Start with the state
x is the input vector encoded onto the optical modes. Each entry is a complex amplitude z = Re + j·Im. Its magnitude |z| describes relative signal strength, its phase arg(z) is the angle in the complex plane, and |z|² is a relative power-like quantity when the selected backend supports that interpretation.
Amplitudes are not watts unless a backend calibration says so. Phase and the α/θ/β controls are in radians; the UI shows the raw complex value, magnitude, phase, and power separately so they are not confused.
What α, θ, and β do
Every programmable cell is a compiler-native 2×2 operator acting only on its two listed modes. α and β are differential phase offsets in the cell’s input/output phase frames. θ is the internal phase control that changes the mixing balance between the two modes. All three are phase settings, measured in radians—not voltages or heater currents.
Photrion evaluates the exact backend operator convention. The MZI drawing is an explanatory, MZI-like realization; it does not claim a particular fabricated geometry or silently rename these controls.
How the matrix is used
The mesh is a sequence of stage operators. At each stage, a⁽ˢ⁺¹⁾ = Uₛ a⁽ˢ⁾. Terminal modal phases then apply a final phase frame, producing y = Ux. In the Operator view, a matrix entry U[i,j] is the complex contribution from input mode j to output mode i; its magnitude and phase are just two views of that same authoritative value.
Why DFT meshes matter
A DFT target is the unitary transform U[k,j] = exp(−j·2πkj/N) / √N. It redistributes a set of input mode amplitudes into frequency-like output modes. The compiler decomposes that target into the available programmable 2×2 cells and terminal phases, then reports the realized operator and its error.
Photonic hardware is attractive for this class of linear algebra because many optical modes can be transformed in parallel. That is a systems opportunity, not a promise that every workload is faster: coupling, detectors, electronics, losses, calibration, and backend evidence still determine the practical result.
Reading the output card
Output y is the vector after every compiled cell and the terminal phase frame. The complex number on each mode is what the backend predicts at that output port. Compare |a| between modes to see how amplitude was redistributed; compare ∠ to see the relative phase that controls later interference. P is the squared magnitude shown as a relative power-like value, not an absolute detector reading unless the backend provides calibration.
Reading the three matrix views
COMPLEX U shows the signed real and imaginary coefficient for every input→output pair. MAGNITUDE |U| keeps only the coefficient size, so it shows how strongly an input contributes to an output. PHASE ∠U keeps only its angle in radians, so it shows the phase offset of that contribution. They are not three different operators—each is a view of the same realized U. Rows are output modes and columns are input modes, so U[i,j] answers “how does input j appear at output i?”