  The deformation method of transformation optics has been demonstrated to be a
useful tool, especially in designing arbitrary and nonsingular transformation
materials. Recently, there are emerging demands for isotropic material
parameters, arising from the broadband requirement of the designed devices. In
this work, the deformation method is further developed to design
quasi-isotropic/isotropic transformation materials. The variational functional
of the inverse Laplace's equation is investigated and found to involve the
smooth and quasi-conformal nature of coordinate transformation. Together with
the sliding boundary conditions, the inverse Laplace's equation can be utilized
to give transformations which are conformal or quasi-conformal, depending on
functionalities of interest. Examples of designing an arbitrary carpet cloak
and a waveguide with arbitrary cross sections are given to validate the
proposed idea. Compared with other quasi-conformal methods based on grid
generation tools, the proposed method unifies the design and validation of
transformation devices, and thus is much convenient.
