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

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94 2.1.5 Pulse propagation in dispersive media [Ref. p. 134 Table 2.1.7. Dispersion quantities, their defining equations and units. k n: wave vector in the dispersive media, i.e. k n = kn = n 2 π/λ, where λ is the vacuum wavelength. z: a certain propagation distance. c: vacuum light velocity. ω: frequency in radians/second. Quantity Symbol Defining equation Defining equation using n (λ) Phase velocity υ p ω k n Group velocity υ g dω dk n Group delay T g Tg = z υ g = dφ dω , φ ≡ knz Dispersion: 1st order dφ dω c n c n nz c nz c 1 1 − dn λ dλ n ( 1 − dn ) λ dλ n ( 1 − dn ) λ dλ n Dispersion: 2nd order Dispersion: 3rd order d 2 φ dω 2 d 3 φ dω 3 λ 3 z d 2 n 2 π c 2 dλ 2 ( ) −λ 4 z 3 d2 n 4 π 2 c 3 dλ + λ d3 n 2 dλ 3 τ0 Dispersive medium φ( ω) = k . n ( ω ) . L τ n ( ω) L Fig. 2.1.15. Dispersive pulse broadening through a material with positive dispersion. to τ p (z) ≈ d2 φ d ω 2 Δ ω p , (2.1.21) where Δ ω p =2π Δ ν p is the FWHM spectral width (in radians/second) of the pulse intensity. 2.1.5.2 Dispersion compensation Without any dispersion compensation the net GDD for one cavity round trip is usually positive, mainly because of the dispersion in the gain medium. Other components like mirrors may also contribute to this. However, in lasers with > 10 ps pulse duration the dispersion effects can often be ignored, as the total GDD in the laser cavity is typically at most a few thousand fs 2 ,much less than the pulse duration squared (2.1.20). For shorter pulse durations, the GDD has to be considered, and pulse durations well below 30 fs usually require the compensation of Third-Order Dispersion (TOD) or even higher orders of dispersion depending on the thickness of the gain material. In most cases, the desired total GDD is not zero but negative, so that soliton formation Landolt-Börnstein New Series VIII/1B1
Ref. p. 134] 2.1 Ultrafast solid-state lasers 95 Table 2.1.8. Examples of material dispersions calculated from the Sellmeier equations given in Table 2.1.6 and the equations given in Table 2.1.7. Material Refractive index n at a center wavelength of 800 nm Propagation constant k n at a center wavelength of 800 nm Fused quartz n (0.8 μm) = 1.45332 ∂n ∂λ∣ = −0.017 1 800 nm μm ∣ ∂ 2 n ∣∣∣800 1 =0.04 ∂λ 2 nm μm 2 ∣ ∂ 3 n ∣∣∣800 1 = −0.24 ∂λ 3 nm μm 3 ∂k n ∂ω ∣ =4.84 × 10 −9 800 nm s ns =4.84 m m ∣ ∂ 2 k n ∣∣∣800 =3.61 × 10 −26 s 2 fs2 =36.1 ∂ω 2 nm m mm ∣ ∂ 3 k n ∣∣∣800 =2.74 × 10 −41 s 3 fs3 =27.4 ∂ω 3 nm m mm SF10 glass n (0.8 μm) = 1.71125 ∂n ∂λ∣ = −0.0496 1 800 nm μm ∣ ∂ 2 n ∣∣∣800 1 =0.176 ∂λ 2 nm μm 2 ∣ ∂ 3 n ∣∣∣800 1 = −0.997 ∂λ 3 nm μm 3 ∂k n ∂ω ∣ =5.70 × 10 −9 800 nm s ns =5.70 m m ∣ ∂ 2 k n ∣∣∣800 =1.59 × 10 −25 s 2 fs2 = 159 ∂ω 2 nm m mm ∣ ∂ 3 k n ∣∣∣800 =1.04 × 10 −40 s 3 fs3 = 104 ∂ω 3 nm m mm Sapphire n (0.8 μm) = 1.76019 ∂n ∂λ∣ = −0.0268 1 800 nm μm ∣ ∂ 2 n ∣∣∣800 1 =0.064 ∂λ 2 nm μm 2 ∣ ∂ 3 n ∣∣∣800 1 = −0.377 ∂λ 3 nm μm 3 ∂k n ∂ω ∣ =5.87 × 10 −9 800 nm s ns =5.87 m m ∣ ∂ 2 k n ∣∣∣800 =5.80 × 10 −26 s 2 fs2 =58 ∂ω 2 nm m mm ∣ ∂ 3 k n ∣∣∣800 =4.21 × 10 −41 s 3 fs3 =42.1 ∂ω 3 nm m mm can be exploited. Usually, one requires sources of negative GDD, and in addition appropriate higherorder dispersion for shorter pulses. The most important techniques for dispersion compensation are discussed in the following subsections. Different optical elements that introduce wavelengthdependent refraction (i.e. prism pairs, Sect. 2.1.5.2.3) or wavelength-dependent diffraction (i.e. grating pairs, Sect. 2.1.5.2.2) can be used to introduce an additional wavelength dependence to the round-trip phase and thus contribute to the overall dispersion. A wavelength-dependent round-trip phase can also be introduced with GTIs (Sect. 2.1.5.2.1) and chirped mirrors (Sect. 2.1.5.2.4). The challenge in ultrashort pulse generation is dispersion compensation over a large bandwidth to compensate for the dispersive pulse broadening that is occurring in the gain material and other elements inside the laser cavity. Dispersion compensation is important because for example, a 10-fs (1-fs) Gaussian pulse at the center wavelength of 800 nm is broadened to 100 fs (1 ps) after only 1 cm of fused quartz due to second-order dispersion. This follows from (2.1.21) for the regime of strong pulse broadening and Tables 2.1.6–2.1.8. In addition, in femtosecond lasers the pulses are ideally soliton pulses for which a constant negative dispersion over the full spectral width of the pulse balances the chirp of the self-phase modulation. The necessary negative dispersion required for a certain pulse duration follows from (2.1.74). It is required that all higher-order dispersion terms are negligibly small. Landolt-Börnstein New Series VIII/1B1
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References for 2.1 161 04Gra Grange
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2.1 Ultrafast solid-state lasers - ETH - the Keller Group

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