The ellipsoidal coordinate system has the interesting property that every other orthogonal coordinate system in which the three-dimensional Helmholtz equation is separable, is a special case of it. In this work, we explore the solutions to the wave equation in ellipsoidal coordinates in order to visualize the behavior of optical fields with ellipsoidal geometry. We show several parity properties which allow us to create fundamental modes of vibration with different symmetries around the (x; y), (x; z) and (y; z) planes. We discuss the resonant modes of an ellipsoidal cavity and the traveling waves with ellipsoidal geometry. We propose a method to calculate the second linearly independent solution to the ellipsoidal wave equation
We study the propagating and shaping characteristics of the novel one-dimensional Cartesian Parabolic-Gaussian
beams. The transverse profile is described by the parabolic cylinder functions and are apodized by a Gaussian
envelope. Their physical properties are studied in detail by finding the 2<i>n</i>-order intensity moments of the beam.
Propagation through complex ABCD optical systems, normalization factor, beam width, the quality <i>M</i><sup>2</sup> 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.
We study the propagating and shaping characteristics of the novel Whittaker-Gaussian beams (WGB). The transverse profile is described by the Whittaker functions. Their physical characteristics are studied in detail by finding the 2n-order intensity moments of the beam. Propagation through complex ABCD optical systems, normalization factor, beamwidth, the quality M<sup>2</sup> 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.
We review the three types of laser beams - Hermite-Gaussian (HG), Laguerre-Gaussian (LG) and the newly discovered Ince-Gaussian (IG) beams. We discuss the helical forms of the LG and IG beams that consist of linear combinations of the even and odd solutions and form a number of vortices that are useful for optical trapping applications. We discuss how to generate these beams by encoding the desired amplitude and phase onto a single parallel-aligned liquid crystal display (LCD). We introduce a novel interference technique where we generate both the object and reference beams using a single LCD and show the vortex interference patterns.
The vector Mathieu-Gauss beams of integer order are examined as the solutions of the vector paraxial wave equation in elliptical coordinates. The propagation of the vector components and the three-dimensional intensity distribution of focused vector Mathieu-Gauss beams are analyzed for a variety of polarizations. Conditions in which the linearly polarized Mathieu-Gauss beams can be approximated by the scalar solutions of the paraxial wave equation are also discussed.
In this work we present a detailed analysis of the tree families of generalized Gaussian beams, which are the generalized Hermite, Laguerre, and Ince Gaussian beams. The generalized Gaussian beams are not the solution of a Hermitian operator at an arbitrary <i>z</i> 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.
We present the experimental generation and characterization of each one of the four fundamental families of Helmholtz-Gauss beams: cosine-Gauss beams, stationary and helical Mathieu-Gauss beams, stationary and traveling parabolic-Gauss beams, and Bessel-Gauss beams. Both the transverse intensity profile and power spectrum that each one of the beams exhibits upon propagation is observed and compared to the theoretical model with good quantitative agreement. Emphasis is made on the fact that each of the four families of HzG beams is complete and orthogonal, and thus of fundamental relevance.
We study the Ince-Gaussian series representation of the two-dimensional fractional Fourier transform in elliptic coordinates. A physical interpretation is provided in terms of field propagation in quadratic graded index media. The kernel of the new series representation is expressed in terms of Ince-Gaussian functions. The equivalence between the Hermite-Gaussian, Laguerre-Gaussian, and Ince-Gaussian series representations is verified by establishing the relation between the three definitions.
The term Helmholtz-Gauss beam refers to a field whose disturbance at the plane z =0 reduces to the product of the transverse field of an arbitrary nondiffracting beam (i.e. a solution of the two-dimensional Helmholtz equation) and a two-dimensional Gaussian function. In this work, the transverse shape and the propagation of Helmholtz-Gauss beams is experimentally studied for the four fundamental orthogonal families of Helmholtz-Gauss beams: cosine-Gauss beams, Bessel-Gauss beams, stationary and helical Mathieu-Gauss beams, and stationary and traveling parabolic-Gauss beams. The power spectrum of the Helmholtz-Gauss beams is also recorded and its intensity distribution is assessed. Potential applications are discussed.
Ince-Gaussian modes form a complete family of exact and orthogonal solutions of the paraxial wave equation for elliptical coordinates. The transverse distribution of these fields is described by the Ince polynomials and have an inherent elliptical symmetry. These modes constitute a smooth transition from Hermite-Gaussian modes to Laguerre-Gaussian modes. We report the experimental observation of Ince-Gaussian modes directly generated in a stable resonator. By slightly breaking the symmetry of the cavity of a diode pumped Nd:YVO<sub>4</sub> 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.
Propagation of light beams with apparent nondiffracting properties have intrigued the scientific community since they were introduced. In this talk we will introduce the fundamentals of nondiffracting beams and discuss the dynamics of optical vortices embedded in the new two families of nondiffracting beams we have recently discovered, Mathieu and parabolic beams.
Recently, a new class of nondiffracting beams has been demonstrated theoretically. Namely, Parabolic nondiffracting optical wavefields constitute the last member of the family of fundamental nondiffracting wavefields. Additionally, the existence of a new class of parabolic traveling waves associated to these wavefields has been demonstrated along the same lines. We have succeeded in demonstrating experimentally the fundamental odd and even parabolic wavefields in the laboratory. In this work, we present and discuss the experimental generation of higher-order parabolic nondiffracting wavefields. Because these fields show a complex structure, their generation relies in the successful construction of the field.