La Teoría de la constitucionalización del Proceso

Dispersion in temporal and frequency domains of optical pulses within laser cavities is due essentially to two optical processes, these being linear and nonlinear in nature [21]. Linear dispersion occurs as a result of group-velocity dispersion (GVD) of the optical wave packets as the pulse propagates through intracavity dispersive elements, such as a colour- centre crystal for example, as generally the refractive index n of a material is a function of the optical frequency co. Nonlinear dispersion occurs due to self-phase modulation (SPM) of the optical pulse which arises due to the optical Kerr effect [21] or to off- resonance saturable absorption or saturable gain [21] both of which give rise to a frequency chirp across the pulse. These nonlinear processes may both occur within the cavity where the energy density of the optical beam is very high. This is the case at the beam waist within the gain and absorber media in the laser to be described here. Linear frequency chirping

mechanisms or GVD arises within the LiFiF^ colour-centre laser cavity due to the presence of the two infrasil optical access windows situated around the colour centre crystal and from the LiF crystal itself.

Firstly, for linear intracavity dispersion we may consider the propagation of a Gaussian pulse through a dispersive element, assuming that such a pulse envelope is easier to work with mathematically, and that the conclusions drawn are not significantly different from those obtained when a Sech^ modelocked pulse envelope is assumed. Thus a Gaussian pulse profile with a peak frequency of coq, with no frequency chirp, and duration

(FWHM) may be expressed as

E=Eo exp- [(21n2) t^/x^] exp icogt (7.1)

To obtain the output phase of the pulse after passing through a dispersive medium the propagation constant in the medium p may be expanded as a Taylor series about the centre frequency cùq to give

P(o))-p((OQ)-4-0(ci}-O)Q)+^p((o—0)Q)^+... (7.2)

where p(co)=n(co)k, p=— and k the free space propagation constant given by k = ^ . It

dco Xq

may be shown that [21] the dispersion has the effect of broadening the pulse duration by a

factor given by

^in

1-

in L(71n2); y

and to create a frequency chirp across the pulse whose sign is opposite to that of P and which may be expressed as

where fis the length of the optical medium. To obtain values of the linear dispersion within the LiFiFn colour-centre laser cavity the values of for the LiF gain crystal and the infrasil windows were obtained from the three-term Sellmeier expansion [22].

Nonlinear dispersion arises within the laser cavity due to the effects of self-phase modulation which may occur by way of three processes. These are, (i) the optical Kerr effect due to the intensity dependence of the refractive index of the dye solvent, (ii) off- resonance absorption of the saturable absorber and (iii) off-resonance saturation of the optical gain [15,21]. However, under the operating conditions of the laser used here the pulse intensity within the optical gain was much less than that observed within the saturable absorber in order to satisfy S parameter conditions [1] and for these operating conditions it has been shown that the contribution of SPM from the off-resonance gain saturation is only significant for very short optical pulses [16,23].

The main intracavity SPM contribution comes from the optical Kerr effect which as mentioned briefly above arises as a direct consequence of the intensity dependence of the refractive index of the dye solvent which may be expressed as

nzznq+n^I (7.5)

n is the index of refraction, nq the index of refraction at an arbitrary low intensity, n2 is the Kerr coefficient or the nonlinear index (for ethylene glycol n2=3xlO”^^cm^W“l). This nonlinear effect may be considered as instantaneous as it arises due to the polarisation of the electric field in the medium and thus being electronic in nature has time scales in the order of lO'^^s [21,24]. Again, as in the case of linear dispersion, if we consider the propagation of a Gaussian pulse through a dispersive medium, say for example, the ethylene glycol solvent of the saturable absorber dye jet it may be seen that different points

of the pulse envelope will experience a different refractive index due to the variation of the flux intensity I across the pulse and thus a phase change A(j> will be experienced across the pulse. This is given by

A(j> =- fn2l (7.6)

where fis the path length of the medium. The instantaneous frequency change Aco which occurs across the pulse envelope as a function of time is given as

dt dt (7.7)

Thus it may be seen that as the pulse propagates through the medium the intensity dependence of the refractive index leads to a red-shift in the leading part of the pulse and a blue shift on the trailing edge of the pulse. This is illustrated in figure 7.1.

Figure 7.1 Computer generated temporal and spectral envelopes for a pulse illustrating the frequency sweep resulting from a nonlinear Kerr effect

The second major source of SPM that may arise within the laser cavity may arise from off- resonance absorption by the saturable absorber [15,21,23]. The sign of the frequency chirp produced by this effect depends upon the duration and energy of the optical pulse and also upon which side of the absorption peak the laser is operating [15,23]. For absorption occurring on the short wavelength side of the absorption peak the chirp obtained across the pulse has been shown to be generally positive in sign (or up-chirped), for small ratios of pulse energy to saturation energy. For the same pulse parameters but with the laser operating on the long wavelength side of the absorption profile a negative chirp is experienced across the centre of the pulse [15,23]. Alternatively for large ratios of pulse energy to absorber saturation energy a large positive chirp is manifest only on the leading “foot” of the pulse [15,23]. It may be seen therefore that the sign and amplitude of the nett frequency chirp experienced by the pulse will depend upon a convolution of processes and operational parameters and will be the combination of the Kerr effect occurring within the dye solvent and the chirp experienced due to off-resonance absorber saturation. This has been modelled quite effectively by Miranda et al [23] who have treated the propagation of a Gaussian pulse through a saturable absorber and solvent. The absorber is assumed to have a Lorentzian line shape given as:

g ( C O ) = 2 7 t Aco

^^

Aco2 J_ (7.8)

where cog is the centre frequency of the absorption profile and Aco is the half width of the spectral line. The refractive index of the saturable absorber is given as:

nr=f(coo-co) — ag(co) (7.9)

where n^ is the refractive index at the resonance peak of the saturable absorber and a the absorption coefficient given by:

-a j I dt a^ogexp

V "sat

J

ag is the absorption coefficient at the peak of the absorption profile, a the cross-sectional area at the beam waist and the saturation energy of the absorber. Hence an expression may be obtained giving the total phase delay A(})tot across the pulse resulting from the

nonlinear Kerr effect in the solvent plus the chirp arising from the off-resonance absorption as: where A<l>roT=

_X

_X

and thus the total frequency chirp Aco^qt induced by both nonlinear processes may be

obtained from the differentiation of the total phase delay as:

A o > r o T = ^ ^ = - f ( r n 2 f G(co)

cf]

Lastly, it should also be mentioned that a linear frequency chirp may also be brought about within ultra-short pulse laser cavities by dispersion arising from the optical mirrors and the theory of such has been detailed elsewhere [20,25].With the large tuning ranges available from dye and colour-centre lasers the cavity mirrors usually have broadband characteristics and it is with these type of mirrors that dispersion can arise [26]. The manner in which the dielectric layer sequences for the mirrors are deposited is known to play an important role in determining the degree of dispersion expected and several workers have in fact reported optimisation of pulse durations by careful construction of their resonator optics [26]. However, the precise details of the dielectric layer sequences for the mirrors used in the laser cavities to be described here are unknown and thus a quantitative knowledge of the

dispersion introduced by these is not available. However, because dispersive effects arising from dielectric stacks is really only significant for sub-lOOfs pulse durations these mirrors with unspecified deposition detail were used.