Use complex projections and test the order of optical elements

1. Establish the optical mechanism

The introductory Jones extension represents a fully polarized monochromatic field by two complex components. The original normalized input is E=(1,exp(iδ))/sqrt2, so |Ex|²+|Ey|²=1. Relative phase changes the polarization state even though each component intensity remains one half.

An ideal linear polarizer at angle θ transmits the component along the real unit vector u=(cosθ,sinθ). Its matrix is Pθ=u uᵀ. Thus the transmitted complex amplitude along u is (cosθ+sinθ exp(iδ))/sqrt2, and intensity is [1+sin(2θ)cosδ]/2.

A following horizontal polarizer multiplies that intensity by cos²θ. In the reverse order, the first horizontal polarizer retains intensity one half and removes the original y component; the subsequent angled polarizer gives cos²θ/2. These are generally different because P0Pθ and PθP0 need not commute.

For δ=90°, the ideal input is circularly polarized in the stated phase convention and every linear analyzer transmits half the intensity. That special equality does not make the two matrices commute for arbitrary inputs. A single deterministic Jones vector cannot represent unpolarized light; partially polarized ensembles require a different statistical description. No physical polarizer or light-source procedure is required.

2. Work the original evidence

For δ=90° and θ=45°, the first analyzer transmits 0.5 and the second gives 0.25. Reversing the order also gives 0.25 for this particular input. With δ=0° at the same angle, forward order gives 0.5 while reversed order still gives 0.25.

3. Predict and investigate

Does equal output for one input prove that two optical transformations commute?

Carry real and imaginary components through each projection, compare both orders and explain why a changed relative phase alters the result despite unchanged component intensities.

Primary mechanism context. Checked 31 October 2026. Original prose, records and diagrams; no copied source exercises or physical light-source operation. Retain first reasoning and public-feedback exposure in the course evidence workspace when registered.