# Spatial distribution for intensity

The paper Evanescently coupled multimode spiral spectrometer describes a novel approach to designing a spectrometer based on a waveguide. When I approached the authors with a question about the theory behind it I got back a somewhat cryptic response that I don't fully understand. Can anyone explain the following in mathematical terms i.e. explain what equations are being used.

In particular, what does "project" mean in:

Then you project that input field onto the modes of the waveguide.

The full text of the reply I got is:

To get the spatial distribution, you need to assume some spatial distribution of the input field at the input facet of the multimode waveguide. Then you project that input field onto the modes of the waveguide. Next, you use the ODE to find how each waveguide mode will propagate in the spiral, as explained in the supplementary information: "We modeled the wave propagation in the spiral waveguide using the approach in [36, 37]. First we ignored the mixing of waveguide modes with different orders, which allowed us to simulate individual modes separately by solving the following two coupled equations:" (If you don't have Comsol you will need to somehow figure out how to estimate the propagation constants and coupling coefficients for each waveguide mode) After you solve the ODE for each waveguide mode, you know the superposition of the modes at the output of the waveguides. That will give you the spatial distribution at the output.