  Inverse transformation optics is introduced, and used to calculate the
reflection at the boundary of a transformation medium under consideration. The
transformation medium for a practical device is obtained from a two-dimensional
(2D) finite embedded coordinate transformation (FECT) which is discontinuous at
the boundary. For an electromagnetic excitation of particular polarization,
many pairs of original medium (in a virtual space V') and inverse
transformation can give exactly the same anisotropic medium through the
conventional procedure of transformation optics. Non-uniqueness of these pairs
is then exploited for the analysis and calculation of the boundary reflection.
The reflection at the boundary of the anisotropic FECT medium (associated with
the corresponding vacuum virtual space V) is converted to the simple reflection
between two isotropic media in virtual space V' by a selected inverse
transformation continuous at the boundary. A reflectionless condition for the
boundary of the FECT medium is found as a special case. The theory is verified
numerically with the finite element method.
