How to excite a certain mode in a waveguide?

Does anyone know how we can excite a certain mode, e.g. the second order mode, in a waveguide? I'm a little confused. According to the ray-optics theory, modes are related with incident angles, which means by ensuring a certain range of incident angle we can excite a desired mode. However, based on the wave theory where each mode has different spatial power distribution. Thinking in this way, it seems that what we need is to make sure the shape of the power profile of the incident light matches the profile of the desired mode and also that the light is projected on the right position on the end face of the waveguide. And in this case, the incident angle doesn't matter.

It seems strange that these two theories give different results which don't agree with each other. Can anyone see where the problem is? Thanks!

• see Chapter 10 in "PRINCIPLES OF MICROWAVE CIRCUITS" by Montgomery, Dicke, Purcell Commented Mar 6, 2017 at 17:01

The concept of waveguide modes originates from wave theory and therefore wave optics is the right way to approach this question.

In order to excite a mode in a waveguide, you want to maximize the overlap integral between the mode field and your excitation. In other word, you want to match both the spatial intensity and phase profile of the mode (often also called its complex amplitude). A mismatch in the excitation profile excites other modes as well – or leads to partial reflection.

The connection to ray optics is found via the phase fronts of the wave field. Phase fronts are spatial iso-phase hyperplanes analyzed for a fixed point in time. The rays in the ray optics interpretation are lines normal to the phase fronts.

Higher order modes typically have tilted phase fronts (with respect to the optical axis or the direction of propagation in the waveguide), and thus a corresponding description in ray optics terms gives a mode specific tilted angle of incidence.

In this sense, both theories are consistent, even though ray optics are not a good 'language' to discuss waveguide modes.

• Hi Ed Cetera, thanks for your helpful answer, but I'm still not very clear about some of the points. Could you please explain them a little further? Commented Mar 7, 2017 at 14:40
• Can we still use plane waves to consider this problem? If so, why higher order modes have tilted phase fronts if the plane waves incident in the direction normal to the end face of the waveguide? It seems that the incident angle should be zero in this case. Commented Mar 7, 2017 at 14:49
• You seem to have a very specific configuration in mind but don't share any details. I can merely guess that you want to know how to excite a higher order mode in a waveguide using an incident plane wave?!? As I wrote in my original answer, you always have to match the intensity and phase profile of the target mode in order to excite it. If you have just a plane wave, your input intensity is fixed. Thus you can only change the phase distribution at the waveguide input by tilting the incident plane wave. Choose the tilt that fits the mode profile best (i.e. maximizes the overlap integral). Commented Mar 8, 2017 at 8:38
• Hi, actually what I'm considering is to use a LED or laser to excite a certain mode in a waveguide. I'm not sure whether we should use a plane wave or other modal to represent these light sources when doing the analysis. I would be grateful if you can give me some advice. Also, you mentioned we can change the phase distribution at the waveguide. So what measures do we normally take to achieve this? Is it some kind of like adjusting the incident angle? Thanks! Commented Mar 8, 2017 at 13:48

based on the wave theory where each mode has different spatial power distribution. Thinking in this way, it seems that what we need is to make sure the shape of the power profile of the incident light matches the profile of the desired mode and also that the light is projected on the right position on the end face of the waveguide.

This is how you would excite a specific mode in a waveguide.

It seems strange that these two theories give different results which don't agree with each other.

Remember that ray optics is a simplified model compared to wave optics. It's very common in physics to use a simplified model for many day-to-day analyses, but know that that model is incorrect in certain cases. When we have to analyze those cases, we have to resort to the more complex model.

For example, Newtonian mechanics predicts that if a constant force is applied, a mass can be accelerated indefinitely. Special relativity says that that's not true. We know that Newtonian mechanics is just a simplified subset of special relativity, and when velocities become a substantial fraction of c we have to consider the more complex model given by special (or general) relativity.

To add to Ed Cetera's correct answer: wave optics is always the correct approach, and ray optics works perfectly well when the Eikonal Equation is sound: I explain the conditions for this to be sound in this answer here.

In fiber optics, ray tracing works well when fibers are heavily multimoded, i.e. the $V$ parameter is very large. For a one moded or few moded fiber, ray optics are utterly inapplicable.

See Chapter 10 of Snyder & Love, "Optical Waveguide Theory", Chapman & Hall, 1983