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

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104 2.1.6 Mode-locking techniques [Ref. p. 134 Table 2.1.11. Predicted pulse duration for the different ModeLocking (ML) techniques. The parameters used here are summarized and defined in Table 2.1.10. The pulse is transform-limited for proper dispersion compensation and is either an unchirped Gaussian or soliton pulse. ML technique Eq. Pulse shape Pulse duration (FWHM) Active ML: Amplitude loss modulation (2.1.49) Gauss τp =1.66 × 4 √ Dg Ms √ 2g =1.66 × 4 M √ 1 ωmΩg Passive ML: Slow saturable absorber and dynamic gain saturation (2.1.62) soliton τp ≈ 1.76 × 4 π 1 Δνg Slow saturable absorber for solid-state lasers and strongly saturated absorbers (S >3) (2.1.70) numerical simulations τp,min ≈ 1.5 Δνg √ g ΔR for τA 30 τp Fast Saturable Absorber (FSA) (2.1.65) soliton τp =1.76 4Dg γAEp Fully saturated ideal fast saturable absorber (2.1.67) τp,min = 1.76 only transform-limited soliton pulses for a well-defined intracavity group delay dispersion (assuming negligible higher-order dispersion): |D|/δL = Dg/γA Ωg √ 2g q0 ≈ 1.12 Δνg √ g ΔR for |D| δL = Dg γA Soliton mode-locking (2.1.74) soliton τp =1.76 2 |D| δLEp transform-limited soliton pulses for the total intracavity group delay dispersion (assuming negligible higher-order dispersion) (2.1.76) τp,min =1.7627 ( ) 3/4 1 √ φ −1/8 6 Ωg s ( τAg 3/2 q0 )1/4 ≈ 0.45 ( 1 Δνg ) 3/4 ( τA ΔR ) 1/4 g 3/8 φ 1/8 s Landolt-Börnstein New Series VIII/1B1
Ref. p. 134] 2.1 Ultrafast solid-state lasers 105 T R ∂A(T,t) ∂T = ∑ i ΔA i =0, (2.1.23) where A is the pulse envelope, T R is the cavity round-trip time, T isthetimethatdevelopsona time scale of the order of T R , t is the fast time of the order of the pulse duration, and ΔA i are the changes of the pulse envelope due to different elements in the cavity (such as gain, loss modulator or saturable absorber, dispersion etc.) (Table 2.1.10). Equation (2.1.23) basically means that at steady state after one laser round trip the pulse envelope cannot change and all the small changes due to the different elements in the cavity have to sum up to zero. Each element is modeled as a linearized operator, which will be discussed in more detail below. The pulse envelope is normalized such that |A (z,t)| 2 is the pulse power P (z,t): E (z,t) ∝ A (z,t)e i[ω0t−kn(ω0)z] with |A (z,t)| 2 ≡ P (z,t) , (2.1.24) where E (z,t) is the electric field, ω 0 the center frequency in radians/second of the pulse spectrum and k n = nk with k =2π/λ thewavenumberwithλ the vacuum wavelength and n the refractive index. Before we discuss the different mode-locking models we briefly discuss the linearized operators for the differential equations. 2.1.6.2.1 Gain A homogeneously broadened gain medium is described by a Lorentzian lineshape for which the frequency-dependent gain coefficient g (ω) isgivenby g g (ω) = ( ) 2 ≈ g (1 − Δ ) ω2 ω − ω0 Ωg 2 for ((ω − ω 0 )/Ω g ) 2 ≪ 1 , (2.1.25) 1+ Ω g where Δ ω = ω − ω 0 , g is the saturated gain coefficient for a cavity round trip, and Ω g is the Half Width at Half Maximum (HWHM) of the gain bandwidth in radians/seconds. In the frequency domain the pulse envelope after the gain medium is given by Ã out (ω) =e g(ω) Ã in (ω) ≈ [1 + g (ω)] Ãin (ω) for g ≪ 1 , (2.1.26) where Ã (ω) is the Fourier transformation of A (t). Equations (2.1.25) and (2.1.26) then give [ Ã out (ω) = 1+g − g ] [ Ωg 2 Δ ω 2 Ã in (ω) ⇒ A out (t) = 1+g + g ∂ 2 ] Ωg 2 ∂t 2 A in (t) , (2.1.27) where we used the fact that a factor of Δ ω in the frequency domain produces a time derivative in the time domain. For example for the electric field we obtain: ∂ ∂t E (t) = ∂ ∂t { 1 2π ∫ } Ẽ (ω)e iωt dω = 1 2π ∫ Ẽ (ω)iω e iωt dω (2.1.28) and ∂ 2 { ∂2 1 E (t) = ∂t2 ∂t 2 2π ∫ } Ẽ (ω)e iωt dω = 1 ∫ 2π Ẽ (ω) [ −ω 2] e iωt dω, (2.1.29) and similarly for the pulse envelope: ∂ ∂t A (t) = ∂ { ∫ } 1 Ã (ω)e iΔωt dω = 1 ∫ ∂t 2π 2π Ã (ω)iΔω e iΔωt dω (2.1.30) Landolt-Börnstein New Series VIII/1B1
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2.1 Ultrafast solid-state lasers - ETH - the Keller Group

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