4.1 Introduction
The experimental work of the previous chapter demonstrated the importance of quadratic phase compensation in the CPM dye laser cavity, in particular, in relation to its interaction with self-phase modulation (SPM) as a pulse shortening process analogous to soliton compression in optical fibers. This new pulse shaping mechanism, acting in tandem with the more conventional pulse shaping processes of saturable gain and saturable loss was found to induce remarkable changes in the operating characteristics of the laser. Indeed, while for certain well defined conditions, stable sequences of hypershort pulses could be generated, fractional changes in quadratic phase were found to be sufficient to result in the onset of severe instabilities. Under other conditions, pulse sequences could be obtained that bore a remarkable resemblance to higher-order soliton evolutions. Significantly, under optimal dispersion-balanced conditions, it was found that the laser could operate in the absence of pulse collision in the absorber jet without noticeably affecting the steady-state pulse duration, in contradiction to previous assumptions of passive modelocking.
While these behaviours have been tentatively explained in reference to other work, especially the theoretical studies of Martinez et al. and the work of Salin et al. and Avramopoulos et al. a full understanding of this laser system is far from being complete. In particular, there remain unanswered questions over the exact nature of the complex pulse evolution regime, the significance of conventional shaping at steady state and especially, the origins of the observed pulse asymmetry.
It is the purpose of this theoretical study, to gain a deeper understanding of the underlying pulse shaping dynamics inherent in the generation of pulses from the dispersion- compensated CPM dye laser, in order both to elucidate the experimental behaviours reported in Chapter 3 and to provide a spring-board fi*om which finther optimisation of the laser system can be realised. In accordance with more recent studies the model includes
Chapter 4: Computer Simulation of the CPM Dye Laser
both conventional pulse shaping (amplitude shaping) and solitonlike pulse shaping (phase shaping) processes, as well as extending the consideration of dispersion effects to include the higher-order cubic phase term. While the influence of cubic phase distortion has been considered in some detail in relation to nonlinear pulse propagation in optical fibers little has been reported to date on its effects in the CPM dye laser. With the advent of the prism system and its successful compensation of quadratic phase in the CPM dye laser, this higher-order phase distortion is likely to be a significant factor in fiirther optimisation of the laser system and so it is investigated here.
The analysis is organised as follows. In the subsequent section, the laser model and the equation system are presented. The next two sections explore the amplitude shaping and phase shaping mechanisms in isolation to assess the salient features of each, before a full numerical analysis is undertaken in Section 4.5. This is followed by a discussion which relates the findings to experiment and other work, and finally a conclusion, which summarises the key results with regard to the CPM dye laser described in Chapter 3 as well as proposing measures for further optimisation of this laser system.
4.2 Numerical Model
A schematic representation of the numerical model used for the CPM dye laser simulation is shown in Figure 4.1. Essentially, it consists of a unidirectional ring cavity in which the modelocked pulse encounters each discrete element once per round-trip. In a single cavity transit the pulse undergoes amplitude shaping through saturable gain, saturable absorption, linear loss and bandwidth limitation and phase shaping through the processes of self-phase modulation (SPM) and dispersion. With the exception of the dispersion and spectral filtering, the interaction of the pulse with each element is calculated in the time domain.
Rather than considering the transient buildup process, by starting with the spontaneous emission, a seed pulse is injected into the cavity to start the evolution. This model follows the development of this pulse to steady-state as it propagates unidirectionally around the ring cavity. For the CPM dye laser, in which the pulse changes only slightly (a few percent) in each round-trip, this technique allows a substantial saving of computer time. In no case that was tested were the initial conditions found to have any effect on the long term pulse behaviour.
Chapter 4; Computer Simulation of the CPM Dye Laser
Amplifier Absorber Linear Loss
IFFT
Dispersion Spectral Filtering FFT SPM
Figure 4.1: Schematic diagram of tlie numerical model of the CPM dye laser.
The evolution of the pulse in the laser cavity can be obtained by expressing the electric field as :
E(t) = ^(t)exp{i[cOot + (j)(t)]} (4.1)
where co ^ is the central frequency, (|)(t) is a time-dependent phase and ^(t) is the pulse amplitude.
The seed pulse injected into the cavity to initiate the simulation process, has a zero time-dependent phase and a hyperbolic secant amplitude profile defined by:
^(t) = ^ 0 sech(1.76t/At) (4.2)
where At is the pulse duration (FWHM). is scaled so that the cumulative energy of the pulse:
U(t) = (l/U f)E ^(t)^dt (4.3)
yields a total energy:
Uo = U(-Hx)) (4.4)
where the pulse energy has been normalised to the saturation energy UJ of the gain medium.
Values for the pulse duration and pulse energy of the initial seed pulse are given in Table 4.1. A hyperbolic secant amplitude profile is chosen because it is the solution of previous theoretical calculations for a CPM laser The final pulse, however, is
Chapter 4: Computer Simulation of the CPM Dye Laser
found to be independent of the initial parameters, such as duration, energy, and pulse shape.
Consider the evolution of this seed pulse as it interacts with each of the discrete elements shown in the ring cavity configuration of Figure 4.1.
Amplifier, Absorber and Linear Loss
Saturable amplification and saturable absorption are the phenomena responsible for pulse formation and these effects occur at the gain (Rh6G) and absorber (DODCI) jets, respectively. The pulse also encounters a linear loss that includes scattering, diffraction and output coupling from the various cavity components.
It was seen in Chapter 1, that in order for stable pulse modelocking to occur in passive systems with a slow saturable absorber, three general conditions must be satisfied First, the recovery time of the gain medium x® must be of the order of the time interval between two consecutive pulses. In the CPM dye laser where there are two counter- propagating pulses in the cavity, this time interval will be x"^/2 where x"^ is the cavity round-trip time (assuming that the counter-propagating pulses meet in the absorber jet and the gain medium is located at a distance of one quarter of the cavity perimeter from the absorber so as to provide the pulses with equal gain). In this case;
x'‘/2x® % 1 (4.5)
Secondly, the absorber recovery time x** must be less than the recovery time of the gain medium, or:
(4.6)
Finally, the S-parameter, which is defined as the ratio of the gain saturation energy U® to the absorber saturation energy should satisfy the condition:
S = U f / u : > l + îi/a„ (4,7)
where t| is the total cavity linear loss coefficient and is the small signal absorption coefficient of the saturable absorber (DODCI).
In addition to meeting these requirements, it is assumed that both gain and absorption media can be described by homogeneously broadened two-level systems with energy relaxation times that are long compared to the pulse duration At and that the amplitude of the pulse is not greatly modified in a single pass through either the gain or absorber jets.
Chapter 4; Computer Simulation of the CPM Dye Laser
Following the rate-equation analysis presented by Haus phase-relaxation terms are neglected. This approximation is valid for sufficiently broad gain and absorber bandwidths (1/At « A 0 ®, Aco^) where Am® and Am" are the gain and absorption bandwidths
(FWHM), respectively. Under these conditions, the output electric field from the gain jet E®^t(t)can be written as:
E*„,(t) = Real[B(t)]xEi„(t) (4.8)
where:
B(t) = expj(l + iÔ®jpexp(U(t))/2j (4.9)
and the corresponding electric field from the absorber jet E"^^(t) can be written as:
E;„(t) = Real[A(t)]xEi„(t) (4.10)
where:
A(t) = exp[(l + iô")aexp(S x U(t))/2j (4.11)
While the terms a and p denote the absorption and gain coefficients respectively, 5® = (m®-mo)/Am® and 5" = (m"-mo)/Am® represent the detuning of the electric field of the pulse centred at frequency m ^, from the peak of the gain profile centred at m® and absorption profile centred at m ", respectively.
It should be noted that since we are only interested in the amplitude shaping effects of the gain and absorber media here, then only the real part of the associated gain and loss factors B(t) and A(t) need to be considered. The phase effects arising from the imaginary part of these factors will be treated in the next step of the simulation. In addition, it should be pointed out that only the ground-state absorption is included in this analysis. While the saturable absorber includes a photoisomer which contributes to the absorption at the laser wavelength, the saturation of this species is found to be so deep that it can be neglected in
comparison with the ground-state as a shaping mechanism (see Section 3.5),
The cavity linear losses are lumped together in a single operation, resulting in an output electric field E|,^^(t) given by:
E L (t) = H x E ^(t) (4.12)
where: