In this example, we study a spherical Fresnel lens. The lens has a radius of curvature of 100cm and a diameter of 4.8cm. Due to the large size of this structure, we must use a 2D approximation of the structure. The focal point of the lens can be studied with FDTD far field projection functions.

## Lens design and setup

We'll consider a Fresnel lens based on a simple spherical design. We assume the radius of curvature of the lens is 100cm and a lens diameter of 4.8cm. The lens is made of a material of index 1.5 and is in air. Ideally, the lens would have a shape defined by

$$ y=\sqrt{R^{2}-x^{2}} $$

In our simple Fresnel design, we assume that we can create a discontinuity in the surface of the lens when y changes by more than λ_{0}/(n2-n1). Since we operate at a wavelength of 500nm, n2=1.5 and n1=1, we can create a discontinuity in y when y changes by more than 1 micron.

This can be accomplished in several ways. One method is to create a surface object and define the lens by the following formula

$$ y=\bmod [R(1-\sqrt{1-\frac{x^{2}}{R^{2}}}), \text { 1 micron }] $$

We can choose the units of our surface object equation to be in microns. Therefore the correct formula to use in the custom "equation" field is

mod(1e5*(1-sqrt(1-(u*1e-5)^2)),1)

This object is difficult to visualize in the layout editor because it is on 1 micron in height, and 5 cm wide. However, we will verify using an index monitor that it is correct.

## Results

The structure is defined in the fsp file fresnel1.fsp. After running the file, the script file fresnel1.lsf can be run and will produce the following results.

The index monitor image showing the shape of the Fresnel lens is shown below. Please note that for better viewing we have resized the figure window and zoomed in.

The electric field intensity. Note the sharp lines due to discontinuities in the lens

The phase of the electric field, in degrees. When we look near a region with a discontinuity in the lens we see an additional feature in the phase, as shown below.

The script then does a near to far field projection to calculate the focal length. We do this projection in air, which will take into account the reflection and refraction that occurs at a flat glass-air interface at the back of the lens. We predict that the the focal length should be approximately R/(n2-n1) = 200 mm.

The projection does a low resolution calculation over a range of values of x and y to create the following figure. Please note that the calculation takes several minutes because there are such a large amount of near field data. We see that the focal plane is indeed at approximately -200mm, as predicted. We confirm this by plotting the E field intensity (|E|^{2}) along the x = 0 line. This shows the following result and the peak intensity is at -200mm.

We then perform a high resolution projection at y=-200mm in order to plot the field at the focal plane. We see a highly focussed spot and we can zoom into the center to see that the spotsize is approximately 20 microns.

The results presented above are for TM polarization. The polarization dependence could be studied by repeating the simulation with TE polarization.

While this 2D example will not reproduce exactly the results expected for a 3D Fresnel lens, it can help identify the origins of different features in a real lens, and suggest possible design improvements for a 3D lens.