page_title: Characteristic mode analysis
section_title: Definition
hierarchical_section_title: Characteristic mode analysis / Definition
caption_attribution_description: English: Original structure to be decomposed into characteristic modes.
context_page_description: Characteristic modes form a set of functions which, under specific boundary conditions, diagonalizes operator relating field and induced sources. Under certain conditions, the set of the CM is unique and complete and thereby capable of describing the behavior of a studied object in full.
context_section_description: For the simplicity, only the original form of the CM — formulated for perfectly electrically conducting (PEC) bodies in free space — will be treated in this article. The electromagnetic quantities will solely be represented as Fourier's images in frequency domain. Lorenz's gauge is used.
  The scattering of an electromagnetic wave on a PEC body is represented via a boundary condition on the PEC body, namely
  with  representing unitary normal to the PEC surface,  representing incident electric field intensity, and  representing scattered electric field intensity defined as
  with  being imaginary unit,  being angular frequency,  being vector potential
   being vacuum permeability,  being scalar potential
   being vacuum permittivity,  being scalar Green's function
  and  being wavenumber. The integro-differential operator  is the one to be diagonalized via characteristic modes.
  The governing equation of the CM decomposition is
  with  and  being real and imaginary parts of impedance operator, respectively:  The operator,  is defined by
  The outcome of (1) is a set of characteristic modes , , accompanied by associated characteristic numbers . Clearly, (1) is a generalized eigenvalue problem, which, however, cannot be analytically solved (except for a few canonical bodies). Therefore, the numerical solution described in the following paragraph is commonly employed.
caption: Example of a scatterer 
    
      
        
          Ω
        
      
      {\displaystyle \Omega }
    
   composed of a perfect electric conductor.
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