  The linear canonical transform (LCT) was extended to complex-valued
parameters, called complex LCT, to describe the complex amplitude propagation
through lossy or lossless optical systems. Bargmann transform is a special case
of the complex LCT. In this paper, we normalize the Bargmann transform such
that it can be bounded near infinity. We derive the relationships of the
normalized Bargmann transform to Gabor transform, Hermite-Gaussian functions,
gyrator transform, and 2D nonseparable LCT. Several kinds of fast and accurate
computational methods of the normalized Bargmann transform and its inverse are
proposed based on these relationships.
