Luminosity of angled collision of strongly focused beams with different Gaussian distributions

The geometric luminosity of angled collision of strongly focused beams with different Gaussian distributions or asymmetric beams is discussed generally on the basis of pseudo-isotropic beam model. The discussion is simplified by converting the angled collision to the head-on collision in a moving frame. There is a transition angle of collision, above which the hourglass effect becomes less effective. The resultant reduction factor of luminosity can be applied to any case of angled collision of asymmetric beams. Numerical examples are given on the angled collision of laser and electron beams for γ-ray generation, and on the electron and positron beam collision in storage rings.