any change in the refractive index of a laser active medium can lead to serious degradation of beam quality, laser beam modes, laser performance and variation in the intensity distribution. alteration in the refractive index of laser active medium is especially notable in high power lasers. it is clear that in the laser beam production, the pumping agent induces a great amount of heat which is loaded on the laser active medium. this induced heat load, subsequently the induced change of refractive index, is enormous. heat load in laser active medium can, at least in certain circumstances, switch the stable laser resonator to an unstable one. besides, highly heated crystals (laser active media) pose serious technical and maintenance problems. another effect of induced heat load on laser crystals is the induced stress within the crystal. stress can deform the crystal so that another considerable change of refractive index is faced by the crystal. the non-uniform heating of laser active medium leads to non-uniform index of refractive index. such medium is named graded index (grin) medium. therefore, these inductions must well be taken into account when one is dealing with design and construction of high power lasers. special types of laser beams are very much attractive in various scientific and industrial applications. this thesis is devoted to an investigation on the generation and propagation of these special types of laser beams, namely helmholtz-gauss and paraxial beams in a grin medium that appears as a result of a highly suffered medium from induced heat load. the procedure with which this goal is achieved will be given shortly. the more important results of our investigation, however, are briefly discussed presently. 3 1. the induced heat load causes the change in the refractive index of laser active medium. this change in turn induces a thermal lens in the medium. 2. the focal length of thermally induced lens decreases with increasing the power of pumping agent. 3. the change in mg output profile is almost independent of the two types of input power distributions used in this work (gaussian and super gaussian). 4. increasing the power of pumping agent causes contraction of the near field mg beam intensity profile without changing its shape. 5. at z=5 cm which is chosen as a measure of middle field, by increasing the power of input power, the intensity profile contract and the central lobe disappears giving rise to the side lobes. 6. the far field profiles show the appearing of the central lobe along with disappearing of the side ones due to increasing of the pump power. 7. in case ig beams the effect of increasing the pump power is the contraction of the near field intensity profiles without changing their shape for different modes of these beams. 8. the middle field intensity profiles of the modes with central lobe suffer two types of change: contraction of the profile and disappearing of the central lobe giving rise to the side ones. 9. the middle field intensity profiles of the modes without central lobe just contract with increasing the pump power. 10. increasing of the pump power cause the disappearing of the side lobes and appearing of the central ones in the case of the far field intensity profiles of ig beams with central lobe. 11. the effect of increasing the pump power on the far field profile of ig beams without central lobe is just the contraction or expansion of the profile. the aforementioned procedure now follows. in chapter 2 a detailed review of the literature including a historical outline of thermo-optical effects in diode-pumped solid state lasers and a historical view of special types of laser beams will be presented. in particular, a thorough review of the history of the development of the mg and ig beams are also presented in this chapter. 4 in order to model the effects of induced heat load on the propagation of special types of laser beams we consider a laser system which produces mathieu-gauss or ince-gaussian beams. a schematic illustration of the laser system is shown bellow: as shown in the illustration, a diode laser with central wavelength of 808 nm is used as the pumping agent. a 5 mm long, nd:yag crystal with 1.5 mm radius is used as the laser active medium. maximum absorption of the crystal is at about 808 nm, the central wavelength of the pumping agent. one end of the laser crystal is coated so as to act as one mirror of the laser cavity. another mirror is placed 80 mm in front of the first one. so the laser free length is 75 mm. the out-put laser beam is a mathieu-gauss or ince-gaussian beam with central wavelength of 1064 nm. at this wavelength the laser active medium is transparent, so the only source of generating heat in the laser crystal is the induced heat of the pumping agent. taking the above considerations into account, chapter 3 presents a detailed study of heat equation. taking the induced heat loads as sources we solve the heat equation, giving the temperature as a function of position. a discussion of the effects of temperature (as a function of position) on the refractive index is also presented in this chapter. a thermal model for lensing effects, along with the corresponding abcd transfer matrix, for an nd:yag laser, are also presented in this chapter. since the central part of this research is comprised of mathieu-gauss and ince- gaussian beams, as the outputs of a solid state laser, suffered from the induced heat loads, chapter 4 is devoted to a discussion of such laser beams. the figure 1-1 schematic illustration of an end pumped nd:yag laser 5 properties of their generation and propagation through an abcd optical systems, is also reviewed in this chapter. in chapter 5, use is made of the temperature distributions, as calculated in the third chapter, to investigate the effects of input power. since the shape of the output beams is of considerable interest, we further investigate these profiles of different locations on the cavity axis, with due attention to the input power. the procedure and the results are also given in this chapter. this thesis is concluded in chapter 6 with an outline of the results, some further remarks and some proposals for future works.