Polarization Relation Cross-Scans

projection-axis-three-row-scan

A three-row scan varies the projection axis angle while holding the rest fixed, showing how the cos^2(theta) factor and transmitted intensity change row by row. The arithmetic is exact only for the stated rows and axis angles in the ideal polarization model. Real beams add finite bandwidth, imperfect polarization, alignment error, detector calibration, and measurement uncertainty.

Aligned, slanted, and crossed axes show the passed field component as a component ledger. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Aligned axis passes the whole field component

With axis factor 1, the incoming 10 volts per metre remains 10 volts per metre along the polarizer axis.

E∥=10 V/m⋅1=10 V/mE_{\parallel}=10\ \text{V}/\text{m}\cdot1=10\ \text{V}/\text{m}
Projection scan rowThe aligned row keeps the full axis field.incidentField=10 V/maxisMode=alignedparallelFactor=1parallelField=10 V/mperpendicularField=0 V/macceptedBit=1 bit

A three-four-five axis passes six volts per metre

The middle row uses projection factor 3/5 and passes 6 volts per metre from the same incident field.

E∥=10 V/m⋅35=6 V/mE_{\parallel}=10\ \text{V}/\text{m}\cdot\frac{3}{5}=6\ \text{V}/\text{m}
Projection scan rowThe slanted row exposes the component split.incidentField=10 V/maxisMode=three-four-fiveparallelFactor=3/5parallelField=6 V/mperpendicularField=8 V/macceptedBit=1 bit

Crossed axis passes zero field component

Crossed orientation changes the projection factor to 0, so the axis component is 0 volts per metre.

E∥=10 V/m⋅0=0 V/mE_{\parallel}=10\ \text{V}/\text{m}\cdot0=0\ \text{V}/\text{m}
Projection scan rowThe crossed row makes the zero component explicit.incidentField=10 V/maxisMode=crossedparallelFactor=0parallelField=0 V/mperpendicularField=10 V/macceptedBit=1 bit