Example 2. Diffraction

Load diffract.scm and respective input beam given in diffract.pls. Parameters of the distribution can be seen if you double click "Source" and chose "Flat-topped beam". It is the circular beam of the diameter 2r= 1.2 cm with the wavelength 1000 nm. The energy has been chosen to correspond to the flux 1 J/cm². The circle diameter ratio to the calculation grid scale (L₀=1.5 cm) is 0.8 that provides most accurate result for a plane wave diffraction. Do not believe when somebody claims that it is necessary to stipulate large guard zone around the beam (i.e. choose L₀>>2r) to avoid the aliasing. It is much better to use the correct calculation procedure instead. In FRESNEL this is done automatically for a beam propagating in free space or through any centered sequence of convergent or divergent lenses.

For the beam propagation several various distances were chosen that correspond to different Fresnel numbers N_f = r²/lZ: N_f = 6 at Z = 6m; N_f =5 at Z= 7.2m; N_f = 4 at Z = 9m (Target). We remind that the first element distance is measured from Source, while for each following it corresponds to the distance from previous element. Looking through the dialog boxes of the elements used in scheme one can see these distances. Run the program and when it stops after N_f = 6 you will see the beam as a small light spot inside the black square of the calculation grid.
Do not be surprised! Pressing the - button below the beam image you will see that the full scale of the calculation grid is 20.48 cm (the step of sampling points 0.04 cm). Such a scale was chosen automatically by the program to minimize the calculation error. Pressing the energy density button you will return back to 2D picture of the beam. We recommend now to press and to use two-pane Beam Window. If you did so, first, press X-profile button in the left pane, and then press Zoom button 8. The image size is now OK, but the resolution seems to be not sufficient.
Do not hurry in making conclusion. Wait a little bit. Recall several facts that the theory gives for the near field (N_f >1) diffraction onto a circular diaphragm. The number of main maxima should be equal to the Fresnel number (N_f) as we can see in the cross section. The energy density at r= 0 should be equal to null for even N_f. In our case it is approximately 4.6 10^-6 J/cm² that compared with 1 J/cm² for the initial beam gives not bad calculation accuracy. Our next stop is at odd N_f =5 and the theory predicts the energy density 4 times higher than the initial one at r= 0. Leave the Beam Window setting unchanged then on next stop, like it is now, you will see only central part of the calculated field. If you like Zoom can be switched off to observe full image.
Press Run and continue the calculations. In a moment you will get the 2D picture and the plot for the distance 7.2 m (N_f=5). As it should be, the energy density at r = 0 is four times higher than the initial one (with the accuracy 10^-4), and the number of the main maxima is 5.
Press Run again. After calculations you will stop at z= 9m (N_f=4) just before the Target. Do not hurry, please. As we could suspect the energy density for r = 0 is very low again. And as before the spatial resolution is not sufficient to observe the picture in details. As soon as the information concerning the complex electric field is quite sufficient (as for our previous stops at N_f=6 and 5) we can reconstruct the energy density distribution with much better spatial resolution. Before we proceed save current beam distribution as temp1.pls by pressing Save button in the Beam Window.
Next, switch off Zoom and check the parameters set for the Target. You will see that Magnification=8 have been chosen beforehand. This very operation will be performed if you press Run. After the procedure of the 2D Fourier interpolation you can observe the calculation result with 8 times higher resolution both in the plot and in the 2D distribution.
It is worth to compare the output picture with that we have had before magnification. To do this load temp1.pls as input beam. Switching on Zoom=8 you will get for the latter picture the same scale (3.84 cm) as for the output beam. Obviously, the image after the magnification (their profiles are given in the figure below) contains more details indistinguishable without this procedure. How precise is the procedure itself we shall discuss in successive examples.

Target:
Magnification=1, Zoom=8	Magnification=8, Zoom=1

Those who want to evaluate the accuracy of the propagation calculations in diffract.scm are invited to load diffract.pls again. After this press Run options button on the Scheme toolbar and change the "precision" check level to 0.001. Now if you run the scheme, on some steps of calculations you will see Precision warning. Answer each time Yes. The value of the current precision indicates that within the calculated field at least one point has the accuracy of the order of that is shown. Usually, it is worthless to hope that after performing complete calculations (propagation through series of the optical elements) the accuracy is essentially better than maximum value of the current precision evaluated when calculations are in progress. Though there are exemptions. Total accuracy can be sometimes better sometimes worse than indicated. I.e. the current precision value serves just to give an approximate estimation of the accuracy. Nevertheless, in above example of diffract.scm the current precision value corresponds to the calculation accuracy very well. It has been checked by performing direct calculations of the Fresnel-Kirchhoff integral. Thus, all beam distributions are quite adequate to real diffraction on the circular sharp edge diaphragm with the accuracy of about 2-5× 10^-3.
If you wish you can now modify the diffract.scm or create new one. When creating a new scheme you will need to load input beam diffract.pls distribution, arrange propagation distance/distances, and choose Magnification value on Target. It is recommended for diffract.pls to use the Magnification value depending on the distance between the Source and the Target. Usually to observe the distribution with sufficient resolution it is quite enough to have the Magnification = 8.

Presented example was chosen to demonstrate that:

• Program FRESNEL performs automatically the adjustment of the calculation scale to provide highest accuracy.
• User has no need in deciding which propagation method should be applied in each particular case as it is should be done often in other programs.

Above two advantages are very noticeable (in FRESNEL Pro version) when the propagation and diffraction are studied for a complex system of lenses and other optical elements with the arbitrary distances between them.
Program FRESNEL performs the evaluation of the calculation precision for each step of the propagation indicating the degree of confidence to the obtained result.
The Magnification procedure permits to analyze results with required spatial resolution.
Same procedure is applied automatically when the radiation passes through the apertures of the optical elements. In many cases this results in the essential enhancement of the calculations accuracy.