In the context of the paraxial regime, usually valid for optical frequencies and also in the microwave spectrum of guided waves, the propagation of electromagnetic fields can be analyzed through a paraxial wave equation, which is analogous to the nonrelativistic Schrödinger equation of quantum mechanics but replacing time t with spatial coordinate z. Considering that, here it is shown that for lossless media in optical frequencies it is possible to construct a Lagrangian operator with an one-to-one correspondence with nonrelativistic quantum mechanics, which allows someone to use the same mathematical methods and techniques for solving problems. To demonstrate that, we explore a few applications in optics with increasing levels of complexity. In the spirit of a Hamiltonian formulation, the ray-tracing trajectories of geometric optics in paraxial regime are obtained in a clear manner. Following that, the gauge symmetries of the optical-field Lagrangian density is discussed in a detailed way, leading to the general form of the interaction Hamiltonian. Through the use of perturbation theory, we discuss a classical analog for a quantum NOT gate, making use of mode coupling in an isotropic chiral medium. At last, we explore the optical spin Hall effect and its possible applications using an effective geometric optics equation derived from an interaction Hamiltonian for the optical fields. We also predict within the framework of paraxial optics a spin Hall effect of light induced by gravitational fields.

O objetivo deste trabalhoé apresentar os conceitos básicos relacionados aos fenômenos de difração e dispersão, permitindo ao leitor perceber as diferenças fundamentais entre os mesmos. Utilizam-se como ferramentas matemáticas fundamentais as noções gerais da equação de ondas, as equações de Maxwell para ondas planas uniformes e a transformada de Fourier para mostrar que a difração está associadaà superposição de ondas monocromáticas com vetores de onda em diferentes direções, tratando-se de um fenômeno espacial, enquanto a dispersão correspondeà superposição de ondas de diferentes frequências e tem portanto um caráter temporal. Embora aênfase seja dada para as ondas eletromagnéticas, os principais resultados podem ser prontamente generalizados para ondas de qualquer natureza (som, ondas em fluídos ou ondas de matéria no caso da mecânica quântica) Palavras-chave: difração, dispersão, ondas eletromagnéticas.The aim of the present work is to present the basic concepts related to the phenomena of diffraction and dispersion, allowing an understanding of fundamental differences between them. To do this, it was used as the main mathematical tools the general notions of wave equation, Maxwell's equations for uniform plane waves and Fourier transforms, showing that diffraction is a consequence of superposition of monochromatic waves with wave-vectors in distinct directions, being in essence a spatial phenomenon, while dispersion corresponds to the sum of waves having different frequencies producing a phenomenon with an essential temporal character. Despite the fact that we emphasized electromagnetic waves the main results can be promptly generalized to waves of any nature (sound, waves in fluids or matter waves in quantum mechanics)

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A similarity transformation is an equivalence relation between square matrices which preserves determinant, trace and eigenvalues, playing a key role in quantum mechanics in simplifying complex hamiltonian systems and improving analytical results attainable from the use of perturbation theory. As a prototypical example, the conventional BCS theory of superconductivity is usually derived from a similarity transformation of the original electron-phonon hamiltonian, written in second quantized version. Here we discuss the general method for writing the similarity transformation operator in second quantized form, allowing one to recast a hamiltonian describing an interacting fermion-boson system into an effective theory in which only the desired degrees of freedom are kept after the transformation.

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