FRESNEL Quick Guide |
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/cm^{2}. The circle diameter ratio to the calculation grid scale (L_{0}=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 L_{0}>>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} = r^{2}/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/cm^{2} that compared with 1 J/cm^{2} 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.
Z, m |
1-2 |
2-4 |
4-8 |
8 and more |
M |
2 |
4 |
8 |
16 |
Presented example was chosen to demonstrate that:
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.