GSOLVER makes use of the piecewise constant approximation as the means to define arbitrarily complex grating structures. The concept is familiar from elementary calculus where a smooth function is approximated by a piecewise linear approximation. The examples illustrate several different types of grating structures where this concept is employed.
Simple surface relief gratings such as Blaze, Sinusoidal, Triangular, and Binary (to name a few) are constructed by a thin layer stack approximation. Each layer consist of a certain number of index transitions where the index transition point is placed at the appropriate location so that the stack approximates the surface relief. GSOLVER has several tools to automate this process. By increasing the number of layers any degree of accuracy may be imposed.
In the figure below is a representation of a four level approximation to a sawtooth grating profile. The four layers have equal thickness. Each layer has one index transition per layer.
The following figure is a representation of the third layer (the first layer is on the substrate) of this grating. The properties of the layer include the position of the index transition, thickness, and values of the index in each region.
One way to get started with a complicated grating is to draw the grating profile on a piece of graph paper. Then identify regions of constant index values appropriately. The number of index transitions, layer thickness, and index values are independent from layer to layer.
For crossed gratings each layer in the piecewise constant approximation consist of a 'checkerboard' like pattern. By making the grid fine enough any grating pattern may be built. Often regions of a fine grid in a layer are all assigned the same index value for example a circular 'dimple' would be
where 'squares' inside the circle are assigned one material index, and those outside are assigned the second material index. By applying several layers with differing circle dimensions, a cone, or similar structure may be approximated.
Crossed gratings require significantly greater
computer resources to compute.