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Chapter 2 Femtosecond Laser Pulses Chapter 2 Femtosecond Laser Pulses Since the birth of the laser, nearly 40 years ago, scientists have been con- tinually interested in generation of ultrafast laser pulses in the picosecond and femtosecond time domain. The recent development of all-solid state fem- tosecond lasers, tunable in the visible and near-infrared spectral regions, has already shown an impact on spectroscopic investigations in di®erent areas in physics, chemistry and biology. This chapter gives a brief introduction into the world of femtosecond laser pulses and the most frequently used techniques to measure such ultrashort pulses. The operating principle of Ti:sapphire oscillators and Ti:sapphire ampli- ¯ers together with the generation of parametric e®ects is described in the ¯rst part of this chapter. The second part is devoted to the diagnostic techniques used in the present work which allow the measurement of these ultrafast events. 2.1 Generation of Femtosecond Laser Pulses The central aim of this section is to give a concise introduction to nonlinear optics and to provide basic information about the most-widely used tunable femtosecond laser sources, in particular tunable Ti:sapphire oscillators and Ti:sapphire ampli¯ers or optical parametric ampli¯ers. 2.1.1 The Ti:sapphire Oscillator In 1982, the ¯rst Ti:sapphire laser was built by Moulton [9]. The laser tunes from 680 nm to 1130 nm, which is the widest tuning range of any laser of its 14 Femtosecond Laser Pulses class1. Nowadays Ti:sapphire lasers usually deliver several Watts of average output power and produce pulses as short as 6.5 fs [11]. Time-Frequency Relationship A pulse can be de¯ned as a transient in a constant background. The shape of this pulse is the shape of this transient. Intuitively, the pulse shape can be represented by a bell-shaped function, such as a Gaussian function. It is known that the Fourier transform of a Gaussian function is also a Gaussian function. The general time and frequency Fourier transforms of a pulse can be written 1 +1 E(t) = E(!)e¡i!td! (2.1) 2¼ Z¡1 and +1 E(!) = E(t)ei!tdt (2.2) Z¡1 where E(!) and E(t) represent the frequency and time evolution of the elec- tric ¯eld of the pulse, respectively. Since half-maximum quantities are ex- perimentally easier to measure, the relationship between the duration and spectral bandwidth of the laser pulse can be written as [12] ¢º¢t > K (2.3) where ¢º is the frequency bandwidth measured at full-width at half- maximum (FWHM) with ! = 2¼º and ¢t is the FWHM in time of the pulse and K is a number which depends only on the pulse shape (see Table 2.1). Thus in order to generate a laser pulse within femtosecond time domain one needs to use a broad spectral bandwidth. If the equality is reached in (2.3) one speaks about a Fourier-transform-limited pulse or simply a transform- limited pulse. One can also calculate the minimum time duration of a pulse giving a spectrum with ¢¸ (nm) at FWHM, central wavelength ¸0 (nm) and the speed of light (m/s) c: ¸2 ¢t > K 0 (2.4) ¢¸ c ¢ 1Other laser active media which provide laser pulses in the femtosecond time domain are Cr3+:LiSAF, Cr3+:LiCAF, Cr4+:YAG, Cr4+:Forsterite. More about in Ref. [10]. 2.1 Generation of Femtosecond Laser Pulses 15 Function E(t) K 2 Gauss e¡(t=t0) =2 0.441 Hyperbolic secant 1=cosh(t=t0) 0.315 2 Lorentz 1=[1 + (t=t0)] 0.142 Table 2.1: Di®erent mathematical descriptions of a laser pulse and corre- sponding expression of the electric ¯eld. The constant K which determines the time-bandwidth product is given for each function as well. Absorption transitions of sapphire crystals doped with Titanium (Ti3+) ions occur over a broad range of wavelengths from 400 to 600 nm (see Fi- gure 2.1). The emission band extends from wavelengths as short as 600 nm to wavelengths greater than 1 ¹m. This makes them suitable materials in constructing laser active media for generating femtosecond pulses. The long wavelength side of the absorption band overlaps with the short wavelength end of the emission spectrum. Therefore the laser action is only possible at wavelengths longer than 660 nm. Additionally, the tuning range is a®ected by mirror coatings, losses in the laser cavity, pump power, and pump mode quality [13]. The optics existing in the Ti:sapphire oscillators used in the present work (see also Chapter 4, sections 4.2.1 and 4.2.2) allow an unprob- lematic tunability between 750 nm and 850 nm, which corresponds to the wavelength range where the emission spectrum has a maximum. Group Velocity Dispersion The Group Velocity Dispersion (GVD) is de¯ned as the propagation of dif- ferent frequency components at di®erent speeds through a dispersive medium. This is due to the wavelength-dependent index of refraction of the dispersive material. GVD causes variation in the temporal pro¯le of the laser pulse, while the spectrum remains unaltered. A transform-limited pulse is also called short pulse or unchirped pulse. It is said that the initial short pulse will become positively chirped (or upchirped) after propagating through a medium with "normal" dispersion (e.g. silica glass). This corresponds to the situation when higher frequencies travel slower than lower frequencies (blue slower than red). The opposite situation, where the pulse travels through a medium with "anomalous" dispersion, leads to a negative chirp (or downchirp). Here the bluer frequencies propagate faster than the redder frequencies. Sources of GVD are glass, prism sequences, di®raction gratings, etc. 16 Femtosecond Laser Pulses Absorption Emission . .u a y / sit en Int Wavelength / nm Figure 2.1: Absorption and emission spectra of the Ti:sapphire laser. The broad character of the absorption/emission spectra is due to the strong coupling between the vibrational energy states of the host sapphire crystal and the electronic energy states of the active Ti3+ ions [13]. The Kerr Lens Mode-locking For simplicity, consider the spatial propagation of a Gaussian laser beam in a nonlinear material. The intensity pro¯le of the beam is a function of its radius r and of a shape parameter g [12]. At high intensities, the refractive index depends nonlinearly on the propagating ¯eld. The lowest order of this dependence can be written as follows [12]: 1 n(r) = n + n I(r) (2.5) 0 2 2 where n2 is the nonlinear index coe±cient and describes the strength of the coupling between the electric ¯eld and the refractive index n. According to Ref. [12] the intensity is: 2 I(r) = e¡gr (2.6) Hence, the refractive index changes with intensity along the optical path and it is larger in the center than at the side of the nonlinear crystal. This leads to the beam self-focusing phenomenon, which is known as the Kerr lens e®ect (see Figure 2.2). This process is enhanced along the optical path 2.1 Generation of Femtosecond Laser Pulses 17 Crystal length r Self-focusing Beam diameter n(r) Intensity Figure 2.2: The Kerr lens e®ect and self-focusing. The index of refraction varies with intensity along the beam diameter. Depending on the sign of the nonlinear term n2 in the expression (2.5) the index of refraction in- creases or decreases towards the center of the laser beam [12]. For positive n2 the laser beam self-focuses. because focusing the beam increases the focal "power" of this "lens". The increase of the focusing stops when the diameter of the beam is small enough and the linear di®raction is large enough to balance the Kerr e®ect. Consider now a seed beam with a Gaussian pro¯le propagating through a nonlinear medium, e.g. a Ti:sapphire crystal, which is pumped by a cw radiation. As aforementioned, for high intensity light, due to the intensity de- pendent refractive index, the Kerr lens e®ect occurs. When the laser operates in its most usual regime (free-running laser), it can oscillate simultaneously over all the resonance frequencies of the cavity. These frequencies make up the set of longitudinal modes of the laser. For the stronger focused frequen- cies, the Kerr lens favors a higher ampli¯cation. Thus the self-focusing of the seed beam can be used to suppress the cw operation, because the losses of the cw radiation are higher. Forcing all the modes to have equal phase (mode-locking) implies that all the waves of di®erent frequencies will inter- fere (add) constructively at one point, resulting in a very intense, short light pulse [14]. The pulsed operation is then favored and it is said that the laser is mode-locked. Thus the mode-locking occurs due to the Kerr lens e®ect induced in the nonlinear medium by the beam itself and the phenomenon is known as Kerr-lens mode-locking. 18 Femtosecond Laser Pulses a) net gain Dn Frequency b) c Resonator's eigenfrequencies 2L Frequency c) amplified modes Frequency Figure 2.3: . The Kerr Lens Mode-Locking (KLM) principle. The axial modes of a laser cavity are separated by the intermode frequency spacing º = c=2L. (a) The net gain curve (gain minus losses). In this example, from all the longitudinal modes in the resonator (b), only six (c) are forced to have an equal phase [12]. The modes are separated in frequency by º = c=2L, L being the resonator length, which also gives the repetition rate of the mode-locked lasers: 1 c ¿ = = (2.7) rep T 2L Moreover the ratio of the resonator length to the pulse duration is a measure of the number of modes oscillating in phase.
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