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The bowing of the energy of the three lowest band-to-band transitions in$\beta$-(Al$_{x}$Ga$_{1-x}$)$_2$O$_3$ alloys was resolved using a combineddensity functional theory (DFT) and generalized spectroscopic ellipsometry(GSE) approach. The DFT calculations of the electronic band structure of both,$\beta$-Ga$_2$O$_3$ and $\theta$-Al$_2$O$_3$, allow extracting of the linearportion of the energy shift in the alloys, and provide a method for quantifyingthe role of coherent strain present in the$\beta$-(Al$_{x}$Ga$_{1-x}$)$_2$O$_3$ thin films on (010) $\beta$-Ga$_2$O$_3$substrates. The energies of band-to-band transitions were obtained using thespectroscopic ellipsometry eigenpolarization model approach [A. Mock et al.,Phys. Rev. B 95, 165202 (2017)]. After subtracting the effects of strain whichalso induces additional bowing and after subtraction of the linear portion ofthe energy shift due to alloying, the bowing parameters associated with thethree lowest band-to-band transitions in monoclinic$\beta$-(Al$_{x}$Ga$_{1-x}$)$_2$O$_3$ are found.

Strain and composition dependencies of the near bandgap optical  transitions in monoclinic (Al$_x$Ga$_{1-x}$)$_2$O$_3$ alloys with coherent  biaxial in-plane strain on (010) Ga$_2$O$_3$