Magic-tee (hybrid-T) — ideal scattering matrix (S) Select the correct 4×4 ideal scattering matrix for a magic-tee with ports 1 and 2 as collinear arms, port 3 as the E-arm (difference), and port 4 as the H-arm (sum).

Introduction / Context:A magic-tee is a four-port hybrid junction combining an H-plane tee (sum) and an E-plane tee (difference). In the ideal, lossless, perfectly matched case, its scattering matrix encodes equal-magnitude 3 dB splits, specific phase relationships, and isolation between E and H arms.

Given Data / Assumptions:

• Port numbering: 1 and 2 collinear; 3 = E-arm; 4 = H-arm.
• Ideal matching → all diagonal Sii = 0.
• Lossless, symmetric splits → nonzero off-diagonals have magnitude 1/√2.

Concept / Approach:Exciting the H-arm produces equal in-phase outputs at ports 1 and 2 and zero at the E-arm; exciting the E-arm produces equal and opposite outputs at ports 1 and 2 and zero at the H-arm. These constraints uniquely set the sign pattern of the S-matrix given the port order.

Step-by-Step Solution:

Verification / Alternative check:Inject at port 4: outputs (1,2) = (+1/√2, +1/√2); at port 3 = 0. Inject at port 3: outputs (1,2) = (+1/√2, −1/√2); at port 4 = 0. Reciprocity and symmetry are satisfied.

Why Other Options Are Wrong:

• B/C/D have incorrect sign patterns or violate E/H isolation or symmetry conditions.

Common Pitfalls:Confusing port numbering conventions; missing the minus sign on the difference port; forgetting the 1/√2 magnitudes.

Final Answer:S = (1/√2) * [[0, 0, 1, 1], [0, 0, -1, 1], [ 1, -1, 0, 0], [ 1, 1, 0, 0]]