  We calculate the operating parameters of a transition edge sensor that is
mounted on a thin dielectric membrane with the assumption that the phononic
heat transport in the membrane is ballistic. Our treatment uses the correct
phonon modes from elasticity theory (Lamb-modes), and spans the transition from
3D to 2D behavior. The phonon cooling power and conductance have a global
minimum as function of membrane thickness, which leads to an optimal value for
the membrane thickness with respect to noise equivalent power at a fixed
operating temperature. The energy resolution of a calorimeter will not be
affected strongly, but, somewhat counterintuitively, the effective time
constant can be reduced by decreasing the membrane thickness in the 2D limit.
