Optimisation of Phase Shaping Process

Pulse Evolution Under Optimal Conditions

It will be seen below that optimal solitonlike pulse shortening is obtained when the SPM and dispersion are defined by the following parameter values: Leg = 0.0125, (|)"(ffl) = -20fs^ and (j)"'(® ) = 0. Figures 4.17(a),(b), and (c) illustrate the evolution of the pulse under these optimal conditions by showing the development of pulse duration, normalised pulse energy and integral pulse asymmetry respectively over 500 resonator round-trips.

Unlike the pulse evolution of the amplitude shaping model (see Figure 4.3), characteristic oscillations are apparent in the pulse parameters once minimum pulse duration is reached. However, these gradually disappear as the pulse approaches steady- state. Such behaviour is indicative of strong phase shaping resulting from the dynamic interplay between SPM and quadratic phase. In this case, the amplitude shaping is strong enough so that these oscillations are gradually damped, enabling steady-state behaviour to ensue. The competition between these two pulse shaping mechanisms will be considered in more detail later on.

Although the evolution of the pulse parameters diverges from those obtained in the amplitude shaping model after about 75 round-trips, the initial build-up is almost identical. This implies that amplitude shaping is the dominant pulse shaping mechanism in the initial

250 -1 200 - 150 100 500 300 400 100 200 0 Number of Transits (a) 0.5 - 100 200 300 400 500 0 Number of Transits (b) 0.25 0.2 0.15 0.1 0.05 -0,05 -0.1 0 100 200 300 400 500 Number of Transits (c)

Figure 4.17: Evolution of (a) pulse duration, (b) normalised pulse energy and (c) integral pulse asymmetry.

Pulse Temporal Profile: Number of Transits=0 Pulse-Duration=200fs

Intensity (arb)

T im e (fs)

Pulse Temporal Profile: Number of Transits=75 Pulse-Duration=80fs

Intensity (arb)

T im e (fs)

Pulse Temporal Profile: Number of Transits=500 Pulse-Duration=25fs

in tensity (arb)

T im e (fs)

Pulse Spectral Profile: Number of Transits=0 Spectral-Bandwidth=1.6THz

Intensity (arb)

F re q u e n c y (TH z)

Pulse Spectral Profile: Number of Transits==75 S pectral-Bandwidth=2. 9THz

Intensity (arb)

■CO -Ai -36 -24 -12 0 12 24 36 48 60

F re q u e n c y (THz)

Pulse Temporal Profile: Number of Transits=125 Pulse-Duration=23fs

in tensity (arb)

T im e (fs)

Pulse Spectral Profile: Number of Translts=125 Spectral-Bandwldth=11.3THz

in tensity (arb)

F re q u e n c y (TH z)

Pulse Spectral Profile: Number of Translts=500 Spectral-Bandwidth=12.2THz

Intensity (arb)

F re q u e n c y (THz)

Chapter 4: Computer Simulation of the CPM Dye Laser

stages of the pulse evolution. However, once the pulse duration reaches values a little below lOOfs, the divergence in the evolution of the pulse parameters suggests that phase shaping becomes increasingly significant, eventually superseding the amplitude shaping process as the dominant pulse shaping mechanism.

Figure 4.18, shows a series of temporal and spectral pulse profiles for various points in the evolution of the pulse. Both temporal and spectral profiles are shown normalised to unit maximum value. This sequence of diagrams supports the contention given above. For while the pulse shape and duration are similar to the amplitude shaping model at 75 round- trips, showing the characteristic positive integral asymmetry indicative of this process, the pulse asymmetry is completely reversed at 125 round-trlps and the duration is almost halved. The final steady-state pulse is nearly symmetrical and can be accurately expressed by a sech^ pulse shape, commensurate with that expected for an N = 1 soliton. The steady- state pulse parameters are characterised by a duration At = 25fs, a normalised energy

Uq = 1.3 and an integral asymmetry F = -0.05 .

In order to understand the rather complex changes in the pulse in the region dominated by the phase shaping mechanism, it is necessary to recall that the SPM is comprised of two types of chirp of competing sign: a negative chirp due to absorber saturation and a positive chirp arising from the optical Kerr effect. For the parameters of this model, the positive chirp is dominant over the central part of the pulse and the negative chirp is consigned to the leading edge due to a deep saturation of the absorber. Note that the dominant positive chirp is evident in the nature of the spectral asymmetry for the pulses at 75 and 125 round- trips respectively (see Figure 2.10).

Once the phase shaping mechanism becomes dominant, solitonlike pulse compression occurs with the positive Kerr effect chirp balanced by negative quadratic phase. However, pulse broadening will occur further out in the leading edge of the pulse due to the quadratic phase and the absorber saturation chirp being of the same sign. This is the reason for the broader leading edge and hence negative integral asymmetry shown for 125 round- trips. Note that the characteristic broader trailing edge effected by the amplitude shaping mechanism is almost completely suppressed and the residue is what causes the spectral splitting in the associated spectrum. As this pulse approaches steady-state, although the amplitude shaping mechanism does not take part in pulse shortening, it serves to clean up the leading and trailing edges of the pulse, the absorber saturation eroding the asymmetry

Chapter 4: Computer Simulation of the CPM Dye Laser

of the leading edge and gain saturation removing the residue at the trailing edge. Hence an almost symmetrical modelocked pulse results, which bears close resemblance to an N = 1 soliton but as will be seen later is characterised by a small residual positive chirp.

Optimisation Analysis

Optimisation of the phase shaping process essentially involves identifying the SPM and quadratic phase values which will produce the shortest pulse. For the purposes of this analysis, only SPM arising from the optical Kerr effect will be optimised with respect to quadratic phase, since experimentally, this was found to be the dominant SPM mechanism. Numerically, this SPM is modelled by an effective length Leg of ethylene glycol.

Figures 4.19(a),(b) and (c) present the usual steady-state parameters of pulse duration, normalised pulse energy and integral asymmetry, respectively as functions of the effective length of ethylene glycol and quadratic phase. In each case, the curves show the extent of stable steady-state operation. Beyond this, distinctive instabilities are found to occur, which will be described below.

It can be seen that for zero Kerr effect chirp (Leg = 0), adding negative quadratic phase results in a broadening of the pulse duration. Comparing with the results of the amplitude shaping model shown in Figure 4.8(a) (where (j)®“‘(t) = (j)‘‘®(t) = 0), reveals that the broadening is almost identical, implying that the absorber saturation chirp is insignificant over the central part of the pulse. As we have seen earlier, this is due to the deep saturation of the absorber which effectively limits the chirp to the leading edge of the pulse. Hence, the pulse broadening is almost purely dispersive and results in pulses with negative chirp values similar to those shown in Figure 4.9.

For zero dispersion, increasing Kerr effect chirp results in a steady decrease in the pulse duration. The shortest pulse as one might expect is now positively chirped as a result of the Kerr effect SPM. The pulse shortening mechanism involved arises from the dynamic interplay between the spectral broadening action of the SPM and the bandwidth limiting action of the spectral filter

In contrast, the combination of both quadratic phase and Kerr effect chirp of the opposite sign allows solitonlike pulse shortening to take place. For increased values of quadratic phase, the shortest pulses occur at maximum possible values of Leg, just before instabilities set in. As can be seen from Figure 4.19(a), the decrease of the pulse duration is saturated at large Leg enabling optimal phase shaping to be obtained for a Kerr effect chirp

80 1 70 - -20fs' 60 - 50 - .-10fs' 40 - Ofs' 30 - 0.0025 0 0.005 0.0075 0.01 00125 0.015

Effective Length of Ethylene Glycol

(a)

0

Ofs‘ -10fs‘ -20fs' -SOfs"

0.5 -

0 0.0025 0.005 0.0075 0.01 0.0125 0.015

Effective Length of Ethylene Glycol

(b) 0.25 0.2 0.15 0fs‘ 0.1 0.05 -1 Ofs^ D) -20fs= _-30fe® -0.05 -0.1 0.0025 0 0.005 0.0075 0.01 0.0125 0.015

Effective Length of Ethylene Glycol

(C)

Figure 4.19: Steady-state (a) pulse duration, (b) normalised pulse energy and (c) integral pulse asynunetry as a function of Kerr effect SPM (modelled by an effective length of etliylene glycol ) for various

Chapter 4: Computer Simulation of tlie CPM Dye Laser

characterised by Leg = 0.0125 and a quadratic phase of {j>"(co) = -20fs^, yielding a minimum pulse duration of 25fs. Stronger phase shaping than this produces no further pulse shortening. The shortest pulses again have a slightly positive chirp.

Outside the range of steady-state pulse behaviour, instabilities are found to occur, which are characterised by fluctuations of the pulse energy as well as of the pulse shape and duration. Near the boundary of the stability range the laser parameters were found to oscillate periodically with the number of cavity round-trips. This is illustrated in Figure 4.20, for one set of laser parameters.

The occurrence of such regimes is characterised by a Kerr effect chirp exceeding a certain value and by too small a negative quadratic phase. In the extreme case, the energy fluctuations are so large that self-quenching results and the laser is finally driven below threshold. If one increases the amplification here, the laser goes into a regime where both pulse edges experience a net gain, leading to a continuous increase of the pulse duration and eventually to a breakup of the stable pulse regime. It should be noted that the instabilities invariably begin to appear at energies Uq <1.3 (see Figure 4.19(b)). For the parameters chosen in this simulation, at such energies, the gain depletion is not strong enough and the net gain behind the pulse edge becomes positive. The result is a broadening of the trailing edge and the occurrence of satellites which give rise to instabilities.

The phase shaping mechanism represents the essential source of the instabilities described here which finally limit the achievable pulse duration. The transition from stable steady state pulse formation to this fluctuating behaviour can be understood in terms of the competition between the phase shaping mechanism and the amplitude shaping mechanism.

In the absence of dispersion, amplitude shaping has the property of causing the pulse duration to decrease until the finite bandwidth of the effective spectral filter in the laser oscillator prevents a further increase of the pulse bandwidth. Increasing the strength of the amplitude shaping hence produces a stronger bandwidth limitation. In contrast, strong phase shaping is found to result in the formation of stable higher-order solitonlike evolutions (if the bandwidth limitation is sufficiently weak). The presence of amplitude shaping will consequently have a disruptive effect on the higher-order soliton evolutions because it necessarily introduces some degree of bandwidth limitation. Similarly the

35 1 30 - B 1 8 25 - 1000 1200 1400 1600 1800 2000 Number of Transits (a) 0,8 - 0.6 1000 1200 1400 1600 1800 2000 Number of Transits (b) 0.5 -I 3OL

I

Figure 4.20: Periodically oscillating parameters: (a) pulse duration, (b) normalised pulse energy and (c) integral pulse asynmretry, obtained near the boundary of the stability regime

Chapter 4: Computer Simulation of the CPM Dye Laser

higher-order soliton evolutions produced by the phase shaping will have a disruptive effect on the tendency of the amplitude shaping to produce stable trains of single pulses.

Consider now how these competing mechanisms relate to the results of the simulation. At the edge of the stable steady-state regime, a minimum steady-state pulse duration is achieved. Here, the amplitude shaping mechanism is just strong enough to contain the tendency of the phase shaping to initiate higher-order solitonlike evolutions, and so a steady-state pulse results (see Figure 4.17). However, a further increase of SPM from this point, disturbs this delicate balance between amplitude and phase shaping and the bandwidth limiting effect of the amplitude shaping is no longer sufficient to suppress the now enhanced phase shaping process. Consequently, the tendency of the amplitude shaping to produce stable steady-state pulses will be seriously dismpted. Nevertheless, the amplitude shaping is still sufficiently strong to introduce significant bandwidth limitation and so it will in turn effectively disrupt the tendency for stable higher-order solitonlike pulses to develop. As a result, intermediate complex pulse evolutions are generated.

Although this complex pulse behaviour can be considered to evolve initially from a higher-order soliton (unless (1)"(®) = Ofs^), this solitonic evolution cannot be maintained due to the losses to the spectral filter which become too large to be replaced by the saturable gain. Not only do these losses not permit the continuation of the soliton evolution but they critically deplete the pulse energy. As we observed earlier, below such an energy threshold (Uq < 1.3) there is insufficient saturation of the gain, which serves to promote the development of a secondary pulse at the trailing edge of the first pulse.

It should be noted that the fluctuating parameters associated with these complex pulse evolutions (see Figure 4.20) bear a close resemblance to that of higher-order solitonlike behaviour (see Figure 4.14(b)) even though the pulse evolutions are actually quite different. Further, the secondary pulse development will manifest as a triple-humped intensity autocorrelation, again commonly associated with higher-order soliton formation (N = 3). Hence, as we observed in the phase shaping analysis, the normal pulse data can be quite ambiguous in determining whether solitonlike or complex pulse evolutions are evident. Note however, that the fluctuating pulse parameters in the complex regime are not always periodic. Indeed for sufficient Kerr effect SPM, complicated multipulsing can result where no detectable periodic or stable behaviour is manifest.

Chapter 4; Computer Simulation of the CPM Dye Laser