**2. Theory**

Figure 1 displays a general schematic of electronic energy levels of laser active centers embedded in an active medium and of saturable absorption centers embedded into solid state passive optical Q-switch cell laser oscillator operated in passive optical Q-switching regime [11-18,26]. As laser active centers, rare earth (RE) triple ions included by substitution into crystalline or poly micro-crystalline structure of the solid state active medium are used to a large extent. The four electronic energy levels of a laser active center can be observed together with the transitions between them, transitions which constitute the laser emission

© 2012 Lăncrănjan et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

cycle of the active medium. The pumping transition, sometimes denoted as pumping absorption, from level "am0" to level "am1" together with the laser emission transition from level "am2" to level "am3" are presented. The pumping transition which assures attaining the population inversion between levels "am2" and "am3" can be technologically realized, considering basically the longitudinal or transverse/perpendicular versus laser beam propagation direction, by absorption of pumping radiation using one of two possibilities: longitudinal, along the active medium axis or transverse, through its side surfaces. Usually longitudinal pumping means a distribution of population inversion density which has an important consequence upon the laser dynamics. The transitions "am1" "am2" and "am3" "am0" are fast non-radiative transitions [26]. The laser emission transition "am2" "am3" has a characteristic fluorescence lifetime *g* in the range from μs to ms. The threelevels structure of electronic energy laser active medium is possible with the main characteristic difference that the level "am3" is identical with the level "am0" [5,6,7].

**Figure 1.** The schematic of a solid state laser active medium and passive optical Q-switch cell electronic energy levels. Active medium - levels: am0 - ground level; am1 - pumping level; am2 - upper laser level; am3 - lower laser level; Saturable absorber - sa1 - ground level; sa2 - upper saturable absorber level; sa3 - saturable absorber lower excited level; sa4 - saturable absorber upper excited level [26].

As can be seen in the schematic describing the functional cycle of the saturable absorber material, the absorption transition from electronic energy level "sa1" to electronic energy level "sa2" is the most important for passive optical Q-switching process, relying on its nonlinear characteristic absorption coefficient, *(I).* The *(I)* coefficient varies with the intensity *I* of the light beam incident on the saturable absorber according to an equation as [5,7]:

$$\alpha\left(I\right) = \frac{\alpha\_0}{1 + \frac{I}{I\_{sat}}}\tag{1}$$

Numerical Simulation of Passively Q-Switched Solid State Lasers 291

optical Q-switch cell appears as transparent (bleached) at the laser wavelength, allowing the laser radiation to circulate inside resonator. It has to be understood that simultaneously with the bleaching process, in the active medium the population inversion is accumulated, the laser active centres being stored into upper laser level. Practically, the assemble of the pumped active medium and passive optical Q-switch cell placed inside a resonator formed by two mirrors represent a laser oscillator with poor optical quality and high threshold condition. When the high energy threshold condition is attained for this low quality oscillator, a large population inversion is stored in the active medium and enhanced fluorescence emission intensity at laser wavelength is obtained. Because of the passive optical Q-switch cell bleaching effect, which is fast enough with in comparison to the upper laser level fluorescence lifetime, the optical quality of the resonator is fast growing to high values allowing laser emission [11,12,19-23]. As a consequence, the population inversion accumulated in the active medium is converted into emission of short laser pulses. The laser oscillator behaves as in free running regime in a very short early period quality and after

that as in Q-switching regime characterised by emission of a short laser pulse burst.

laser resonator after the bleaching of the passive optical Q-switch cell [13-16].






presented in Figure 2 and in Figure 5, are the following:

**2.1. Laser rate equations** 








As can be noticed, the possible situation of a non-saturable parasitic absorption of the passive optical Q-switch cell is presented. This parasitic absorption can take place between levels "sa3" and "sa4", levels "sa3" being populated by excitation transfer process from level "sa2" [20,21,27]. The non-saturable parasitic absorption can represent an embarrassment for Q-switching process in the case of a good spectral match between laser emission wavelength and "sa3" "sa4" absorption because imposing a lower quality of the

The parameters characteristic for a passively optical Q-switched operated solid state laser system and appearing in the equations describing its functionality, as schematically

*Isat* represents the saturation value of the light beam intensity, the value for which the absorption coefficient is reduced with 50%. The function as a passive optical Q-switch cell is obtained by absorptive transition "sa1" "sa2" which is characterized by a fluorescence lifetime of the upper level long enough to store the saturable absorption centres on the upper level and to depopulate the lower level at a fast rate. In this situation, the passive optical Q-switch cell appears as transparent (bleached) at the laser wavelength, allowing the laser radiation to circulate inside resonator. It has to be understood that simultaneously with the bleaching process, in the active medium the population inversion is accumulated, the laser active centres being stored into upper laser level. Practically, the assemble of the pumped active medium and passive optical Q-switch cell placed inside a resonator formed by two mirrors represent a laser oscillator with poor optical quality and high threshold condition. When the high energy threshold condition is attained for this low quality oscillator, a large population inversion is stored in the active medium and enhanced fluorescence emission intensity at laser wavelength is obtained. Because of the passive optical Q-switch cell bleaching effect, which is fast enough with in comparison to the upper laser level fluorescence lifetime, the optical quality of the resonator is fast growing to high values allowing laser emission [11,12,19-23]. As a consequence, the population inversion accumulated in the active medium is converted into emission of short laser pulses. The laser oscillator behaves as in free running regime in a very short early period quality and after that as in Q-switching regime characterised by emission of a short laser pulse burst.

As can be noticed, the possible situation of a non-saturable parasitic absorption of the passive optical Q-switch cell is presented. This parasitic absorption can take place between levels "sa3" and "sa4", levels "sa3" being populated by excitation transfer process from level "sa2" [20,21,27]. The non-saturable parasitic absorption can represent an embarrassment for Q-switching process in the case of a good spectral match between laser emission wavelength and "sa3" "sa4" absorption because imposing a lower quality of the laser resonator after the bleaching of the passive optical Q-switch cell [13-16].
