Diffractive optical elements
(DOEs) have found wide applications such as laser beam focusing, communication,
and others, and they are being considered as potential solutions to a number
of optical design problem that are difficult or impossible to solve with
the conventional refractive and reflective elements.
In the last decades, the scalar
diffraction theory, based on a simplification of Maxwell's equations, were
usually developed to model the diffractive optical elements. However, the
polarization between the vector components of electromagnetic field is
neglected. While the size of diffractive optical elements is comparable
to the wavelength of illumination, scalar diffractive theory is not valid
and more rigorous models of diffraction must be used.
In theory, Maxwell's equation can determine exactly the diffracted field of any diffractive structure. In practice, it is not possible to obtain exactly solutions for the majority of cases. Thus, the solutions of Maxwell's equations have to be calculated numerically. Various approaches are developed to solve the electromagnetic equations.
A semi-infinite finite-difference time-domain (SIFDTD) method is presented and applied to the analysis of diffractive elements. The method discretizes Maxwell's equations with a second order central difference scheme, with electric and magnetic being interleaved in time and space on staggered grid. The electric and magnetic field components are successively updated by repeated implementation of the finite difference equivalents of curl equations. The scheme is an explicit scheme and no matrix inversion is required.
Here, several applications of FDTD-based rigorious numerical electromagnetic method to analysis of the diffracted optical elements are presented, including two dimensional lenses and beam spliitter, axially symmetric lens, and three dimensional lenses.
2D lenses
1. Offaxis lens
Two dimensional propagation plot of the electric field amplitude of an
8-level lens with a focal length of 100um and an aperture of 80um for an
incident plane wave having TE polarization and an off-axis angle
of 30 .
2. beamsplitter
Two dimensional propagation plot of the electric field amplitude of Analysis of an 8-level 1-to-2 beam splitter for a normally incident, TE polarized, plane wave with a focal length of 100um and an aperture of 59.92um. The splitter angles are designed to be -30 and 30(deg.).
3. Animation
This lens is excited by an plane wave with TM polarization. The focal length is 10um with aperture of 13um. After a short period of time, the source is turned off. We can see exactly what happen when a light hits on a lens!!
Three dimensional figure and two dimensional image of the electric field amplitude on the focal plane of an 8-level lens are shown below. This big lens has a focal length of 500um and an aperture of 504um.
3D lenses
Two dimensional image of the electric field amplitude on the focal plane of an 8-level lens are shown below. Its cross sectional distribution is given in another figure.
