Julio C. Gutiérrez-Vega is associate professor in the Physics Department and heads the Optics Center and the Photonics and Mathematical Optics Group at the Tecnológico de Monterrey, Monterrey, México. http://optica.mty.itesm.mx/pmog/

Julio C. Gutiérrez-Vega received the BS degree in physics (1991) and MS degree in electric engineering (1995) from the Tecnológico de Monterrey. In 2000, he received his PhD degree in optics from the National Institute for Astrophysics, Optics, and Electronics in Puebla, México. He is the author and co-author of more than 145

scientific publications in international journals, conference proceedings, and books. His research activities are focused on the nondiffracting propagation of wavefields, special solutions of the Helmholtz and paraxial wave equation: Mathieu, parabolic, and Ince-Gaussian beams, and laser resonators. Dr. Gutiérrez-Vega is a member of

SPIE, OSA, and APS.

Julio C. Gutiérrez-Vega received the BS degree in physics (1991) and MS degree in electric engineering (1995) from the Tecnológico de Monterrey. In 2000, he received his PhD degree in optics from the National Institute for Astrophysics, Optics, and Electronics in Puebla, México. He is the author and co-author of more than 145

scientific publications in international journals, conference proceedings, and books. His research activities are focused on the nondiffracting propagation of wavefields, special solutions of the Helmholtz and paraxial wave equation: Mathieu, parabolic, and Ince-Gaussian beams, and laser resonators. Dr. Gutiérrez-Vega is a member of

SPIE, OSA, and APS.

**Publications**(58)

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*n*-order intensity moments of the beam. Propagation through complex ABCD optical systems, normalization factor, beam width, the quality

*M*

^{2}factor and its kurtosis parameter are derived. We discuss its behavior for different beam parameters and the relation between them. The Cartesian Parabolic-Gaussian beams carry finite power and form a biorthogonal set of solutions of the paraxial wave equation in Cartesian coordinates.

^{2}factor and its kurtosis parameter are derived. We discuss its behavior for different beam parameters and the relation between them. The WGBs carry finite power and form a biorthogonal set of solutions of the paraxial wave equation (PWE) in circular cylindrical coordinates.

*unwinding*of vortex beams is proposed. This method allows for the continuous tuning of the orbital angular momentum content and longitudinal intensity distribution of the beam. We provide with a closed expression for the orbital angular momentum content of a general superposition of vortex beams and find its relation to the functional parameters of the beams. We compare our theoretical predictions with experimental results with excellent fidelity.

_{2}laser resonator composed by an axicon and a plane mirror. Several low order even and odd Mathieu-Gauss modes were obtained by slightly breaking the symmetry of the cavity. Intracavity elements were used for obtaining odd Mathieu modes. The nondiffracting nature of the Mathieu-Gauss modes is measured indirectly by recording their annular spectra at the focal plane of a converging external lens. The experimental results are corroborated with theoretical predictions.

*z*plane. We derived the adjoint operator and the adjoint eigenfunctions. Each family of generalized Gaussian beams forms a complete biorthonormal set with their adjoint eigenfunctions, therefore, any paraxial field can be described as a superposition of a generalized family with the appropriate weighting and phase factors. Each family of generalized Gaussian beams includes the standard and elegant corresponding families as particular cases when the parameters of the generalized families are chosen properly. The generalized Hermite Gaussian and Laguerre Gaussian beams correspond to limiting cases of the generalized Ince Gaussian beams when the ellipticity parameter of the latter tends to infinity or to zero, respectively. The expansion formulas among the three generalized families and their Fourier transforms are also presented.

*m*, the OAM per photon per unit length can be tuned continuously from (formula available in manuscript). We also observe the experimental propagation of the beam and compare the propagation parameters to our theoretical predictions.

*x*axis. With this formulation we experimentally obtain the spectrum of an elliptic contour in a circular geometry, thus acquiring non-diffracting beam characteristics. Additionally we include the generalization to N-dimensional Dirac delta curves.

**21**, 4 (1996)], for boundary-less one-dimensional beam propagation, to two-dimensional optical wave-fronts. With this formulation the arbitrary choice of physical window size is avoided by mapping the infinite transverse dimensions into a finite-size domain with an appropriate change of variables, thus avoiding the energy loss through the artificial physical boundary that is usually required for the absorbing or the transparent boundary approach. Comparison of analytical solution of propagating wave-fronts and those obtained with the proposed algorithm is given, a discussion of the method advantages and limitations is also provided.

_{2}lasers are significantly affected by the discharge current, as was reported by Witteman [IEEE J. Quantum Electron.

**QE-4**, 786-8 (1968)]. He found that in a sealed laser, with a stable resonator, a spatial mode switching is observed upon increasing the current; due to a modification in the radial profile of the small signal gain. Through an atypical gain profile the lowest-loss bare cavity mode, usually dominant in laser dynamics, may have lower net cavity gain than a mode with higher diffraction losses. Through this work a dynamic differential equation for the homogeneously saturating gain is included in the original dynamic coupled modes method [Appl. Opt.

**29**, 3905-15 (1990)] and applied to a CO

_{2}unstable resonator, with suitable high current small signal gain profiles. By expanding the gain loaded cavity field into the bare cavity oscillation eigenstates, this new model provides a realistic temporal evolution of mode competition, output power and gain saturation within the resonator. We have found that although unstable resonators have excellent transverse mode discrimination the spatial mode switching may also occur, resulting in a significant modification in the output intensity profile. Thus, under certain design parameters, the common assumption of the small signal gain to be constant through the lasing medium may incur in serious inaccuracies for determining the transverse intensity profile and output power. The application of the method is fully described, and the results and their connection to relevant physical properties of gas lasers are discussed.

_{4}laser and its pump beam configuration we were able to generate single high order Ince Gaussian modes with very high quality. The observed transverse modes and nodal patterns have the proposed elliptic structure and exhibit remarkable agreement with the theoretical predictions.

_{2}laser resonator. The cavity is composed by a plane output mirror and a total reflective axicon, this configuration had been studied previously by Gutierrez-Vega et al [J.Opt.Soc.Am.A

**20**, 2113-22 (2003)]. Bessel-Gauss beams are produced directly from the cavity. The use of a reflective axicon instead of a refractive one results in reduction of surface-induced aberrations, minimizing absorption and increasing the non-diffracting distance. This results in a higher power non-diffracting laser beam with potential scientific and industrial applications. In order to characterize the resonator, we have obtained its output transverse intensity distribution. Additionally, we have numerically and experimentally studied the effects of mirror tilt on the output transverse mode structure. We have made numerical simulations of the misaligned resonator modes based on Bowers’s method [Appl. Opt.

**31**, 1185-98 (1992)]. Direct comparison of numerical and experimental results allow us to estimate the diffractive losses of the modes on the misaligned cavity and their dependence on the aligned bare cavity eigenmodes, thus providing valuable information of the output power dependence on mirror misalignment. Relevant experimental parameters and numerical procedure are fully described.

_{2}unstable confocal resonator is fully described, results and their connection to relevant physical properties of gas lasers, such as spiking and relaxation oscillations are discussed. Results of the numerical implementation of the DCM method with dynamic gain are in very good agreement with experimental measurements reported previously.

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