The high power and high radiation frequency of microwave sources require the use of multi-mode (oversized) electrodynamical systems. The highest-power microwave sources (gyrotrons and generators with relativistic electron beams) operate, as a rule, at very high modes of a circular waveguide. Correspondingly, there arises the problem of conversion of these modes to the waves of the simplest structure, which are convenient for further transportation and application. In a more general approach, this problem can be formulated as a synthesis of three-dimensional waveguiding or antenna systems which convert the initial vector field to the required one.
A significant progress has been achieved at IAP in the development of mode conversion methods (G. G. Denisov, A. V. Chirkov, D. I. Sobolev). In particular, wave converters are capable of transforming such complex modes as ТЕ_25.10, ТЕ_28.12, and ТЕ_34.19 to Gaussian beams (eigenmodes of mirror waveguides) with an efficiency of 95—97%, while ensuring that the structures formed in the waveguide cross sections are optimal for this or that assembly unit, e.g., for the output window of the gyrotron. This made it possible to realize simple and reliable versions of industrial gyrotrons (1 MW/170 GHz gyrotron for ITER, multi-frequency gyrotron for ASDEX-Upgrade), which have record-breaking coefficients of conversion of the operating mode to the output radiation in the form of the Gaussian beam.
Original methods have been developed for designing of multi-mode systems: measurement of spatial structures of wave flows in the thermal-imaging measurements, reconstruction of phase fronts from amplitude distributions, analysis and synthesis of multi-mode systems on the basis of scalar and vector integral equations.
Since transportation of high-power wave beams requires wide-cross-section waveguides (closed oversized or open mirror ones), the necessity arises to develop the matching, controlling, and metering elements for such transmission lines. The list of such elements designed at IAP includes directed quasioptical and waveguide couplers, mirror and waveguide converters of field structures, dividers and summators of wave flows, multiplexors of wideband signals, universal polarizers, controlled switches of wave flows, passive and active compressors of microwave pulses, and absorbing load for megawatt millimeter radiation.

Quasioptical resonance diplexer, which sums radiation from two gyrotrons (~140 GHz/~0.5 MW) and switches the total flow between output channels with a frequency of up to 20 kHz

An important recent result is the method for controlling of a flow of microwave radiation generated by a complex of gyrotrons whose frequencies are controlled by varying voltages (M. I. Petelin et al.). The proposed method allows one to sum up the powers of a set of gyrotrons and scan the total wave flow adaptively (for suppression of hydrodynamic instabilities in plasmas).
Development of new components and optimal diagnostic methods using analysis of the radiation mode content resulted in creation of transmission lines or their components for high-power microwaves in several controlled fusion facilities (T-10, TEXTOR, LHD, Aditia, ASDEX, W7-AS), accelerators (CERN), technological setups for microwave processing of materials and CVD deposition of diamond films, sources of multi-charge ions, radar systems, and sophisticated FEMs.

Example of a difficult problem from electrodynamics of multi-mode systems: synthesis of the quasioptical system of a multifrequency megawatt gyrotron. Scheme and structures of the fields at different points within the electrodynamic system. The three-dimensional surface of the waveguide and mirrors is synthesized using the original method developed at IAP on the basis of calculations of A. Chirkov