The Lagrange invariant can be defined as a product of numerical aperture and diameter of the optical fiber. With a lens system that re-images the light from one fiber end to the other, one can change the light cone converging toward the other fiber (NA) and create different spot size.
However, the product of NA and spot diameter or Lagrange invariant remains the same as its name implies.
When coupling multimode fibers with different diameter and/or numerical aperture (NA) the following situations can be encountered:
-Input fiber with smaller diameter and smaller NA couples well with output optical fiber with larger diameter and NA. Common connectors like FC/PC, SMA and others can be used. However, in the opposite direction the connection is very lossy.
-The coupling between optical fibers of similar values of Lagrange invariant but different NA and core diameters cannot be butt-coupled without introducing a significant loss in both directions. However, their coupling can be effectively optimized using a lens system with the appropriate magnification. Then, the connection has a high transmission in both directions. For the lab situations where different types of fibers have to be connected, these NA converters can be effective match makers. They can also be used to increase the filling of the fiber's NA resulting in larger illuminating cone on the other end of the fiber.