  Based on operations prescribed under the paradigm of Complex Transformation
Optics (CTO) [1-5], it was recently shown in [5] that a complex source point
(CSP) can be mimicked by a parity-time ($\mathcal{PT}$) transformation media.
Such coordinate transformation has a mirror symmetry for the imaginary part,
and results in a balanced loss/gain metamaterial slab. A CSP produces a
Gaussian beam and, consequently, a point source placed at the center of such
metamaterial slab produces a Gaussian beam propagating away from the slab.
Here, we extend the CTO analysis to non-symmetric complex coordinate
transformations as put forth in [6] and verify that, by using simply a
(homogeneous) doubly anisotropic gain-media metamaterial slab, one can still
mimic a CSP and produce Gaussian beam. In addition, we show that a
Gaussian-like beams can be produced by point sources placed {\it outside} the
slab as well [6]. By making use of the extra degrees of freedom (real and
imaginary part of the coordinate transformation) provided by CTO, the near-zero
requirement on the real part of the resulting constitutive parameters can be
relaxed to facilitate potential realization of Gaussian-like beams. We
illustrate how beam properties such as peak amplitude and waist location can be
controlled by a proper choice of (complex-valued) CTO Jacobian elements. In
particular, the beam waist location may be moved bidirectionally by allowing
for negative entries in the Jacobian (equivalent to inducing negative
refraction effects). These results are then interpreted in light of the ensuing
CSP location.
