Q4Microwave Theory And Techniques

Setup: a directional coupler is a 4-port device with an input port (1), a through port (2), a coupled port (3), and an isolated port (4). For an ideal coupler, three physical conditions must hold: (i) matched at every port (Sii=0 for all i), (ii) lossless (the S-matrix is unitary, S†S=I), and (iii) reciprocal (S is symmetric, Sij=Sji). For a symmetric directional coupler with the conventional port-numbering convention, the isolated port has zero coupling from the input (S14=0 in one common numbering, or equivalently the isolation condition S12=S34=... depending on labelling) — the exact zero-entries pattern follows from the device's physical symmetry.

Applying unitarity to a symmetric, matched, reciprocal coupler, the general S-matrix reduces to only two independent nonzero magnitudes, the through coupling |S12| and the cross coupling |S13| (using the convention where port 4 is isolated from port 1), related by the fundamental power-conservation constraint:

3 dB (equal-split) condition: a 3 dB coupler is specifically designed so that exactly half the input power emerges from each of the through and coupled ports, i.e. |S12|²=|S13|²=1/2, giving |S12|=|S13|=1/√2. Unitarity additionally requires a specific 90° relative phase between the through and coupled outputs for the matrix to remain unitary (this quadrature-phase condition is what distinguishes a true 3 dB hybrid coupler, such as a branch-line or Lange coupler, from other power-splitting arrangements) — conventionally written with the through path real and the coupled path imaginary:

Verification: each row/column has magnitude-squared sum (1/2)+(1/2)=1 (power conservation, unitarity satisfied), the matrix is symmetric (S=Sᵀ, confirming reciprocity), the diagonal is all zero (confirming each port is matched), and port 1's row has a zero exactly at the position of the isolated port (4 in this labelling), confirming the required port-to-port isolation of an ideal directional coupler. This is the standard reference S-matrix used to characterize any real 3 dB coupler (branch-line, Lange, or 3 dB directional coupler) against ideal behaviour when evaluating measured coupler performance.