Luminosity optimization near the beam-beam limit by increasing bunch length or crossing angle

We discuss the choice of bunch length and crossing angle near the beam-beam limit in a storage-ring collider. First, we derive expressions for the tune shifts of either bunched or continuous round beams which are induced by a single collision with arbitrary crossing angle and bunch length and for the associated luminosities. Then, considering two collision points with alternating planes of crossing, we demonstrate that, if the total beam-beam tune shift is held constant, the collider luminosity increases as a function of bunch length and crossing angle. This implies a corresponding increase in the bunch intensity. As an illustration, we present numerical examples for a Large Hadron Collider upgrade and for the Very Large Hadron Collider.