  Huygens' principle (HP), extinction theorem (ET), and Franz's /
Franz-Harrington formulation (FHF, which is a mathematical expression of
surface equivalence principle) are the important components of electromagnetic
(EM) theory, and they are generalized from the following aspects.
  1) Traditional HP, ET, and FHF in homogeneous isotropic environment are
generalized to inhomogeneous anisotropic lossy environment. 2) Traditional FHF
for homogeneous isotropic material system is generalized to inhomogeneous
anisotropic lossy material system in this paper, and will be further
generalized to metal-material combined system in future works. 3) The Huygens'
surface in traditional HP and ET is a single closed surface. In this paper, it
is generalized to the "Huygens' surface" which is constructed by multiple
closed surfaces. In future works, it will be further generalized to the
"Huygens' boundary" which includes some lines and open surfaces. 4) For a
material body, traditional FHF has only ability to express the external
scattering field and the internal total field (the summation of scattering and
incident fields) in terms of the equivalent sources on material boundary, and
it is generalized to formulating the internal scattering and incident fields in
this paper.
  In addition, the relationships among HP, ET, and FHF are studied, and it is
proved that HP and ET are equivalent to each other.
