Laboratory measurements of super-resolving Toraldo pupils for radio astronomical applications

Abstract

The concept of super-resolution refers to various methods for improving the angular resolution of an optical imaging system beyond the classical diffraction limit. Although several techniques to narrow the central lobe of the illumination Point Spread Function have been developed in optical microscopy, most of these methods cannot be implemented on astronomical telescopes. A possible exception is represented by the variable transmittance filters, also known as “Toraldo Pupils” (TPs) since they were introduced for the first time by G. Toraldo di Francia in 1952 (Toraldo di Francia, Il Nuovo Cimento (Suppl.) 9, 426, 1952). In the microwave range, the first successful laboratory test of TPs was performed in 2003 (Mugnai et al. Phys. Lett. A 311, 77–81, 2003). These first results suggested that TPs could represent a viable approach to achieve super-resolution in Radio Astronomy. We have therefore started a project devoted to a more exhaustive analysis of TPs, in order to assess their potential usefulness to achieve super-resolution on a radio telescope, as well as to determine their drawbacks. In the present work we report on the results of extensive microwave measurements, using TPs with different geometrical shapes, which confirm the correctness of the first experiments in 2003. We have also extended the original investigation to carry out full-wave electromagnetic numerical simulations and also to perform planar scanning of the near-field and transform the results into the far-field.

Appendix A: Plane-rectangular NF-FF transformation

Among the NF-FF transformation techniques (planar, cylindrical or spherical), the plane-rectangular scanning is the simplest and most efficient from the analytical and computational viewpoints. Such a technique is particularly suitable for highly directive antennas since the pattern can be reconstructed only in a cone with an apex angle less than 180^∘. In the plane-rectangular scanning (Fig. 15), the probe is mounted on a x-y positioner so that it can measure the NF amplitude and phase on a plane-rectangular grid. From these data, measured for two orthogonal orientations (horizontal, H, and vertical, V) of the probe, or equivalently the transmitting feedhorn (by applying a 90^∘ rotation around the longitudinal axis in the second set), and taking into account the probe spatial response, one can compute the antenna FF pattern [7, 19].

Fig. 15

Schematics of the plane-rectangular scanning

Itcan be easily recognized that the NF tangential components of the field (E _x and E _y ) cannot be obtained when performing the measurement by means of a real probe. In fact, the probe sees the center of the diffracting pupil (which constitutes our “antenna under test”, or AUT) from different directions when moving in the scanning plane. Moreover, even at a fixed position the probe sees each point of the AUT from a different angle. As a consequence, the FF of the AUT cannot be accurately recovered from the measured NF data by employing the uncompensated NF-FF transformation. The basic theory of probe compensated NF measurements on a plane as proposed in Refs. [8, 19] is based on the application of the Lorentz reciprocity theorem. The key relations in the reference system used in the present work are:

$$ E_{\theta}(\theta,\phi) = \frac{1}{\Delta} [I_H {E_{\phi}}_V^{\prime}(\theta,-\phi) - I_V {E_{\phi}}_H^{\prime}(\theta,-\phi) ] $$

(2)

$$ E_{\phi}(\theta,\phi) = \frac{1}{\Delta} [I_H {E_{\theta}}_V^{\prime}(\theta,-\phi) - I_V {E_{\theta}}_H^{\prime}(\theta,-\phi) ] $$

(3)

where:

$$ {\Delta} = {E_{\theta}}_H^{\prime}(\theta,-\phi) {E_{\phi}}_V^{\prime}(\theta,-\phi) - {E_{\theta}}_V^{\prime}(\theta,-\phi) {E_{\phi}}_H^{\prime}(\theta,-\phi) $$

(4)

and

$$\begin{array}{@{}rcl@{}} I_{V,H} &=& A \cos\theta \, e^{j \beta d \cos\theta} \times \\ && {\int}_{-\infty}^{\infty} {\int}_{-\infty}^{\infty} V_{V,H}(x,y) \, e^{j \beta x \sin\theta \cos\phi } \, e^{j \beta y \sin\theta \sin\phi } \mathrm{d}x \mathrm{d}y \end{array} $$

(5)

where A is a constant and β is the free-space wavenumber. Namely, the antenna FF is related to: (i) the 2D Fourier transforms I _V and I _H of the output voltages V _V and V _H of the probe for the two independent sets of measurements; and (ii) the FF components ${E_{\theta }}_{V}^{\prime }$, ${E_{\phi }}_{V}^{\prime }$ and ${E_{\theta }}_{H}^{\prime }$, ${E_{\phi }}_{H}^{\prime }$ radiated by the probe and the rotated probe, respectively, when used as transmitting antennas. According to Ref. [25], the FF components of the electric field, ${E_{\theta }}_{V}^{\prime }$, ${E_{\phi }}_{V}^{\prime }$, radiated by an open-ended rectangular waveguide (of sizes a ^′ and b ^′ along the x and y axis, respectively) excited by a TE ₁₀ mode are:

$$ {E_{\theta}}_V^{\prime} = f_{\theta}(\theta; a^{\prime},b^{\prime}) \, \sin\phi \frac{e^{-j\beta r} }{r} $$

(6)

$$ {E_{\phi}}_V^{\prime} = f_{\phi}(\theta; a^{\prime},b^{\prime}) \, \cos\phi \frac{e^{-j\beta r} }{r} $$

(7)

where the function f _ϕ (𝜃; a ^′,b ^′) is discussed in Ref. [25]. Similar equations can be found for the ${E_{\theta }}_{H}^{\prime }$, ${E_{\phi }}_{H}^{\prime }$ field components.

According to (5), in order to obtain the FF pattern all over the hemisphere in front of the AUT, the measurement plane should be infinite but, of course, this is not possible in a practical setup. The dimension of the plane should be such that the field becomes negligible at its edges, thus, minimizing the error associated with this truncation. Due to this so-called truncation error, the calculated FF using the planar NF data is valid only up to a critical angle 𝜃 _c ≃ 16.7^∘ outward from the aperture of the AUT (see Fig. 16):

Fig. 16

Definition of the critical angle, 𝜃 _c , and its relation to the geometry of the measurement setup. From Fig. 9, D = 9 cm and d = 10 cm

$$ \theta_c = \arctan \left( \frac{L-D} {2d} \right)\!. $$

(8)