Can fractal-like spectra be experimentally observed in aperiodic superlattices?

We numerically investigate the effects of inhomogeneities in the energy spectrum of aperiodic semiconductor superlattices, focusing our attention on Thue-Morse and Fibonacci sequences. In the absence of disorder, the corresponding electronic spectra are self-similar. The presence of a certain degree of randomness, due to imperfections occurring during the growth processes, gives rise to a progressive loss of quantum coherence, smearing out the finer details of the energy spectra predicted for perfect aperiodic superlattices and spurring the onset of electron localization. However, depending on the degree of disorder introduced, a critical size for the system exists, below which peculiar transport properties, related to the pre-fractal nature of the energy spectrum, may be measured.

Far field of binary phase gratings with errors in the height of the strips

Diffraction gratings are not always ideal but, due to the fabrication process, several errors can be produced. In this work we show that when the strips of a binary phase diffraction grating present certain randomness in their height, the intensity of the diffraction orders varies with respect to that obtained with a perfect grating. To show this, we perform an analysis of the mutual coherence function and then, the intensity distribution at the far field is obtained. In addition to the far field diffraction orders, a "halo" that surrounds the diffraction order is found, which is due to the randomness of the strips height.

Binary gratings with random heights

We analyze the far-field intensity distribution of binary phase gratings whose strips present certain randomness in their height. A statistical analysis based on the mutual coherence function is done in the plane just after the grating. Then, the mutual coherence function is propagated to the far field and the intensity distribution is obtained. Generally, the intensity of the diffraction orders decreases in comparison to that of the ideal perfect grating. Several important limit cases, such as low- and high-randomness perturbed gratings, are analyzed. In the high-randomness limit, the phase grating is equivalent to an amplitude grating plus a “halo.” Although these structures are not purely periodic, they behave approximately as a diffraction grating.

Diffraction by random Ronchi gratings

In this work, we obtain analytical expressions for the near-and far-field diffraction of random Ronchi diffraction gratings where the slits of the grating are randomly displaced around their periodical positions. We theoretically show that the effect of randomness in the position of the slits of the grating produces a decrease of the contrast and even disappearance of the self-images for high randomness level at the near field. On the other hand, it cancels high-order harmonics in far field, resulting in only a few central diffraction orders. Numerical simulations by means of the Rayleigh–Sommerfeld diffraction formula are performed in order to corroborate the analytical results. These results are of interest for industrial and technological applications where manufacture errors need to be considered.