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upchirp {blue shift). Other groups have reported pulse compression to near transform limited pulse widths using similar structures [3] (upchirp), a hybridly mode-locked laser with an external quantum well absorber [4] (upchirp), and optically pumped active mode-locking of VCSEL in an external cavity with a downchirp [5] or an upchirp [6]. It should be noted that while actively mode-locking can result in either an upchirp or downchirp, structures using a saturable absorber have only shown an upchirp.
A qualitative explanation for the chirping is based on self-phase-modulation due to the gain and absorber saturation. The mode-locking mechanism is an interaction between pulse compression (spectral broadening) and pulse broadening (spectral nar- rowing). The pulse compression is obtained by a modulation of the net gain of the device such that the center of the pulse sees net gain, while the tails of the pulse see a net loss. The pulse broadening mechanism are usually group velocity disper- sion (GVD) and the finite spectral bandwidth of the gain medium (gain dispersion).
Under stable mode-locked operation the pulse compression per round-trip is equal to the pulse broadening per round-trip and the pulse reproduces itself. For semiconduc- tor lasers however there is an additional mechanism that contributes to the spectral broadening, namely the self-phase-modulation (SPM) due to the change in electron density induced by the pulse and the dependence of the refractive index on the elec- tron density. Stable mode-locking is then obtained when the spectral broadening per round-trip due to pulse compression and SPM is equal to the spectral narrowing per round-trip due to the finite gain bandwidth. The amount of chirp increases with
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each round-trip until equilibrium is reached, resulting in chirped pulses with large time-bandwidth products.
While this qualitative treatment explains how SPM can result in chirped pulses, a more rigorous analysis is necessary to explain the strong linearity and the sign of the chirp induced by SPM. Different models have been developed for passively mode-locked lasers using a time domain analysis [7 ,8,9,10,11 ], resulting in analytic expressions for the pulse shape, the pulse energy and the pulse width by making several approximations. These models however do not include SPM as they were developed for dye lasers where the refractive index does not depend on the inversion level. Other drawbacks of these models are that they assume symmetric pulse shapes and pulse energies much smaller than the saturation energy of the absorber and gain, while both assumptions are not valid for semiconductor lasers. The leading edge of the pulse is determined by saturation of the absorber, while the trailing edge is determined by the saturation of the gain, and as the saturation energies for absorber and gain are different, asymmetric pulse shapes should be expected. For semiconductor lasers where the gain and the absorber are strongly saturated, the pulse energy is of the same order of magnitude as the saturation energy of the active media (12).
More recently a model based on the traveling wave rate equations was developed for active mode-locking (13), where the effects of SPM are neglected, and for passive mode-locking of two-section lasers (3) where SPM is briefly discussed. Several models of pulse propagation through laser amplifiers have been developed, including SPM (14), gain dispersion (finite gain bandwidth) (15] and other fast gain dynamics (16,17).
Recently fast gain dynamics were discovered in semiconductor laser amplifiers using pump-probe measurements techniques [18,19,20,21]. These dynamics were attributed to carrier heating induced by free carrier absorption and two photon absorption [17, 21], and have time constants associated to intraband scattering that are on the order of 100fs. It was found that the fast gain dynamics have little effect on picosecond pulses, and are only important for subpicosecond pulses [16,17]. For the passively mode-locked two-section lasers studied here, the pulses propagating in the cavity are on the order of picoseconds, such that these fast gain dynamics can be neglected. The only time constants are then the carrier recovery time which is on the order of 1 ns for the gain section, and 100 ps for a reverse biased absorber. As these recovery times are much larger than the pulse widths, gain and absorption recovery can be neglected during passage of the pulse and will only be taken into account once the pulse has passed.
To include the effects of SPM and gain dispersion the frequency dependence of the gain and refractive index has to be included. As pointed out in [16] this frequency dependence was erroneously introduced in [15] as the frequency dependence of the a-parameter was ignored. In the next section a simple model will be used to calculate the frequency dependence of the real and imaginary part of the complex susceptibility X· Once the correct frequency dependence of xis determined, the traveling wave rate equation for the complex magnitude of the optical field will be derived together with a rate equation for the gain and the absorption. Solution of these coupled nonlinear rate equations will be discussed and applied two-section laser structures as used in
178 the experiments of the previous chapter.