I am trying to determine whether it is possible to use a reversed lens to defocus a collimated beam into a ball lens in order to increase the BFL of the ball lens. What formulas would I need to use to calculate the lens required and its position?

You can use a negative focal length lens (Negative Achromatic Lens, Plano-Concave Lens, or Double Concave Lens) in combination with a Ball Lens to increase the back focal length of the ball lens. The focal length of the negative lens would have to be smaller in absolute value than the positive focal length of the ball lens in order to extend the back focal length. Another method to increase back focal length is to simply use a ball lens with as small an index of refraction as you can find. Our N-BK7 ball lenses have the smallest index and generally the longest focal lengths of any of our ball lenses. View Understanding Ball Lenses for additional information and useful equations.