Spectroscopy for Materials Characterization. Группа авторов

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of laser modes oscillating in‐phase within the laser cavity [19, 20]. The most widespread type of femtosecond mode‐locked laser in modern spectroscopy is the Ti:sapphire laser. A typical Ti:sapphire oscillator emits laser pulses with a central wavelength tunable around 800 nm, typical duration of 10–100 fs, energy of 1–100 nJ pulse⁻¹, and repetition rate of ∼80 MHz. By using an external amplifier, these pulses can be then amplified up to μJ or mJ per pulse with a proportional reduction of the repetition rate. Because of the very short duration, these numbers imply intensities as high as tens of GW per cm², which can easily be achieved even without focusing. These intense, amplified pulses are then available to feed a range of experiments in nonlinear optics and spectroscopy such as those described in this chapter. The details of mode‐locking will not be further discussed here, and the rest of this section will be devoted to describing some general properties of propagating femtosecond light pulses.

      The time dependence of the oscillating electric field in an amplified ultrashort pulse is described by . The wave amplitude follows a Gaussian envelope, with the shape factor γ proportional to the squared inverse of the duration of the pulse: γ ∝ Δt ⁻². To obtain the spectral content of the pulse, one can calculate the Fourier transform of E(t), which is shown to be again a Gaussian function, centered at the frequency ω ₀ and with a frequency width Δω proportional to γ ^1/2 [19]. This entails a strict relation between the time duration of the pulse and its spectral width, according to Heisenberg's uncertainty principle . This leads to the fundamental consequence that ultrashort laser pulses are intrinsically non‐monochromatic: in order to have pulse durations in the femtosecond range, it is compulsory to have a broad enough spectral distribution, typically in the tens of nanometers. When the equivalence of the uncertainty principle is verified, the pulse is named Fourier transform‐limited. This condition can only be perfectly achieved by a Gaussian pulse, and guarantees the shortest possible pulse for a given spectral width. For example, a transform‐limited Gaussian pulse with 10 fs duration, peaking at 800 nm, shows a spectral bandwidth of 94 nm. As discussed hereafter, one consequence of the broad bandwidth of femtosecond pulses is that their propagation is affected by strong dispersion effects [19, 20]. On the other hand, their extremely high peak intensities, due to the short duration, lead to intense nonlinear optical effects.

      3.2.1.1 Dispersion Effect: Group Velocity Dispersion

      When dealing with optical pulses with femtosecond pulse durations, it is important to consider the effects of group velocity dispersion (GVD). The latter affects the duration of a light pulse which traverses any media, because of the frequency dependence of the refractive index n(ω). GVD is defined as:

      (3.1)

where v _g is the group velocity. The latter can be written as:

      (3.2)

GVD is measured in fs² mm⁻¹ and, in a transparent region, is typically positive because of the characteristic dependence of n on frequency. During propagation, every spectral component of the pulse acquires a different delay, resulting in a temporal broadening of the pulse without any spectral changes [21]. To visualize the effect of GVD on a Gaussian pulse passing through a medium, a simulation is shown in Figure 3.1. From top to bottom, three pulses are shown, representing a transform‐limited Gaussian with FWHM = 5 fs centered at 550 nm, and the same pulse after passing through 1 or 2 mm SiO₂, respectively. As evident from the figure, GVD substantially enlarges the pulse duration, increasing it to several hundreds of femtoseconds. The pulse duration Δt, that is the FWHM of the Gaussian intensity profile, broadens to Δt _b given by:

      (3.3)

where L is the propagation length inside the material [19]. Besides broadening, GVD causes a frequency chirp, that is a time dependence of the instantaneous frequency of the pulse, given by . In Figure 3.1, the chirp is evident from the comparison of the frequencies of the two sides of the pulse, showing that the instantaneous frequency is redder in the front part of the pulse and bluer in the back [19, 20], which is exactly the effect of a positive GVD. In particular, the instantaneous frequency acquires an approximately linear time dependence, ω(t) = ω ₀ + αt, because the phase of the wave acquires a quadratic time term.

Figure 3.1 Panel (a): Simulation of a gaussian pulse centered at 550 nm with FHWM = 5 fs (first curve from the top) after propagation through a SiO₂ medium of 1 mm thickness (second curve) and 2 mm (third curve). Panel (b and c): zooms of the tails of the black pulse (squares). Each wavelength is delayed by a different phase, resulting in a longer pulse with a positive chirp (the redder frequencies are faster than the bluer).

      Pulse broadening and chirp acquired by femtosecond pulses during their propagation in optical setups need to be put under control in order to preserve good time resolution. One way to do it is to limit the use of transparent optical components, preferring the use of reflective optics only. Some special methods exist to manipulate the chirp, such as what is called a pulse compressor, built by using a pair of prisms or gratings. In a pulse compressor, one can add negative GVD (redder part of a pulse propagates slower than the blue part) which compensates the effect of pulse broadening in a transparent media, recompressing the pulse [19, 20].

      Last but not least, dispersion also affects the temporal overlap of two pulses centered at different wavelengths which pass through the same medium, because their group velocities are generally different: this effect is called group velocity mismatch, or GVM. Thereby, if the pulses are initially overlapping in time at a certain point in space, they overlap no more after some propagation distance within a dispersive medium. The GVM effect, for example, can be very important in the generation of pulses through nonlinear effects because it can limit the effective interaction length between the two pulses.

      3.2.2 Nonlinear Optics: Basis and Applications

      3.2.2.1 Second Harmonic Generation and Sum Frequency Generation

      Second harmonic generation (SHG) is a nonlinear optical phenomenon in which two photons of the same frequency, interacting in a nonlinear material, are converted in a single photon with doubled frequency [22]. The polarization of a medium excited by an electrical field can be expressed as [19, 20]:

      (3.4)

      where, in general, χ ⁽ⁿ⁾ is a tensor. The first term of the equation describes the phenomena usually encountered in linear optics, while the other describes nonlinear effects at different orders in the electric field. Under certain conditions (χ ⁽²⁾ ≠ 0, as generally occurs in

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