Coupled Mode Theory: Info on Cylindrical Waveguides

[Claude Bile]

I need information on coupled mode theory, specifically with reference to cylindrical waveguides. The more info the better. I have some references, but they are quite scattered, I am yet to find a reference that really encapsulates the 'heart' of coupled mode theory.

If anyone has a good reference (web based or otherwise), on coupled mode theory, I would be greatful.

If somebody that is familiar with Coupled Mode Theory has some time on their hands, you are welcome to post info on the theory itself, however I gather that the theory is rather complex.

Regards, Claude.

 

[pervect]

I need information on coupled mode theory, specifically with reference to cylindrical waveguides. The more info the better. I have some references, but they are quite scattered, I am yet to find a reference that really encapsulates the 'heart' of coupled mode theory.

If anyone has a good reference (web based or otherwise), on coupled mode theory, I would be greatful.

If somebody that is familiar with Coupled Mode Theory has some time on their hands, you are welcome to post info on the theory itself, however I gather that the theory is rather complex.

Regards, Claude.

I never heard of coupled mode theory. I can talk a little bit about the behavior of weakly and strongly coupled resonant transformers, which are coupled resonators, though they are lumped systems. This makes them easier to analyze, too.

For the strongly coupled resonant transformer, the voltage ratio depends on the turns ratio of the primary and secondary. For a weakly coupled case (like a typicial Tesla coil), the voltage ratio depends much more critically on the tuning of the primary & secondary than on the turns ratio.

More insight into the resonant transformer by writing down the differential equations

For the coupled inductors let the voltages across the primary and secondary be V1 and V2, the currents into the primary and secondary be I1 and I2. Orient the currents as follows.

V1-----I1----> <-----I2------V2

(I tried to draw a better pciture, couldnt' figure out how to turn off formatting)

For the coupled inductors we can write

$$\begin{array}{l} V_1 = L_1 \frac {dI_1}{dt} + M \frac {dI_2}{dt}\\ V_2 = M \frac {dI_1}{dt} + L_2 \frac {dI_2}{dt} \end{array}$$

To complete the circuit, you need to add a capacitor C₁ across the primary, and a capacitor C₂ across the secondary, and some series resistances R₁ and R₂ in series with the primary and secondary inductors

You can write down the diffeq's with the lapalace transform in terms of the current into the primary and secondary

$$\begin{array}{l} -I_1 / s C1 = L_1 s I_1 + R_1 I_1 + M s I_2 \\ -I_2/ s C2 = M s I_2 + L_2 s I_1 + R_2 I_2 \end{array}$$

The rest is a matter of analyzing the behavior of these diffeq's - you can substitute s=jw in the usual way to get the behavior vs frequency.

Well, there's one more piece of info you need to know

The strongly coupled case is represented by $$M = k \sqrt{L_1 L_2}$$ with k approximately equal to unity. The weakly coupled case is where k is less than unity. Qualitiatively, you should find that for the strongly coupled case, the bandwidth will be very wide, and the voltage will be a function of $$L_2 / L_1$$, which is equivalent to the turns ratio.

The weakly coupled case is much more similar to your coupled resonators. The bandwidth will be much narrower. I don't recall the expression for the voltage ratio offhand though.

I hope this is of some use, and not too far afield.

 

[Claude Bile]

Thankyou for the reply pervect.

Coupled Mode Theory is similar to what you presented, but it applies to optical systems, spcifically how power couples between two modes, whether they be in a single waveguide (say a multimode optic fibre), or modes from two adjacent waveguides (evanescent coupling).

The mathematics is quite similar, for that reason your post is quite informative as to the maths side of it, however most of the physics is encapsulated in the coupling coefficients, specifically, I need to know under what conditions one gets strong coupling to other modes.

Once again pervect, thanks.

Claude.