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We observed the $A^1\Sigma^+\sim b^3\Pi\to a^3\Sigma^+/X^1\Sigma^+$laser-induced fluorescence (LIF) spectra of the RbCs molecule excited from theground $X^1\Sigma^+$ state by the Ti:Sapphire laser. The LIF radiation from thecommon perturbed levels of the singlet-triplet $A\sim b$ complex was recordedby the Fourier-transform (FT) spectrometer with the instrumental resolution of0.03~cm$^{-1}$. The relative intensity distribution in the rotationallyresolved $A\sim b\to a^3\Sigma^+(v_a)/X^1\Sigma^+(v_X)$ progressions wasmeasured, and their branching ratio was found to be about of 1$\div$5$\times$10$^{-4}$ in the bound region of the $a^3\Sigma^+$ and $X^1\Sigma^+$states. The experiment was complemented with the scalar and full relativisticcalculations of the $A/b - X/a$ transition dipole moments (TDMs) as functionsof internuclear distance. The relative systematic error in the resulting\emph{ab initio} TDM functions evaluated for the strong $A - X$ transition wasestimated as few percent in the energy region, where the experimental LIFintensities are relevant. The relative spectral sensitivity of the FTregistration system, operated with the InGaAs diode detector and CaFbeam-splitter, was calibrated in the range $[6~500,12~000]$~cm$^{-1}$ by acomparison of experimental intensities in the long $A\sim b\to X(v_X)$ LIFprogressions of the K$_2$ and KCs molecules with their theoretical counterpartsevaluated using the \emph{ab initio} $A - X$ TDMs. Both experimental andtheoretical transition probabilities can be employed to improve the stimulatedRaman adiabatic passage process, $a\to A\sim b \to X$, which is exploited for alaser assembling of ultracold RbCs molecules.

The branching ratio of intercombination $A^1\Sigma^+\sim b^3\Pi\to  a^3\Sigma^+/X^1\Sigma^+$ transitions in the RbCs molecule: measurements and  calculations