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

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92 2.1.5 Pulse propagation in dispersive media [Ref. p. 134 overlap of the laser mode with the pump profile in the gain medium [93Pic]. The Kerr lens provides the strongest advantage for the pulsed operation when the cavity is operated close to the stability limit. Optimization guidelines for SAM produced by the Kerr lens in different cavities can be found in [95Mag]. Unfortunately, the transverse Kerr effect couples the mode-locking process with the laser cavity mode. In contrast, the use of only the longitudinal Kerr effect in mode-locking totally decouples the mode-locking process from the laser mode. This allows optimum cavity design for scaling the laser to higher powers and to higher pulse repetition rates without being constrained by the Kerr lens. 2.1.4.4.4 Nonlinear polarization rotation In fiber lasers a different Kerr-effect-based effective saturable absorber has been used to generate pulses as short as 38 fs [92Hof] – the shortest pulses generated directly from a fiber laser so far. An effective fast saturable absorber is obtained with a Kerr-induced nonlinear polarization rotation in a weakly birefringent fiber combined with a polarization-dependent loss. Previously, a similar idea has been used to “clean up” high-intensity pulses by reducing the low-intensity pulse pedestals [92Tap, 92Bea]. 2.1.4.5 Nonlinear mirror based on second-harmonic generation The second-order nonlinear susceptibility χ (2) nonlinearities can also be used to construct effective saturable absorbers [88Sta]. A nonlinear mirror based on this principle consists of a frequencydoubling crystal and a dichroic mirror. For short pulses, a part of the incident laser light is converted to the second harmonic, for which the mirror is highly reflective, and converted back to the fundamental wave, if an appropriate relative phase shift for fundamental and second-harmonic light is applied. On the other hand, unconverted fundamental light experiences a significant loss at the mirror. Thus the device has a higher reflectivity at higher intensities. This has been used for mode-locking e.g. with up to 1.35 W of average output power in 7.9-ps pulses from a Nd 3+ :YVO 4 laser [97Agn]. The achievable pulse duration is often limited by group-velocity mismatch between fundamental and second-harmonic light. 2.1.5 Pulse propagation in dispersive media 2.1.5.1 Dispersive pulse broadening Dispersion compensation is important in ultrashort pulse generation. When a pulse travels through a medium, it acquires a frequency-dependent phase shift φ(ω) =kn(ω)L, wherek is the wave number, n(ω) the refractive index and L the medium length. A phase shift which varies linearly with the frequency corresponds to a time delay, without any change of the temporal shape of the pulse. Higher-order phase shifts, however, tend to modify the pulse shape and are thus of relevance for the formation of short pulses. The phase shift can be expanded in a Taylor series around the center angular frequency ω 0 of the pulse: φ (ω) =φ 0 + ∂φ ∂ω (ω − ω 0)+ 1 ∂ 2 φ 2 ∂ω 2 (ω − ω 0) 2 + 1 ∂ 3 φ 6 ∂ω 3 (ω − ω 0) 3 + ... Landolt-Börnstein New Series VIII/1B1
Ref. p. 134] 2.1 Ultrafast solid-state lasers 93 Table 2.1.6. Sellmeier equations for different materials. The wavelength λ is given in units of μm. Material Defining Sellmeier equation Constants Fused quartz n 2 =1+ Aλ2 + Bλ2 + Cλ2 λ 2 − λ 2 1 λ 2 − λ 2 2 λ 2 − λ 2 3 A =0.6961663 λ 1 =0.0684043 B =0.4079426 λ 2 =0.1162414 C =0.8974794 λ 3 =9.896161 SF10 glass a 0 =2.8784725 n 2 = a 0 + a 1λ 2 + a2 λ + a3 2 λ + a4 4 λ + a5 a 1 = −0.010565453 6 λ 8 a 2 =3.327942 × 10 −2 a 3 =2.0551378 × 10 −3 a 4 = −1.1396226 × 10 −4 a 5 =1.6340021 × 10 −5 Sapphire n 2 =1+ a1λ2 + a2λ2 + λ 2 − b 1 λ 2 − b 2 a 1 =1.023798 a 2 =1.058264 a3λ2 a 3 =5.280792 λ 2 − b 3 b 1 =0.00377588 b 2 =0.0122544 b 3 = 321.3616 Here, the derivatives are evaluated at ω 0 . ∂φ/∂ω is the group delay T g , ∂ 2 φ/∂ω 2 the Group Delay Dispersion (GDD), ∂ 3 φ/∂ω 3 the Third-Order Dispersion (TOD). The GDD describes a linear frequency dependence of the group delay and thus tends to separate the frequency components of a pulse: For positive GDD, e.g., the components with higher frequencies are delayed with respect to those with lower frequencies, which results in a positive “chirp” (“up-chirp”) of the pulse. Higher orders of dispersion generate more complicated distortions. Material dispersion is normally described with Sellmeier equations for the refractive index as a function of the wavelength, i.e. n (λ). With the Sellmeier equations (Table 2.1.6) all the necessary dispersive quantities can be calculated (Table 2.1.7). An example is given in Table 2.1.8. The broader the bandwidth of the pulse (i.e., the shorter the pulse duration), the more terms of this expansion are significant. GDD which acts on an initially unchirped Gaussian pulse with Full-Width-at-Half-Maximum (FWHM) pulse duration τ 0 , increases the pulse duration according to [86Sie] √ ( ) τ p (z) 4ln2d2 φ/d ω = 1+ 2 2 , (2.1.20) τ 0 τ 2 0 where it is assumed that the incident pulse is transform-limited, i.e. the time–bandwidth product of the Gaussian pulse is τ 0 Δ ν p =0.4413, where Δ ν p is the Full-Width-at-Half-Maximum (FWHM) spectral width of the pulse intensity (Fig. 2.1.15). Only second-order dispersion (i.e. ∂ 2 φ/∂ ω 2 ) and higher orders are broadening the pulse. The first-order dispersion gives the group delay, i.e. the delay of the peak of the pulse envelope. It is apparent that the effect of GDD becomes strong if GDD >τ 2 0 . Similarly, TOD becomes important if TOD >τ 3 0 . It is important to note that for dispersive pulse broadening (which is in the linear pulse propagation regime) the spectrum of the pulse remains unchanged, only the spectral content of the pulse is redistributed in time. With positive dispersion the long-wavelength part of the spectrum is in the leading edge of the pulse and the short-wavelength part in the trailing edge of the pulse, i.e. “red is faster than blue” (Fig. 2.1.15). In the regime of strong pulse broadening, i.e. d 2 φ/d ω 2 ≫ τ 2 0 , we can reduce (2.1.20) Landolt-Börnstein New Series VIII/1B1
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References for 2.1 167 07Li 07Maa 0
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2.1 Ultrafast solid-state lasers - ETH - the Keller Group

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