3.8. Number of wave guide modes With increasing V parameter (3.18) not only the number of allowed values for Je(X) increases but also the num- ber of roots X < V continues to rise. In order to estimate the number of modes for a given V parameter, begin by assuring yourself that J(x)= (2/mx)1/2 cos(x-(+1/2)/2) is a good approximation for large x. Use this approximation to calculate explicitly the number M of allowed modes and show that it rises with M4V/. Calculate the mode number for a step- index fibre with NA=0.2 and core diameter 2a = 50 m. How does the mode number change if the cladding is replaced with air, i.e. if a pure solid 50 m diameter glass fibre is used?