Ideal 3-port circulator — scattering (S) matrix form For an ideal lossless, matched three-port circulator with ports numbered 1→2→3 in the direction of circulation, which S-matrix correctly represents its behavior?

Introduction / Context:
A three-port circulator is a non-reciprocal passive device that routes power directionally: from port 1 to port 2, port 2 to port 3, and port 3 to port 1 (for one chosen sense of rotation). In the ideal, lossless, perfectly matched case, its scattering matrix has zeros on the diagonal (no reflections) and unit magnitude on the single “forward” off-diagonal in each row, encoding the circulation.

Given Data / Assumptions:

• Ideal circulator: lossless, perfectly matched (S₁₁=S₂₂=S₃₃=0), perfectly isolated reverse paths.
• Port order 1→2→3 defines the circulation direction.

Concept / Approach:
By definition, with 1→2→3 circulation: power injected into port 1 exits port 2; into port 2 exits port 3; into port 3 exits port 1. The S-matrix rows describe outputs given an input at each port. Therefore:

which encodes 1→2→3 circulation when read with the standard convention used in this option set.

Step-by-Step Solution:

Verification / Alternative check:
Switching port numbering (clockwise vs counter-clockwise) permutes the columns/rows; the chosen option matches one consistent numbering. Any other arrangement implies different circulation or wrong mapping.

Why Other Options Are Wrong:

• B/D: Represent the opposite circulation or inconsistent mapping relative to the stated 1→2→3 definition.
• C: Places a “1” on S33, implying self-transmission for port 3, which is not a circulator behavior.

Common Pitfalls:
Mismatching the port numbering versus circulation sense; forgetting that ideal ports are matched so the diagonal must be zero.

Final Answer:
[[0, 0, 1], [1, 0, 0], [0, 1, 0]]