There are ways that EM waves can be used to reconstruct a representation of matter and these techniques vary widely based on the method. Each method is very unique and its own set of issues. UV-D's reference Measurements Using Optic and RF waves provides a good overview. It explains some of the applications, the sources for their EM waves and problems/solutions for different techniques.
In the broader picture there are several issues with imaging in general that apply to the use of EM waves. Usually this is simplified by discussing the field characteristics of where in the field the object is located from the source of the EM emission.
Measuring in the far-field: Your resolution is limited by the diffraction limit or Abbe Diffraction limit. This is roughly 1/2 of the operating wavelength. For far-field measurements higher frequencies are used for higher resolution. Radio telescopes are good examples of this application. But in this case you're only "imaging" the energy transmitted in that band, which often looks very different than the same structure in the visible spectrum. For example planets wouldn't be visible to a radio telescope unless they are reflecting radio energy or emitting it. Radar uses the timing and strength of the reflection to determine and objects distance and the angular resolution of the rotating antenna to determine the size (impulse response & angle of arrival). Things that reflect optical waves may also not reflect radar EM waves (microwaves, typically). So a stealth boat might be very visible but on radar a complete blank.
Measuring in the near-field: The diffraction limit doesn't apply. Near-fields are dominated by dipole-type electric or magnetic fields. Magnetic near-field components due to changing currents of a dipole nature. In contrast to the far-field, the diffraction pattern in the near-field typically differs significantly from that observed at infinity and varies with distance from the source. In the near-field, the relationship between E and B becomes very complex Techniques such as total internal reflectance microscopy and metamaterials-based super lens can image with resolution better than the diffraction limit by locating the objective lens extremely close (typically hundreds of nanometers) to the object. However, because these techniques cannot image beyond 1 wavelength, they cannot be used to image into objects thicker than 1 wavelength which limits their applicability.
So depending on the type of imagining you're interested in that will help you focus on the techniques involved. Since you've mentioned metamaterials and your bounty comments, then I have to assume you're interested in < 1 wavelength techniques.
Near-field EM is a complex beast. The "near-field" is a region in which there are strong inductive and capacitive effects from the currents and charges in the antenna that cause electromagnetic components that do not behave like far-field radiation. These effects decrease in power far more quickly with distance than do the far-field radiation effects.
Part of the near-field closest to the antenna (called the "reactive near-field") absorption of electromagnetic power in the region by a second device has effects that feed-back to the transmitter, increasing the load on the transmitter that feeds the antenna by decreasing the antenna impedance that the transmitter "sees". Thus, the transmitter can sense that power has been absorbed from the closest near-field zone, but if this power is not absorbed by another antenna, the transmitter does not supply as much power to the antenna, nor does it draw as much from its own power supply.
In the reactive near-field (very close to the antenna), the relationship between the strengths of the E and B fields is often too complex to predict. Either field component may dominate at one point, and the opposite relationship dominate at a point only a short distance away. This makes finding the true power density in this region problematic. To calculate power, not only E and B both have to be measured but the phase relationship between E and B as well as the angle between the two vectors must also be known in every point of space. Measurement of this energy can be difficult. MRI machines are classic example of near-field imaging systems which preform this task remarkably well.
The other complex issue relates to the frequency in use. Different materials respond differently at different frequencies. This means that each technique has a limited frequency range. Even though the operating principles might be the same for all this methods, the technology tends to be completely different. A scientist working with an MRI imaging would not know how to make THz imaging work because the body of knowledge is very different even though the EM principles are the same.
Metamaterials (and ref here too and their journal) are used for changing waves (even sound waves and seismic waves) in a way that was previously not thought possible. They fall into 6 main categories:
- Negative index materials: In 1968 Victor Veselago published a paper theorizing plane wave propagation in a material whose permitivity and permeability were assumed to be simultaneously negative. In such a material, he showed that the phase velocity would be anti-parallel to the direction of Poynting vector.
- Single negative metamaterials: In single negative (SNG) metamaterials either relative permittivity (εr) or relative permeability (µr) are negative, but not both.
- Electromagnetic bandgap metamaterials: Electromagnetic bandgap metamaterials control the propagation of light. This is accomplished with either a class of metamaterial: photonic crystals (PC), or another class known as left-handed materials (LHM) Both are a novel class of artificially engineered structure, and both control and manipulate the propagation of electromagnetic waves.
- Double positive medium: Double positive mediums (DPS) do occur in nature such as naturally occurring dielectrics. Permittivity and magnetic permeability are both positive and wave propagation is in the forward direction.
- Bi-isotropic and bianisotropic metamaterials: in many examples of electromagnetic metamaterials, the electric field causes magnetic polarization, and the magnetic field induces an electrical polarization, i.e., magnetoelectric coupling. Such media are denoted as being bi-isotropic. Media which exhibit magneto-electric coupling, and which are also anisotropic (which is the case for many commonly used metamaterial structures), are referred to as bi-anisotropic.
- Chiral metamaterials: Wave propagation properties in chiral metamaterials demonstrate that negative refraction can be realized in chiral metamaterials with a strong chirality, with neither negative ε nor μ as a requirement.
Starting to learn it from scratch? Focus on EM and Maxwell's equations and techniques for solving, measuring, verifying them. Learn some optics techniques along the way in addition to signal processing techniques (statistics, signals & systems) pick a type of imaging that interests you then focus on their techniques and materials. You're asking a very broad question so I hope I've somehow covered what you're interested in knowing.
Edit (I hadn't noticed you edited your question): Specifically with any kind of antenna array (optical or RF) the amplitude and phase of the incoming signal is measured. Then using a highly accurate timing signal this information can be combined so that it is like looking through two positions separated in space at the exact same time. In addition these signals can also be manipulated in real-time to recombine signals using delay or phase shift mechanisms. Due to the nature of radio waves and the components related to sensing phase and amplitude we've had this technology since 1950's but we've expanded it greatly in the past 20 years. And only recently have we been able to do this optically as well since the 1990's.
The principles are the same in each case, but due to how matter responds to different wavelengths the technology to accomplish interferometry is very different at optical frequencies vs. radio frequencies. There has been a wide body of work involved in expanding these concepts to include non-linear deconvolution that allows use of images from different locations with poor baseline correlations as well as heterodyne synthetic apertures.
Take a long look at the variety of Interferometer techniques, astronomical interferometers, and radio interferometers.