VCVS constraint stamp has phase-dependent back-reflection (S11 ≠ 0)

[cdaunt]

Problem

The standard VCVS (voltage-controlled voltage source) stamp used for unidirectional optical components has an implicit phase-dependent input reflection coefficient.

A VCVS with V_out = T · V_in is stamped as:

return {
    "o1": s.i_fwd,      # shared branch current at input
    "o2": -s.i_fwd,     # shared branch current at output
    "i_fwd": V_o2 - T * V_o1,  # constraint
}

Converting to S-parameters, this device has:

S11 = (1 - T) / (1 + T)        S21 = 2T / (1 + T)

For |T| ≈ 1, |S11| = |tan(φ/2)| — the reflection sweeps from 0 to ∞ as phase rotates. This is not physical for a waveguide or phase shifter, which should have S11 = 0 and S21 = T.

Impact

In a thermal MZI circuit (mmi1x2 → heater → mmi2x2 → detectors), the artificial back-reflection from the heater VCVS propagates backward through the mmi1x2's Y-matrix. Because the mmi1x2 is near-lossless, its Y-matrix is ill-conditioned (det(S+I) = 1 - 2t² ≈ 0.067 for 0.3 dB loss), amplifying the reflected current ~15×. This causes:

• Non-physical power oscillation at intermediate nodes (between mmi1x2 output and heater input)
• Node voltage V(W1) depends on heater phase: V(W1) ∝ 1 / (T(I) + T_fixed + 2) instead of being constant
• Visible amplitude modulation on what should be a pure phase rotation

Fix

Replace the shared branch current at the input port with a matched input admittance (Y_in = 1/z₀ = 1):

return {
    "o1": signals.o1,    # matched input: Y_in = 1, gives S11 = 0
    "o2": -s.i_fwd,      # output current from constraint
    "i_fwd": V_o2 - T * V_o1,
}

This decouples the input from the output loading. The KCL at the input node no longer contains T(I), so the node voltage is constant across the sweep. Power conservation holds: input power |V|² equals output power |T|² · |V|².

Scope

This affects any VCVS-based component where T varies during a simulation (e.g., thermal phase shifters, tunable couplers). Components with fixed T are less affected because the reflection coefficient is constant, but still present incorrect S11 values.

The TunableBeamSplitter in circulax/components/photonic.py uses the same shared-branch-current pattern and would have the same issue.

The near-lossless SAX component Y-matrices (via s_to_y) amplify the problem but are not the root cause — the VCVS stamp itself is the source of the artificial reflection.