It looks like there is a big discontinuity in fiber sizes moving from fiber 1 to fiber 2 with core size of 4um to 3um, respectively. In such cases, fundamental mode in the first fiber will be coupled to higher order modes in the fiber 2. This should explain most of your questions. Here are my response to your queries:
1. How to interpret the increase in the value of |E|^2 after the fiber joint? and 3. How to visualize this increase in |E|^2 near the core of the fiber in a line plot? And what causes this? Is this the standing pattern due to the waves reflected at the output facet of the fiber? How to avoid it in simulations?
This can be due to three reasons: a) higher order modes might be scattering of the device b) you are visualizing total electric field which has higher order modes with complicated profiles c) modes are more confined in the smaller fiber.
Also, note that you are using only 6 mode in each EME section/fiber, which might not be sufficient enough to completely capture coupling to all the higher order modes. I recommend you to use a higher number of modes to propagate.
For example, fundamental mode in fiber 1 is coupled to fundamental mode and fifth mode in fiber 2 (you can check this by visualizing internal s matrix from EME result view):
and then clicking on view data in the visualizer:
You can learn more about the EME tool by checking the link below:
2.The value of transmission (S21)^2 is close to the expected analytical value although with some amount of discrepancy. Is there a way to improve accuracy of the result?
Yes, a) by increasing number of modes b) performing 3D simulations as in current 2D simulations the spherical shape of fibers are not considered. Software assumes that your geometry is identical in the z-direction (an infinitely wide waveguide).
4.Can the same configuration be modelled using EME 3D prop mode? How to extract the transmission in that case?
I am not quite sure if I understood this question, but you can use 3D simulations and then sum over S1j^2 (assuming mode 1 in fiber 1 and all the modes in fiber 2 denoted by j).
Also, I recommend you to use a larger EME simulation region and use metal for y-PML boundary conditions.
Please keep me update with your thoughts. It is also a good idea to discuss one question to make sure that it is answered properly.