We use Hamiltonian optics to design and analyze beam propagation in two-dimensional (2D) periodic structures with slowly varying nonuniformities. We extend a conventional Hamiltonian method, adding equations for calculating the width of a beam propagating in such structures, and quantify the range of validity of the extended Hamiltonian equations. For calculating the beam width, the equations are more than 10(3) times faster than finite difference time domain (FDTD) simulations. We perform FDTD simulations of beam propagation in large 2D periodic structures with slowly varying nonuniformities to validate our method. Beam path and beam width calculated using the extended Hamiltonian method show good agreement with FDTD simulations. By contrasting the method with ray tracing of the bundle of rays, we highlight and explain the limitations of the extended Hamiltonian method. Finally, we use a frequency demultiplexing device optimization example to demonstrate the potential applications of the method.
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[Show abstract][Hide abstract] ABSTRACT: We use analytical and numerical techniques to design a cylindrical lens with a gradient index of refraction. In our device, we design the desired index distribution by using a photonic crystal with slowly-varying lattice parameters.