This experiment aims to simulate and analyze the intensity behaviour of beams, within the near field (Fresnel region), of selected apertures.
propTF vs. propIR
Propagation of beams, with different shapes, were studied by simulation. Two different types of propagator functions were used, i.e, the Fresnel transfer function (TF) propagator, and the impulse response (IR) propagator. Voelz shows that the two methods are conceptually the same, both of them uses the Fresenel diffraction equation, however, results alter once they are applied computationally due to technical limitations. propIR often leaves computational artifacts, but it is useful for simulating propagation over large distances [1].
Methodology
Square, circular, Gaussian, single-slit, and double-slit apertures were synthesized (see the first two images (a & b) of each Figure). Beams having light source wavelength 
Square Beam
Fig. 1 shows the results obtained from using a square beam. Notice the diffraction intensity pattern at the observation plane. Line scan at the cross section will give Fig. 1e or 1f. The three peaks are consistent with the diffraction pattern it corresponds to. Outputs from are two methods look relatively similar.
Figure 1: Square beam propagation simulation.
Circular Beam
Figure 2, on the other hand shows what will happen if the beam passes through a circular aperture. The irradiance has three peaks, similar to square beam, except for the towering height of the middle peak above the rest. This dominance of the middle peak is consistent with the irradiance of the middle region of the diffraction pattern (Fig. 2c & 2d). Again, no signicant differences were seen between propTF and propIR.
Figure 2: Circular beam propagation simulation.
Gaussian Beam
The two previous apertures are binary, but what if the intensity at the source plane is radially decreasing. Fig. 3a-b 
Figure 3: Gaussian beam propagation simulation.
Single-Slit Beam
The last two physical phenomena were obtained from slits. For a single slit in the Fresnel approximation, irradiance pattern at the observation plane is just similar to a circular aperture, except that it is longitudinal. The corresponding irradiance profile shows peaks that are close to each other, as if they have merged. At near zero irradiance, sinusoid function is still evident.
Figure 4: Single slit – beam propagation simulation.
Double-Slit Beam
Double slit gives similar result except that we have two towering intensities with the same height. Irradiance at the middle region is increased due to an occurring constructive superposition between the two beams.
Figure 5: Double slit – beam propagation simulation.
Where is the difference between propTF and propIR.
Unfortunately, results from Figures 1-5 doesn’t support our claim that the two propagation methods should have different outputs, somehow. Maybe our propagation distance is not enough, or maybe those artifacts that we are trying to look for cannot be observed from the intensity spatial/intensity domain. Maybe it’s in the phase. Further studies may opt phase retrieval and comprehensive theoretical grounds regarding intensity and phase.
Reference:
[1] D. Voelz. Computational Fourier Optics: A MATLABR Tutorial. Chapter 5. SPIE Press. Washington, USA. 2011.
Hinumduman 