We study the effect of elastic anisotropic biaxial strain on the light emitted by neutral excitons confined in different kinds of semiconductor quantum dots (QDs). We find that the light polarization rotates by up to 80 degree and the excitonic fine structure splitting varies by several tens of $mu$eVs as the strain is varied. By means of a continuum model we mainly ascribe the observed effects to substantial changes of the hole wave function. These results show that strain-fields of a few permill magnitude are suffcient to dramatically modify the electronic structure of QDs.
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