Finding a proper Scattering Matrix for an 3x3 MMI

[aignermax]

Objective

Design a 3x3 Multimode Interference (MMI) device capable of reciprocal electromagnetic field propagation.

Scattering Matrix Requirements

• Energy Conservation: The output vector (outputVector = M * InputVector) should preserve the power of the input vector, particularly when the field elements of the input vector include phase information.
• Symmetry Principle: The scattering matrix should satisfy M * inputVector * inputVector = inputVector, ensuring the device operates symmetrically.

Reciprocal Behavior

• When an electromagnetic field with a specified phase and intensity enters at port A1 and exits at port B1, reversing the MMI (i.e., rotating the device or reversing the input and output) should result in the field entering at B1 and exiting at A1 with identical phase and intensity.
• This requirement emphasizes the device's symmetrical operation, indicating that the optical path within the MMI is reversible.

Cascading MMI Devices

• Cascading two MMI devices should demonstrate the principle of reversibility in electromagnetic propagation within the system. Specifically, the output vector from the second MMI should match the original input vector of the first MMI, thus reverting to the initial state.

here is a paper on nxn MMIs
MMI-Paper nxn MMIs oe-24-20-22481.pdf

[aignermax]

so according to the paper, the formula should be like this:

it turned out, that this formula is also wrong, the q mod 2 should be a 0 (even) and the q+1 mod 2 should be a 1, then it works.
you can fiddle around with the code using this online editor here, that returns the exact same values for q=4 as in the paper.

https://dotnetfiddle.net/mycn0T

[aignermax]

The formula works for a 4x4 MMI -> I have tested the energy conservation using a unittest and also the results are what to be expected) but not for a 3x3 MMI...

[aignermax]

it could be like that:

[aignermax]

evtl this one
(r:0,5774,ϕ: PI0,7854) (r:0,5774,ϕ: PI2,8798) (r:0,5774,ϕ: PI*-0,2618)
(r:0,5774,ϕ: PI2,8798) (r:0,5774,ϕ: PI0,7854) (r:0,5774,ϕ: PI*-0,2618)
(r:0,5774,ϕ: PI*-0,2618) (r:0,5774,ϕ: PI*-0,2618) (r:0,5774,ϕ: PI*0,7854)

[aignermax]

ok fixed it, thanks to @Fixolito and @Kayzaks and all the others to help me work myself through that paper!