1. Super resolving pupils: beyond the diffraction limit * Anedio Ranfagni, Daniela Mugnai, Rocco Ruggeri In an attempt to transfer the results obtained with super-gain antennas to optical systems for increasing the resolving power, an original way for overcoming the diffraction limit was proposed in 1952 by Toraldo di Francia. Recent microwave experiments, performed by using a composed pupils set-up, demonstrate the correctness of these predictions. By employing special kinds of pupils, consisting of a paraffin torus placed in the middle of a metallic frame (the outer pupil), we have a composed (three coronae) system suitable to test the improvement in the resolving power. The presence of the paraffin torus (refractive index n=1.49) assures the exact relationships among phases and amplitudes in the three coronae. In particular, the required phase inversion of the wave was obtained since the thickness of the torus (= 34 mm) was about one wavelength: thus the optical path inside the paraffin is about 1.5, while in the surrounding space and in the central bore it is. In figure to left, the measured amplitude patterns relative to the composed pupil is shown: in the presence of the paraffin torus (full points and small crosses), and to the simple pupil (open circles and small squares). The results show a strong narrowing of the beam, from a width of about 6 cm, in the absence of the ``open pupil", to about 3 cm, when the “open pupil" is present. Moreover, we had also a strong increase in the beam intensity with an amplification of the maximum of about a factor two. The open-end detector has been moved in the near field of the pupil, as sketched in the inset. The data marked with open and full circles were obtained from a series of measurements at 9.30 GHz, while the ones marked by small crosses and small squares were obtained from another series of measurements at 9.27 GHz. In figure to right, the measured amplitude pattern obtained in the far field limit (Fraunhofer optics) is shown in the case of composed pupil (open circles and full line), and for simple pupil with the same outer diameter (full points and dashed line). The far field is simulated by putting a paraffin lens with a focal length of about 15 cm beyond the pupil under test. The detector was moved in the focal plane of a paraffin lens placed beyond the pupil, as sketched in the inset. The frequency was taken at 9.37 GHz. The amplitudes, which are reported in a logarithmic scale, show a narrowing of the central peak (taking, for instance, the width at 3 dB of attenuation) of a factor of about two. This case (far field) should be considered to be a more convincing test of the theory described previously, although these results essentially confirm the prior ones by showing the same narrowing of the central peak of a factor of about two. It seems that the theoretical predictions made by Toraldo di Francia on 1952 have been confirmed, to a large extent, by the present experiments. The improvement demonstrated in the resolving power was only of about a factor of two, and this was due to the simplicity of the system adopted. However, we are confident that this improvement can be greatly increased, provided that the pupil consists of an adequate number of coronae, in order to obtain a resolution power well beyond the diffraction limit. ________________________________________________________________________________________ * A. Ranfagni, D. Mugnai, and R. Ruggeri, Phys. Lett, A 311, 77 (2003) ; J. Appl. Phys., in press. Istituto di Fisica Applicata “Nello Carrara” IFAC – CNR, Firenze http://www.ifac.cnr.it In the two upper figures, we compare the pattern-amplitude obtained experimentally in the two cases with their theoretical curves. The amplitudes acting on the three circular coronae were approximately proportional to 1, -0.631, and 0.234, respectively, as required by the theory. These values were roughly satisfied by the amplitude profile in the absence of the paraffin torus, at the values of r =7.5, 22.5, 37.5 mm, which correspond to the mean radius of each corona. We observed a narrowing of the beam by a factor of more than two and an amplification of the maximum of almost the same amount.