Extended Data Fig. 9: Optical and quantum-mechanical analogies to the existence of two-period trajectoids.

a, Illustration of the Bloch sphere representation of a single qubit. The state represented by a red circle on the sphere rotates (orange arrow) around an instantaneous axis \({\bf{n}}(t)\), which is defined by the driving field. b,c, Field pulse equivalent to a single period \(T\) (b, solid curves), its envelope (b dashed curves) and phase shift (c) as functions of time. Shown are two possible analogies to varying the radius \(r\) of the rolling sphere in case of a given field pulse (black): either scaling the applied pulse’s magnitude (green) or stretching the pulse’s functions (envelope and phase shift) in time (blue). d, Illustration of the Poincaré sphere representation of polarization state of light. Squares show polarization states corresponding to respective black points: two circular polarizations at the poles and four linear polarizations at the equatorial plane. e–g, Given almost any (i.e. those having Property \({\boldsymbol{\pi }}\)) sequence of waveplates (dark blue in e), their thicknesses can be scaled (f) by such a factor \(1/r\) that the doubled sequence (g) has no net effect on polarization state of light (yellow helices) passing through it. In this example, curved green arrow shows left-handed circular polarization, curved orange arrows show right-handed elliptic (f) and right-handed circular (g) polarization. See also Supplementary Videos 5, 6.