**1. Introduction**

Propagation of ultrashort optical pulses in a linear optical medium consisting of free space [1–5], dispersive media [6, 7], diffractive optical elements [8–10], focusing elements [11] and apertures [12, 13] has been extensively studied analytically, though only a few isolated attempts have been made on numerical simulation. However, analytical methods have the limitations of not being able to handle arbitrary pulse profiles.

### **2. Ultrashort laser pulses generations**

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 amplifiers or optical parametric amplifiers.

#### **2.1 Titane sapphire oscillator**

In 1982, the first Ti:sapphire laser was built by Mouton [14]. The laser tunes from 680 nm to 1130 nm, which is the widest tuning range of any laser of its class1. Nowadays Ti:sapphire lasers usually deliver several watts of average output power and produce pulses as short as 6.5 fs (**Figure 1**) [14].

At high intensities, the refractive index depends nonlinearly on the propagating field. The lowest order of this dependence can be written as follows:

$$n(r) = n\_0 \frac{1}{2} n\_2 I(r) \tag{1}$$

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 effect induced in the nonlinear medium by the beam itself and the phenomenon is known as Kerr-lens mode-

*Femtosecond Laser Pulses: Generation, Measurement and Propagation*

*DOI: http://dx.doi.org/10.5772/intechopen.95978*

locking.

**Figure 3.**

**5**

*2.1.1 Examples*

Auto TPL Tripler for laser oscillator.

Sprite XT: Tunable ultrafast Ti: sapphire laser (**Figure 3**).

which also gives the repetition rate of the mode-locked lasers:

Where L is length of cavity and T is period.

The modes are separated in frequency by *ν* ¼ *c=*2*L*, L being the resonator length,

*<sup>T</sup>* <sup>¼</sup> *<sup>c</sup>*

*The Kerr lens mode-locking (KLM) principle. (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.*

<sup>2</sup>*<sup>L</sup>* (3)

*<sup>τ</sup>rep* <sup>¼</sup> <sup>1</sup>

n0: linear index refractive.

where *n*<sup>2</sup> is the nonlinear index coefficient and describes the strength of the coupling between the electric field and the refractive index *n*. The intensity is:

$$I(r) = e^{-gr^2} \tag{2}$$

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 selffocusing phenomenon, which is known as the Kerr lens effect (see **Figure 2**).

Consider now a seed beam with a Gaussian profile propagating through a nonlinear medium, e.g. a Ti:sapphire crystal, which is pumped by a cw radiation. For the stronger focused frequencies, the Kerr lens favors a higher amplification. 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 different frequencies will interfere (add) constructively at one point, resulting in a very intense, short light pulse.

**Figure 2.** *The Kerr lens effect and self-focusing.*

*Femtosecond Laser Pulses: Generation, Measurement and Propagation DOI: http://dx.doi.org/10.5772/intechopen.95978*

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 effect induced in the nonlinear medium by the beam itself and the phenomenon is known as Kerr-lens modelocking.
