence of the effective gain we present examples of the averaged power output versus pump power, phase space trajectories and bifurcation diagrams. Finally we discuss the role of the time dependence of the gain and the spontaneous emission.

#### 4.1 Frequency dependence of gain and field

In conventional lasers, lacking FSF, lasing action takes place when the power *P _{in}* that is driving the population inversion exceeds a threshold value,

*P*. As the pump power increases above this value, stimulated emission acts to limit the increase of population inversion: the inversion approaches a limiting value

_{th}*N*and the gain becomes saturated. Any further increase of the pump power will increase the photon number, but not change the atomic inversion. Figure 2 shows the gain and the spectrum when the output is a continuous wave. The effective gain curve is always concave and its value is determined by the inversion [see Eq. (10)]. Without FSF we have

_{sat}*G*(

*v*)≥0 only for

*v*=

*v*

_{0}. In practice a conventional single-mode laser provides amplification within a frequency interval of typically several MHz. When FSF is present (see Fig. 2) the effective gain

*G*(

*v, t*) is positive for an interval Δ

*v*of typically several GHz. This means that the FSF laser acts as a broadband optical amplifier.

_{gain}#### 4.2 Time dependence of field and gain

As with many lasers, the initial (transient) stage of FSF laser operation consists of relaxation oscillations. These eventually die out and the output becomes stationary - though not necessarily steady. We next discuss these two regimes.

### 4.2.1 Transient operation

Figure 4 presents plots of the photon number *M _{i, j}* versus time and frequency, for laser operation (a) without FSF and (b) with FSF. Vertical cuts of this contour plot provide the spectrum at a fixed time. Horizontal cuts, along the time axis, exhibit relaxation oscillations. One notices the following features:

- Without FSF there is the well-known substantial narrowing of the spectral bandwidth as the laser approaches steady state operation. Spectral narrowing is not observed for the laser with FSF.
- When FSF is present the center of the output spectrum is substantially displaced from
*v*_{0}. The spectrum is centered around 16.5 GHz and is not exactly symmetric. - The frequency of the relaxation oscillation is slower by about 25% when FSF is present.

We conclude that the presence of FSF prohibits spectral narrowing. The displacement *v*-*v*
_{0} of the spectral maximum away from *v*
_{0} is caused by the periodic frequency shifting. Its magnitude can be deduced by applying the criterion for a stationary spectrum from [7].

Observation (iii) indicates that the coupling between the electric field and the atomic system is weakened by FSF. We will discuss this aspect in Section *4.3*.

### 4.2.2 Stationary operation

Following an interval of transient behavior, the FSF laser may produce a stationary output. Figure 5 shows a contour plot of photon number versus time and frequency to illustrate some of the representative forms of the stationary behavior. Within the time window shown, from 150 to 200 *µ*s, the operation has become stationary. The behavior exhibits both steady output (first and last frames) and periodic output, the details of which depend on the pump power. As the pump power increases the output characteristics change from a continuous-wave emission (at *P _{in}*=4.0 W) through several pulsations at a different repetition rate (

*P*=4.05–13.25 W) and back to a continuous-wave emission (for

_{in}*P*=13.25 W).

_{in}In Section *4.4.2* we classify five regimes of operation, denoted ①–⑤ and list their thresholds. There we show that the frames of Fig. 5 illustrate each of the five regimes. Anticipating that classification, we make the following observations:

① For *P _{in}*=4.0 W the output is continuous (regime ①). The spectrum is stationary and its center is displaced 16 GHz from

*v*

_{0}with a width (FWHM) of 8 GHz. From Fig. 4 we know that this spectrum is established after an interval of exponentially damped spiking.

② For *P _{in}*=4.5 W the output is a stationary continuation of the transient spiking, here periodic (regime ②). The spectrally integrated output power completely vanishes periodically - the laser turns on and off. At the periodic moments of peak output intensity the spectrum is centered around (

*v*-

*v*

_{0})=16.5 GHz. As

*P*is increased to 8.0 W the repetition rate increases and the center of the spectrum moves towards

_{in}*v*

_{0}.

③ For *P _{in}*=9.87 W (regime ③) we observe a qualitative change in the output characteristics. The center frequency of a pulse spectrum takes one of two distinct values, which depend on Δ

*v*and

*τ*. Here (

_{RT}*v*-

*v*

_{0}) is either about 14 or 21 GHz. This Fig. 5 together with our knowledge about the gain (see Section

*4.1*) proves a claim of Ref. [7]: Following a pulse whose spectrum is centered at 14 GHz the field will undergo frequency shifts while the amplitude dies away. However the gain recovers sufficiently rapidly that the following pulse will retain memory of the earlier pulse. The center of the following pulse is found near 21 GHz.We call this process the

*dual-frequency-scenario*. Such behavior is not observed for a conventional laser (see also Section

*4.2.3*).

④ Further increase of the pump power causes the two distinct maxima to merge in frequency and in time until at *P _{in}*=12.5 W (regime ④) a period-one spiking is again observed, but now at a higher repetition rate. The center frequency of the spectrum returns to (

*v*-

*v*

_{0})=16 GHz.

⑤ When the pump power increases beyond 13.25 W there is no further change in the time dependence of the output. The output is continuous and the spectrum is stationary (regime ⑤), now with (*v*-*v*
_{0})=17.8 GHz and a width (FWHM) of 7.7 GHz.

### 4.2.3 Animation of stationary operation

The animation of Fig. 6 (220kB) shows the time evolution of the spectral density and the effective gain within regime 3 at a pump power of 9.5 W - both are non-stationary in this regime. While the inversion is recovering and the gain curve is rising the number of photons (proportional to the area under the spectral density) is increasing and the spectrum shifts to higher frequencies. The photon number grows exponentially until the energy extracted from the atomic system by stimulated emission is maximized. The inversion is reduced towards unity. Then a pulse is emitted with a center frequency *v*-*v*
_{0}=21 GHz, the inversion is declining and thus the gain is reduced. Because the spectral maximum of the pulse continues to shift to higher frequencies the main part of the spectrum is found outside the amplification bandwidth of the FSF laser and therefore the photon number in this frequency interval is decreasing. In the interval where the effective gain is still positive, though falling, the photon number is increasing. A pulse is emitted with a center frequency *v*-*v*
_{0}=14 GHz. As time progresses the photon number decreases and the cycle starts all over.

The time evolution of the population inversion correlates with that of the photon density. Figure 7 shows the time dependence of the inversion for the same pump powers as used in Fig. 5. From Eq. (10) we know that an oscillation of the inversion accompanies an oscillation of the gain and its bandwidth.

From the existence of the different pulsation characteristics and the oscillation of the inversion we conclude that the gain is not necessarily saturated when laser activity occurs. Gain saturation requires that the pump power be increased to *P _{sat}*=13.25 W. In Section

*4.4.2*we show that this value corresponds to the threshold value of regime ①.

#### 4.3 Average output power

Studies of the time dependence of relevant parameters give valuable insight into the operation of the FSF laser, allowing identification of different regimes of pulsed or continuous-wave output. Complementary information comes from examining the time averaged output power, treating the pump power as a control parameter.

Figure 8 shows the averaged output power *P̅ _{out}*, calculated using Eq. (9), as a function of the input pump power

*P*, for a laser with and without FSF. We note the following properties:

_{in}- the presence of FSF decreases the output power,
- the slope of the curve of output power versus input power is steeper without FSF,
- the FSF curve shows an abrupt decline around
*P*=8.6 W and another feature near_{in}*P*=12 W,_{in} - Without FSF, the lasing action has a well defined threshold, while the onset of lasing action is smooth with FSF.

Observations (i) and (ii) are a direct consequence of the fact that the AOMinduces additional losses during the buildup of the laser field. (The major losses are the removal of output light in zero order from the AOM, and slight absorption of light in the crystal.) Therefore the FSF has a higher threshold and a lower slope of efficiency vs. input power.

Observation (iii), the abrupt decline in the FSF curve, corresponds to the onset of the *dual-frequency-scenario* identified in the previous section. Emission in this regime is less efficient because the pulse from the preceding emission, whose spectrum is centered around 21 GHz, competes with the newly emitted pulse, whose spectrum centers around 14 GHz. The older pulse gains energy from the inversion but is subject to large loss from the filter.

The smooth onset of the FSF laser, noted in (iv), means that unlike a conventional laser, the FSF laser does not have a well defined threshold beyond which the output power increases linearly with the input power. Figure 2 showed the relationship between the gain and the spectrum in the case of a stationary spectrum. Because of the frequency shift, FSF laser operation leads to a finite amplification bandwidth Δ*v _{gain}* (see Section.

*3.3*). At low pump power this gain bandwidth grows with increasing pump power [see Fig. 2 and Eq. (10)]. Because the outputÊgrows exponentially with the gain, the output power increases faster than linearly with the pump power, i.e., a plot of output power versus input power has positive second derivative for low pump power.Ê

#### 4.4 Phase space description

As we shall show next, the techniques of phase-space analysis discussed in Section. *3.4* reveal the complicated variation of *P _{out}* (

*t*) as the pump power is changed.

### 4.4.1 Trajectories: pulsation

Part (b) of Fig. 3 shows a limit cycle of the phase space trajectory for the pulsation observed at *P _{in}*=8 W. Figure 9 shows representative time-domain pulsations of the output power, along with the corresponding limit cycles in phase space. For the dual-frequency-scenario (8.61 W≤

*P*≤9.87 W) one observes in frame (a) and (b) a second pulse emission on the rising edge of the next laser pulse. For

_{in}*P*≥9.2 W a separate pulse emerges. Further increase of Pin causes the limit-cycle to shrink continuously into a single point at

_{in}*P*=13.2 W. This corresponds to another cw emission regime. No chaotic trajectory is observed.

_{in}The phase-space trajectories allow us to test the precision of the integration algorithm. We reduced the integration time-step *dt* to 0.1 *τ _{RT}* and verified that the calculated attractors are independent of the choice of initial conditions.

### 4.4.2 Bifurcation diagrams: pump power

To reveal more clearly the variety of the pulsations and to determine the thresholds for the regimes we make use of the bifurcation diagram technique (see Section *3.3*). Figure 10 displays such a diagram for a laser operated with and without FSF. When FSF is absent, only cw output occurs. The variation of *P _{out}* as a function of

*P*for

_{in}*Ṗ*=0 is a single monotonically rising curve which shows no bifurcation. When FSF is present, a bifurcation diagram reveals a set of five regimes forming five contiguous vertical bands, separated in the figure by vertical lines. Because the locus of points in such a diagram falls into a pattern reminiscent of the outline of a violin, we refer to the overall portrait of FSF dynamics as a

_{out}*violin scenario*. The five regimes, ordered by increasing pump power, are:

① low-power cw output,

② irregularly timed single intense pulses (sustained spiking),

③ low frequency pulsation on a cw baseline,

④ low frequency pulsation without a baseline,

⑤ high-power cw output.

The transition from ① to ② and from ④ to ⑤ are *bifurcations* [28] whereas the two transitions from ② to ③ and from ③ to ④ are not continuous and can not be labeled as standard nonlinear dynamics features. The precise values of the input power that delimit the regimes depend on the AOM-shift per roundtrip and the amplification bandwidth; in Section *4.5* we discuss the variation of the thresholds as parameters are changed.

### 4.4.3 Influence of spontaneous emission

In previous publications [3, 25, 26] the FSF laser was considered as a regenerative amplifier of spontaneous emission. The following numerical result confirms this view. Figure 11 demonstrates that by switching off the spontaneous emission (at *t*=100*µ*s), the previously stationary spectrum (for *P _{in}*=14W) starts to drift toward higher frequencies after a short (≈2

*µ*s) delay which is determined by

*τ*Δ

_{RT}*v*/Δ

_{gain}*v*. After this delay all seed-photons within the amplifierbandwidth have been shifted to the position of the spectral maximum. Then the energy disappears as it is filtered out, here within a few

*µ*s. This shows that, within the present FSF model, stimulated emission alone is not sufficient to keep the laser operating. Instead, continued lasing is achieved by regenerative amplification of the spontaneous emission. This is in contrast to the behavior of the conventional laser, where switching off the spontaneous emission in the numerical model does not affect laser activity. In the FSF case the output is determined by seeding with spontaneous emission. Of course, it can also be influenced by external seed light.

We now consider whether the output characteristic changes when modeling the spontaneous emission as a stochastic process (see Section *3.3*). The result is displayed in the bifurcation diagram in Fig. 12. There is some sensitivity of the details of the scenario to small perturbations: a fluctuation of the maximum and minimum output powers is observable, especially during the dual-frequency scenario of regime ②. When the two pulses compete the system is very sensitive to how the seeding occurs. The transitions from ③ to ④ and from ④ to ⑤ are no longer distinct. However, the repetition rate is not affected. By examining a Fourier transform of the output power at *P _{in}*=9.5 W we found that even for random spontaneous emission the repetition rate remains constant. Furthermore, we found no evidence of chaotic behavior even when spontaneous emission is modeled as a random process.

#### 4.5 Variation of the experimentally controlled parameters

Study of the influence on the dynamics of the four most relevant system parameters (the AOM shift, the free spectral range of the etalon, the etalon reflectivity and the frequency-independent losses) has revealed a number of relationships. We observe universality of the violin-scenario. The pattern is stretched or shifted but no additional regimes occur. For a quantitative analysis we increase one of the parameters while the others remain unchanged. The threshold values of different regimes show the following behavior.

- When the AOM shift is increased the threshold values increase linearly but with a different slope. This expresses the fact that a higher shift rate reduces the coupling strength between the laser field and the inversion. The following linear function describes, for the Ti:Sa FSF laser, how the regime-delineating powers
*P*_{②}and*P*_{⑤}vary with changes in the frequency shift Δ*v*:*P*_{②}=3.97Δ*v*+2.7,*P*_{⑤}=41.49Δ*v*+0.76. Here the power is expressed in Watts and Δ*v*is in kHz. - When the free spectral range (FSR) of the etalon is increased the thresholds decrease monotonically because a larger amplifier bandwidth Δ
*v*increases the coupling between the laser field and the inversion._{gain} - When the etalon reflectivity R is increased the thresholds increase monotonically because the amplifier bandwidth Δ
*v*is reduced, thereby decreasing the coupling between the laser field and the inversion._{gain} - When the frequency-independent
*γ*losses are increased the thresholds also increase because the weaker field is less able to stimulate emission._{cav}

## 5. Results for other gain media

Although the focus of the present work is the Ti:Sa laser system, we have also carried out simulations for three neodymium-based systems: Nd:YAG, Nd:YLF, Nd:YVO_{4}. Results of these simulations reveal some of the five regimes discussed above. The effects described under (i)–(iv) in the preceding section were also confirmed.We conclude therefore that the *violin scenario* is universal for lasers with FSF.

Table I lists the parameter sets used in these simulations. The gain bandwidth of neodymium doped crystals is only some hundred GHz, and therefore it is possible to operate those FSF lasers without an intracavity etalon. Tables 2 and 3 list the values of the regime-delimiting input powers *P _{in}* for Ti:Sa and three neodymium-based gain materials. Values in Table 2 are those for operation with an etalon having a 200 GHz bandwidth. Table 3 presents the values of

*P*for operation without an etalon.

_{in}For the neodymium-doped crystals some portions of the *violin-scenario* were simulated with and without an etalon. Figure 13 shows bifurcation diagrams calculated for a FSF system without intracavity etalon. The etalon does not limit the bandwidth of the gain medium, which is almost identical to the gain bandwidth, but it introduces additional losses. The cw -emission regime ① has not been observed in simulation, probably because the output power is too low. It is surprising that the high frequency pulsation regime ④ is not observed but there is an abrupt change from low frequency pulsation to cw emission. At present we can not offer an explanation of that simulation observation, which is very different from the behavior of the Ti:Sa laser.

We note that the neodymium-doped FSF lasers without an etalon have the lowest thresholds for all the regimes. The threshold of the cw -regime ⑤ for the Nd:YVO_{4} laser is more than a factor 10 lower than the corresponding threshold in the Ti:Sa system. The variation of output power with input power in the cw regime ⑤ is independent of the crystal material for all Nd-doped crystals.

## 6. Comparison with experimental results

The present work provides answers to most of the questions raised in [7]:

*Why is there a regime where the slope of the averaged output power versus the input power is negative?*This is a consequence of the pulse competition in the dual-frequency-scenario discussed in Section*4.3*.*Why is the pulsation for P*This is consequence of the fact that the field inside the laser does not vanish during the dual-frequency-scenario (see Fig. 5). Therefore the next pulse will not be initiated with the same initial conditions. Because of the pulse competition the output power will be different in each period._{in}≈10W different from those at 6 and 8W?*Why are there different regimes of pulsation observable for low pump powers?*The weak dependence of the field on the inversion leads to a non-stationary gain and a time dependent displacement of the spectral maximum (*v*-*v*_{0}) (see Section*4.2.2*) yielding complicated output characteristics.*What determines the dependence of the experimentally observed repetition rate on the pump power? Can the fluctuation of the repetition rate be reproduced when the spontaneous emission is modeled as a random process?*The answer is obtained from a comparison of the bifurcation diagram with experimental data about the repetition rate (see Fig. 4 of [7]). The fundamental frequency of the pulsation grows linearly until the input power reaches 8.5 W. It stays almost constant until 10 W, before a sudden increase to almost twice this value. Those changes correspond exactly to the transitions between the regimes ②, ③ and ④. With increasing*P*we expect an increasing repetition rate because after each pulse emission the inversion recovers faster. The ineffective use of the gain within the dual-frequency-scenario causes the plateau in the plot of the repetition rate versus pump power. The sudden change is a consequence of the transition from a period-two back to a period-one pulsation, as evident in the bifurcation diagram (see Section_{in}*3.4*). In the experiment the repetition rate was not perfectly regular, as is evident from the RFSA-spectrum shown in Fig. 4 of [7]. A possible cause may be the influence of the stochastic nature of spontaneous emission. In our model the random perturbations prohibit strict periodicity of the output power, as shown in Sec.*4.4.3*, but they do not affect the repetition rate.*Is there a well defined threshold for each regime?*The several regimes do have a well defined threshold. The classification of the regimes is provided by the violin-scenario in the bifurcation diagram.

## 7. Conclusions

A FSF laser is, in part, an optical amplifier operating within a restricted amplification (and filter) bandwidth Δ*v _{gain}*. This means that as long as the gain is not saturated the center frequency of the observed broadband spectrum may vary in time. Consequently the presence of FSF introduces an additional relevant variables, namely the displacement of the spectral maximum,

*v*-

*v*

_{0}. In our simulation and in the experiment [7] the spectrum is always nearly Gaussian (or a superposition of two Gaussians).

There exist several regimes of operation. Our classification into five regimes, identifiable by the time dependence of the output power, is compatible with the observations in the experiment [7], but in that work not all the regimes were distinguishable.

Because we observe the same spectral stationarity with and without FSF for all pump powers above the threshold power of regime ⑤ we conclude that within regime ④ the gain becomes saturated.

We have shown that the rate-equation model provides a satisfactory description of most of the observed dynamics of the Ti:Sa FSF laser, as reported in [7]. Only the short-pulse (mode locked) regime observed in [7] is not covered here. The rate equation model, lacking any information about phases, is unable to treat this regime. The classification of the regimes into a violin scenario seems to be universal for FSF lasers, as evidenced by the simulations of the Ti:Sa lasers several neodymium-based laser systems.

Our description of the FSF laser, being based on rate equations, takes no account of phases and coherences. It treats photons of different frequencies independently, competing for gain much as in a biological model the individuals compete for resources. Diffractive propagation effects are also neglected.

We have investigated how the inevitable intensity fluctuations associated with the random nature of spontaneous emission affect the FSF laser behavior; section 3.2 describes our approach and section 4.4.3 discusses some of the results. Our further simulations have revealed that the predicted spectral density is affected by the fluctuations, but the overall scenario of nonlinear operation is not qualitatively altered.

## Acknowledgements

This work was supported by the Stiftung Rheinland-Pfalz für Innovation.We thank H.-J. Korsch (Kaiserslautern) and F. Kowalski (Golden, Colorado) for discussions and criticism.

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