Simple formula for the Jones product and the Pancharatnam connection in optics

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Department of Physics


An inner (Hermitian) product of two polarization states describes the interference of polarized beams, detection and production of polarized radiation, entanglement of polarized photons, etc. Here we pose the question of expressing the inner product γu1†u2 of two Jones vectors (polarization states u1 and u2) explicitly in terms of ellipticities (ϵ1,ϵ2) and tilts (τ1,τ2) of the associated polarization ellipses. To that end, we derive a remarkably simple equation, γ=cos(τ2-τ1)cos(ϵ2-ϵ1)+isin(τ2-τ1)sin(ϵ2+ϵ1). The Poincaré sphere interpretation in terms of distance between polarization states is given, and the Pancharatnam phase is set to arg(γ) to show its invariance under the parity operation.

Publication Title

Physical Review A