2.1 Ultrafast solid-state lasers - ETH - the Keller Group

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102 2.1.6 Mode-locking techniques [Ref. p. 134 shifts of those GTI oscillations so that a special selection of mirrors makes it ultimately possible to obtain the right dispersion compensation. Some tuning of the oscillation peaks can be obtained by the angle of incidence [00Sut]. A specially designed pair of DCMs has been used to cancel the spurious GTI oscillation [01Kae] where an additional quarter-wave layer between the AR-coating and the DCM structure was added in one of the DCMs. Also this design has its drawbacks and limitations because it requires an extremely high precision in fabrication and restricts the range of angles of incidence. After this overview it becomes clear that there is no perfect solution to the challenge of ultrabroadband dispersion compensation. At this point ultrabroadband chirped mirrors are the only way to compress pulses in the one- to two-optical-cycle regime and normally a larger selection of chirped mirrors are required to “match and reduce” the residual unwanted GDD oscillations from all mirrors inside the laser cavity. 2.1.6 Mode-locking techniques 2.1.6.1 Overview Passive mode-locking mechanisms are well-explained by three fundamental models: slow saturable absorber mode-locking with dynamic gain saturation [72New, 74New] (Fig. 2.1.5a), fast saturable absorber mode-locking [75Hau1, 92Hau] (Fig. 2.1.5b) and slow saturable absorber modelocking without dynamic gain saturation which in the femtosecond regime is described by soliton mode-locking [95Kae1, 95Jun2, 96Kae] and in the picosecond regime by Paschotta et al. [01Pas1] (Fig. 2.1.5c). The physics of most of these techniques can be well explained with Haus’s master equation formalism as long as at steady state the changes in the pulse envelope during the propagation inside the cavity are small. At steady state the pulse envelope has to be unchanged after one round trip through the cavity. Passive mode-locking, however, can only be analytically modeled in the weak saturation regime, which is typically not the case in SESAM mode-locked solid-state lasers. However, this formalism still provides a useful approach to describe mode-locking techniques in an unified fashion. Recent numerical simulations show that analytical results with fast saturable absorbers only slightly underestimate numerical solutions and correctly describe the dependence on saturated gain, gain bandwidth and absorber modulation taking into account more strongly saturated absorbers and somewhat longer saturation recovery times in SESAM mode-locked solid-state lasers [01Pas1]. A short introduction to Haus’s formalism is given in Sect. 2.1.6.2 and Table 2.1.10. Afterwards we will describe all mode-locking techniques using this formalism and summarize the theoretical prediction for pulse shape and pulse duration (Table 2.1.11). For solid-state lasers self-Q-switching instabilities in passive mode-locking are a serious challenge. Simple guidelines to prevent those instabilities and obtain stable cw mode-locking are presented in Sect. 2.1.6.8. 2.1.6.2 Haus’s master equations Haus’s master equation formalism [95Hau2] is based on linearized differential operators that describe the temporal evolution of a pulse envelope inside the laser cavity. At steady state we then obtain the differential equation: Landolt-Börnstein New Series VIII/1B1
Ref. p. 134] 2.1 Ultrafast solid-state lasers 103 Table 2.1.10. Linearized operators that model the change in the pulse envelope A (t) for each element in the laser cavity and their defining equations. The pulse envelope is normalized such that |A (z, t)| 2 is the pulse power P (z, t) (2.1.24). kn: wave vector in the dispersive media, i.e. kn = kn = n2π/λ, where λ is the vacuum wavelength. z: the relevant propagation distance for negative dispersion or SPM, respectively. c: vacuum light velocity. ω: frequency in radians/second. Laser cavity element Eq. Linearized operator New constants Constants [ Gain (2.1.32) ΔA ≈ g + Dg ] ∂ 2 A ∂t 2 Dg ≡ g Ω g 2 Dg: gain dispersion (2.1.32) g: saturated amplitude gain coefficient Ωg: HWHM of gain bandwidth in radians/second Δνg: FWHM of gain bandwidth, i.e. Δνg = Ωg/π Loss modulator (2.1.34) ΔA ≈−Mst 2 A Ms ≡ Mω2 m 2 ωm: loss modulation frequency in radians/second 2M: peak-to-peak modulation depth for amplitude loss coefficient Constant loss ΔA ≈−lA l: amplitude loss coefficient Constant phase shift ΔA ≈ iψA ψ: phase shift Fast saturable absorber (2.1.36) ΔA ≈ γA |A| 2 A γA ≡ q0 Isat,AAA γA: absorber coefficient (2.1.18) and (2.1.35) q0: maximum saturable amplitude loss coefficient Isat,A: saturation intensity AA: laser mode area in saturable absorber Dispersion: 2nd order (2.1.41) ΔA ≈ iD ∂2 ∂t A D ≡ 1 2 2 k′′ nz D: dispersion parameter (half of the total group delay dispersion per cavity round trip – Table 2.1.7) (2.1.40) k n ′′ = d2 kn dω 2 SPM (2.1.47) ΔA ≈−iδL |A| 2 A δL ≡ kn2z AL δL: SPM coefficient (2.1.44) n2: nonlinear refractive index AL: laser mode area inside laser material (Note: Here we assume that the dominant SPM occurs in the laser material. Then z is equal to 2 times the length of the laser crystal in a standing-wave cavity.) Landolt-Börnstein New Series VIII/1B1
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2.1 Ultrafast solid-state lasers - ETH - the Keller Group

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