Good day,

I am trying to understand exactly what you are trying to do here. From my understanding, you are trying to couple light from a single-mode fiber to a multimode fiber through some sort of optical system (for instance, a lens)? If that is the case, another important question to ask is: do you need to operate only in the fundamental mode of your MMF or you don't mind have a mixture of modes? 

Another way to ask this question is: do you need the system to work in both directions? i.e. from SMF to MMF and back again?

Generally, coupling efficiency in multimode regime will be driven by what is called the brightness ratio, which is analogous to the beam parameter product in laser science. For a multimode fiber, this quantity is roughly given by the product of numerical aperture and core size:

BMM=RcoreNAcore

with Rcore being the core radius and NAcore is the core numerical aperture. For a step-index single mode fiber, this quantity can be approximated as:

BSMF=U/k0 with k0 being the wavenumber in vacuum and U being the solution of the waveguide eigenvalue equation (see Optical Waveguide Theory, Snyder and Love, 1983).  From there, the maximum achievable coupling in MM regime can be derived from the ratio Ca-b=(Bb/Ba)2 with "a" being the input fiber and "b" the output fiber. Usually, going from SMF to MMF using the definitions here will yield CSMF-MMF>1, which means that all the power will be coupled but there will be a loss in brightness so that the action will be irreversible as the coupling efficiency on the way back would be CMMF-SMF=1/CSMF-MMF

Thank you for the response. I am doing a multimode interference-based fiber optic displacement sensor experimental setup. Here I need to calculate the length of MMF that needed to be spliced to SMF (SMF/MMF interface) . The MMF length is important to be calculated because if length of MMF less than reimaging distance, it will ensures max coupling to the output fiber.

As a rule coupling from SMF to MMF doesn't present a serious problem. Both the NA and the light accepting area of a MMF are larger than those of a SMF.

For further details see Chapter 11 of our book "Fundamentals of Optical Fiber Communications" by Wim van Etten and Jan van der Plaats, Prentice Hall 1991.