Define the dimensionless energy
\[ x=\frac{E}{V_0}, \]and the dimensionless barrier parameter
\[ \boxed{ \beta= \frac{a\sqrt{2mV_0}}{\hbar} }. \]The graph below covers both the tunnelling region \( E < V_0 \) and the above-barrier region \(E > V_0 \).
The exact transmission coefficient is
\[ T= \left[ 1+ \frac{V_0^2} {4E(V_0-E)} \sinh^2(ka) \right]^{-1}, \] where \[ k= \frac{\sqrt{2m(V_0-E)}}{\hbar}. \]Using \(x=E/V_0\) and \(\beta=a\sqrt{2mV_0}/\hbar\),
\[ ka=\beta\sqrt{1-x}. \]Therefore
\[ \boxed{ T_{\rm exact}(x) = \left[ 1+ \frac{ \sinh^2\!\left( \beta\sqrt{1-x} \right)} {4x(1-x)} \right]^{-1} }, \qquad 0 < x < 1. \]Since probability current is conserved,
\[ \boxed{ R_{\rm exact}=1-T_{\rm exact}. } \]If
\[ ka\gg1, \]then
\[ \boxed{ T_{\rm thick} \approx 16 \frac{E}{V_0} \left( 1-\frac{E}{V_0} \right) \exp\left[ -\frac{2a}{\hbar} \sqrt{2m(V_0-E)} \right]. } \]In dimensionless form,
\[ \boxed{ T_{\rm thick}(x) \approx 16x(1-x) \exp\left[ -2\beta\sqrt{1-x} \right]. } \]The corresponding approximate reflection coefficient is
\[ \boxed{ R_{\rm thick} \approx1-T_{\rm thick}. } \]This approximation requires \[ \beta\sqrt{1-x}=ka\gg1. \] It is therefore not reliable close to \(x=1\). The dashed curve in the graph is included for comparison with the exact result.
Define
\[ q= \frac{\sqrt{2m(E-V_0)}}{\hbar}. \]The exact transmission coefficient is
\[ T= \left[ 1+ \frac{V_0^2} {4E(E-V_0)} \sin^2(qa) \right]^{-1}. \]Since
\[ qa= \beta\sqrt{x-1}, \]we obtain
\[ \boxed{ T_{\rm exact}(x) = \left[ 1+ \frac{ \sin^2\!\left( \beta\sqrt{x-1} \right)} {4x(x-1)} \right]^{-1} }, \qquad x>1. \]Again,
\[ \boxed{ R_{\rm exact}=1-T_{\rm exact}. } \]Perfect transmission occurs whenever
\[ qa=n\pi, \]so that
\[ \boxed{ T=1,\qquad R=0. } \]Both exact formulas have the same finite limiting value:
\[ \boxed{ T(E=V_0) = \frac{1} {1+\beta^2/4} } \] and therefore \[ \boxed{ R(E=V_0) = 1- \frac{1} {1+\beta^2/4}. } \]\[ \beta= \frac{a\sqrt{2mV_0}}{\hbar} \]