## 1 Introduction  
In recent years, several physical platforms based on metamaterial aperture antennas (MAAs) have been used to realize computational imaging, as shown in Figure 1. In general, all natural scenes can be compressed on some basis. Scenes can be perfectly reconstructed with significantly fewer measurement modes than the space bandwidth product (SBP) [1; 2; 3]. These measurement modes are composed of radiated field patterns of the metamaterial aperture antenna (MAA) at different frequencies. For image reconstruction schemes, which use an arbitrary set of measurement modes, it is essential that the modes are as orthogonal to each other as possible [4] so that the correlation of the modes or field patterns is small [5; 6; 7]. Since the measurement modes are indexed by frequency, the low correlation of field patterns can be interpreted as far-field patterns with strong frequency diversity. Hence, to obtain a metamaterial aperture that generates the frequency-agile far-field patterns that are as orthogonal as possible, elements distributed on the aperture must have a strong resonance with a high \(Q\)-value. Other approaches to frequency-diverse imaging have also been pursued, including multiply scattering structures, such as mode-mixing cavities and dynamic metamaterial apertures [8; 9; 10]. Fractal models have been used to design fractal antennas with very special properties: about one-tenth of a wavelength and a pre-fractal geometrical configuration [11] could be used in metamaterial miniaturized technology, which is needed for coherent imaging. The fractional signal, especially its geometrical interpretation, which gives a powerful mathematical tool to model the most advanced concepts of modern physics, could help us to build a simpler mathematical model of the relationship between the metamaterial aperture antenna and its radiation feature, which will improve the speed of full-wave simulation [12]. The initial metamaterial aperture which consists of numerous subwavelength-resonant radiators present a trade-off between the \(Q\) factor and radiation efficiency (each resonator distributed on aperture is assigned a resonant frequency randomly selected from the bandwidth of operation) [13; 14], and most of the power is radiated by the elements closest to the feed, resulting in poor aperture efficiency.  
In this paper, the correlation of far-field patterns for metamaterial apertures with different modulations is analyzed, and a trade-off strategy between orthogonality of the far-field patterns and radiation efficiency of the MAA is proposed. The metamaterial aperture is composed of complementary electric-field-coupled elements (CELCs) [15] and complementary Jerusalem cross (CJC) [16] unit cells. The CELC element is strongly resonant with frequency diversity, while the CJC unit cell has high radiation efficiency across the frequency band. By analyzing the filling ratio of the two different elements on the MAA, a relatively low correlation of the far-field patterns and high radiation efficiency is obtained. MAAs with a 20 \(\times\) 20 randomly distributed elements array are theoretically analyzed in the frequency band from 33 to 37 GHz. Numerical simulations show that the radiation efficiency of 60% can be achieved, which yields improved SNRs when using the complementary electric-field-coupled (CELC) element and the CJC element with a random distribution ratio of 4 to 1 on the aperture. A prototype of the MAA with 120 \(\times\) 120 randomly distributed elements whose total size is 250 mm \(\times\) 250 mm \(\times\) 0.5 mm is fabricated to verify the effectiveness of the design and analysis procedure, and to show that the measured far-field patterns across the frequency band from 33 to 37 GHz could be used to effectively reconstruct the target scene with scatterers.