Transcript Propagation in the near field: Super resolving pupils

Istituto di Fisica Applicata “Nello Carrara” IFAC – CNR, Firenze
http://www.ifac.cnr.it
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 .
1.2
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.
1
0.8
0.6
0.4
0.2
0
-6
-4
-2
0
2
4
6
 (cm)
(a)
1.2
1
0.8
0.6
0.4
0.2
0
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.
-6
-4
-2
0
2
4
(cm)
6
(b)
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.
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.
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*A. Ranfagni, D. Mugnai,
and R. Ruggeri, Phys. Lett, A 311, 77 (2003) ; J. Appl. Phys., in press.