# Output Intensity Profile of Multimode Fiber

I am trying to model light that (1) exits a multimode fiber (2) passes through a cylindrical lens system and (3) enters a multimode fiber (and yes I know that a cylindrical lens system is not ideal for fiber coupling).

The radius of curvature for each of the cylindrical lenses in my system is known and fixed, so I know that on one axis the light converges and on the other the light diverges. For this reason I am thinking that the best position for the collection fiber will not be at the focus of the converging axis. Instead I suspect that the optimal location for the collection fiber will be at some point in front of the focus.

I have been trying to evaluate the intensity profile of the light as it exits the lens system. My thought is to place the fiber where the intensity is maximum. I have the transfer matrix for the lens system. If the output from a multimode fiber is Gaussian, then I know how to use the transfer function to get the E-field on the other side of the lens system. From the E-field I can get the intensity profile. The problem is that I am not sure if the output of a multimode fiber is Gaussian, and I have not been able to find whether it is or is not in any of my books or from several hours a google searching.

So, is the output of a multimode fiber Gaussian? Whether yes or no, please provide a reference if you can.

Here is why I do not think that it is.

the light exiting a fiber should obey NA = n*sin(B) (where NA = numerical aperture, n is the index of the medium outside the fiber, and B is the acceptance angle/ exit angle)

so for example with NA=0.22, n=1, B would be 12.71 degrees
this means that if the fiber core is 50 μm then the spot size 1 mm away from the fiber output should be 250 μm.

If I treat the output as Gaussian by letting the beam waist (w0) be the radius of the core then I can use wz = w0*((1+(z/zR)^2)^.5) and zR = (pi*(w0^2))/λ. Using one micron for λ at 1 mm away I get wz = 28 μm.

Obviously there is a big difference between these two models (assuming I did the math correctly). Please show me the steps if I did it wrong.

If the output from a multimode fiber is not Gaussian, then can someone tell me how to model the intensity profile (1) as the light exits the fiber, and (2) after passing though a lens system?

If someone can think of a better way to find the optimal location for the collection fiber, I would love to hear it.

## Answers and Replies

Claude Bile
Science Advisor
If I treat the output as Gaussian by letting the beam waist (w0) be the radius of the core then I can use wz = w0*((1+(z/zR)^2)^.5) and zR = (pi*(w0^2))/λ. Using one micron for λ at 1 mm away I get wz = 28 μm.

This has to be wrong because, in the case where there is not focusing optic (like the output of a fibre), the beam waist cannot possibly reduce. The diameter of the fibre output must be the minimum diameter. Note than wz is a function of z, exactly what value of z are you evaluating wz at to get a value of 28 microns?

In any case, the output of a multimode fibre is most definitely not Gaussian. Look up Optical Waveguide Theory by Snyder and Love for your reference.

Claude.

This has to be wrong because, in the case where there is not focusing optic (like the output of a fibre), the beam waist cannot possibly reduce. The diameter of the fibre output must be the minimum diameter. Note than wz is a function of z, exactly what value of z are you evaluating wz at to get a value of 28 microns?

In any case, the output of a multimode fibre is most definitely not Gaussian. Look up Optical Waveguide Theory by Snyder and Love for your reference.

Claude.

In the definition that I am using the beam waist (w0) is the radius of the fiber core not the diameter, so 28 μm is an acceptable wz for z=1mm.

Any thoughts on how I can model the intensity profile as the light exits the fiber, and after passing though a lens system, or a better way to find the optimal location for the collection fiber?