Zonal Phase Factor
(aka. Zonal Lens, Lens Function, Quadratic Phase, Radial Chirp, etc.)

The button labeled "zonal lens" duplicates action as the menu item (goodies) "Zonal Phase Factor" and has the keyboard equivalent (-Z). This opens a window which (outside of a script) is usually left open for repeated use.

The initial values in the text fields may be made default by setting them in the Preferences window (found on the CoherentOptics menu). I usually set the focal length to the "Magic Distance" (described below) since it gives the shortest focus without aliasing.

These settings of the window will result in the display bellow.

This is a complex array generated from the following formula.

Where the lambda is the wave length, "z" is the distance (radius of curvature of the wave front), delta sub zero is the spacing between pixels of your spacial light modulator, and (j,k) are the coordinates of the pixel. The offset variables δx & δy move the center to a new location. These values do not need to be integers.

In the example shown the numbers have been chosen so that the Niquist limit is just reached at the edges of the image. The condition that is met for this to happen is that where N is the size of the array (number of pixels in each row). The distance satisfying this condition is sometimes called the "magic distance" or "canonical distance".

The image can be shifted horizontally and vertically by entering values for the δx and δy offset.

Like several other functions this one may be used to modify rather than replace the existing contents of the complex array. The "multiply" check box will cause a complex multiplication of the contents by this function.

The complex conjugation check box results in a converging lens (at least within this program). NOTE: The equations on this page show converging lenses as set up in the computer code.

There are times when you want to calculate a lens using a deferent formula. If you check the "FT dual lens " box the lens will be calculated using the following formula.

This formula is useful for fast calculation of the propagation of light by a fast Fresnel Transform. Notice that the offsets no longer move the lens center to their values. Instead they produce a grating like phase pattern which is the Fourier transform of a dot at the location of the offsets. The center is move elsewhere.

If you are doing a fast Fresnel Transform with small values of Δ on the order of a wave length or smaller you will probably want to check the "non-paraxial" box. In that case the following formula will be used.

A short description of the fast Fresnel Transform is that it permits the rapid calculation of the propagation of light from one plane to another. It does so by first taking the Inverse Fast Fourier Transform of the starting image, multiplying the result by one of the two formulas above, and then doing a final Fast Fourier Transform.

Use In a Script 

When the icon is used in a script the actions are much the same but the fields are changeable by variable if you wish.

 

The batch button permits a sequence of entries to be read in from a text file. All of the above fields may be filled from the records in this file. Read more here

The three additional fields and check boxes are so you can fill the step, wave length, and pixel pitch text fields from a table of values kept in a seperat icon . The words "step", "wl", and "del" are symbols which were used to insert the values to their left. You can read more about this in the "script flow" page.