Limits...
Acceleration toward polarization singularity inspired by relativistic E × B drift

View Article: PubMed Central - PubMed

ABSTRACT

The relativistic trajectory of a charged particle driven by the Lorentz force is different from the classical one, by velocity-dependent relativistic acceleration term. Here we show that the evolution of optical polarization states near the polarization singularity can be described in analogy to the relativistic dynamics of charged particles. A phase transition in parity-time symmetric potentials is then interpreted in terms of the competition between electric and magnetic ‘pseudo’-fields applied to polarization states. Based on this Lorentz pseudo-force representation, we reveal that zero Lorentz pseudo-force is the origin of recently reported strong polarization convergence to the singular state at the exceptional point. We also demonstrate the deterministic design of achiral and directional eigenstates at the exceptional point, allowing an anomalous linear polarizer which operates orthogonal to forward and backward waves. Our results linking parity-time symmetry and relativistic electrodynamics show that previous PT-symmetric potentials for the polarization singularity with a chiral eigenstate are the subset of optical potentials for the E×B “polarization” drift.

No MeSH data available.


Related in: MedlinePlus

Phases of PT symmetry in terms of the E×B drift.The evolution of the eigenvalues Δεeig is shown in (a) for their real parts (Re[Δεeig]), and in (b) for imaginary parts (Im[Δεeig]). The Lorentz pseudo-force acceleration for each phase of PT symmetry is shown: (c) before the EP with /B/ > /E/, (d) at the EP with /B/ = /E/, and (e) after the EP with /B/ < /E/. The enlarged plots in (c–e) show the distribution of accelerations near the north and south poles.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC5121894&req=5

f2: Phases of PT symmetry in terms of the E×B drift.The evolution of the eigenvalues Δεeig is shown in (a) for their real parts (Re[Δεeig]), and in (b) for imaginary parts (Im[Δεeig]). The Lorentz pseudo-force acceleration for each phase of PT symmetry is shown: (c) before the EP with /B/ > /E/, (d) at the EP with /B/ = /E/, and (e) after the EP with /B/ < /E/. The enlarged plots in (c–e) show the distribution of accelerations near the north and south poles.

Mentions: Based on the Lorentz pseudo-force equation of Eq. (2), a phase transition61617 between real and complex eigenspectra in PT-symmetric potentials can be interpreted in terms of the E×B drift3445: the competition between electric and magnetic pseudo-forces. Figure 2a,b shows the evolution of real and imaginary eigenvalues Δεeig for the Hamiltonian equation dψe/dz = Hs·ψe, as a function of the imaginary potential (εi = Im[ε1], εc = ε2). First, before the EP where eigenvalues are real and non-degenerate (Fig. 2a, εi < εc), the magnetic pseudo-field is larger than the electric pseudo-field, resulting in the counter-directive acceleration of SOP (lower panels in Fig. 2c) to northern-/southern-hemispheres. At the EP with the coalescence (d point in Fig. 2a,b, εi = εc), the equal magnitude of E = 2·εi·e1/λ and B = −2εc·e2/λ fields derives the suppression of total Lorentz pseudo-force on the southern Poincaré sphere, especially with the zero net force at the south pole (Sn = −e3, dSn/dz = E + Sn × B − (Sn·E)Sn = 2·εi·e1/λ + e3 × 2εc·e2/λ = 0, Fig. 2d). It is emphasized that this force cancellation impedes the acceleration near the south pole of the stationary polarization, deriving the SOP convergence to perfect LCP chirality26. After the EP with amplifying and dissipative states (Fig. 2b, εi > εc), the strong electric pseudo-field dominates the motion equation of the SOP, with the co-directive force (lower panels in Fig. 2e) to opposite hemispheres. In the context of electrodynamics analogy, the phase of eigenvalues in PT-symmetric potentials can thus be divided by the (i) B-dominant (before the EP), (ii) B = E (at the EP) and (iii) E-dominant regime (after the EP). It is worth mentioning that the stable point with the stationary polarization can also be obtained at the north pole by changing the sign of εc (converting the fast and slow axes for the birefringence) or εi (converting the gain and loss axes for the linear dichroism), allowing perfect RCP chirality. In terms of this Lorentz pseudo-force representation of SOP, we also note that PT-symmetric potentials1516 with real-valued ε2 and imaginary-valued ε1 are the special case of the E×B drift with specific field vectors E = 2·εi·e1/λ and B = −2εc·e2/λ, implying the existence of unconventional polarization singularity at other SOPs (e.g. without optical spin) which will be discussed later.


Acceleration toward polarization singularity inspired by relativistic E × B drift
Phases of PT symmetry in terms of the E×B drift.The evolution of the eigenvalues Δεeig is shown in (a) for their real parts (Re[Δεeig]), and in (b) for imaginary parts (Im[Δεeig]). The Lorentz pseudo-force acceleration for each phase of PT symmetry is shown: (c) before the EP with /B/ > /E/, (d) at the EP with /B/ = /E/, and (e) after the EP with /B/ < /E/. The enlarged plots in (c–e) show the distribution of accelerations near the north and south poles.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5121894&req=5

f2: Phases of PT symmetry in terms of the E×B drift.The evolution of the eigenvalues Δεeig is shown in (a) for their real parts (Re[Δεeig]), and in (b) for imaginary parts (Im[Δεeig]). The Lorentz pseudo-force acceleration for each phase of PT symmetry is shown: (c) before the EP with /B/ > /E/, (d) at the EP with /B/ = /E/, and (e) after the EP with /B/ < /E/. The enlarged plots in (c–e) show the distribution of accelerations near the north and south poles.
Mentions: Based on the Lorentz pseudo-force equation of Eq. (2), a phase transition61617 between real and complex eigenspectra in PT-symmetric potentials can be interpreted in terms of the E×B drift3445: the competition between electric and magnetic pseudo-forces. Figure 2a,b shows the evolution of real and imaginary eigenvalues Δεeig for the Hamiltonian equation dψe/dz = Hs·ψe, as a function of the imaginary potential (εi = Im[ε1], εc = ε2). First, before the EP where eigenvalues are real and non-degenerate (Fig. 2a, εi < εc), the magnetic pseudo-field is larger than the electric pseudo-field, resulting in the counter-directive acceleration of SOP (lower panels in Fig. 2c) to northern-/southern-hemispheres. At the EP with the coalescence (d point in Fig. 2a,b, εi = εc), the equal magnitude of E = 2·εi·e1/λ and B = −2εc·e2/λ fields derives the suppression of total Lorentz pseudo-force on the southern Poincaré sphere, especially with the zero net force at the south pole (Sn = −e3, dSn/dz = E + Sn × B − (Sn·E)Sn = 2·εi·e1/λ + e3 × 2εc·e2/λ = 0, Fig. 2d). It is emphasized that this force cancellation impedes the acceleration near the south pole of the stationary polarization, deriving the SOP convergence to perfect LCP chirality26. After the EP with amplifying and dissipative states (Fig. 2b, εi > εc), the strong electric pseudo-field dominates the motion equation of the SOP, with the co-directive force (lower panels in Fig. 2e) to opposite hemispheres. In the context of electrodynamics analogy, the phase of eigenvalues in PT-symmetric potentials can thus be divided by the (i) B-dominant (before the EP), (ii) B = E (at the EP) and (iii) E-dominant regime (after the EP). It is worth mentioning that the stable point with the stationary polarization can also be obtained at the north pole by changing the sign of εc (converting the fast and slow axes for the birefringence) or εi (converting the gain and loss axes for the linear dichroism), allowing perfect RCP chirality. In terms of this Lorentz pseudo-force representation of SOP, we also note that PT-symmetric potentials1516 with real-valued ε2 and imaginary-valued ε1 are the special case of the E×B drift with specific field vectors E = 2·εi·e1/λ and B = −2εc·e2/λ, implying the existence of unconventional polarization singularity at other SOPs (e.g. without optical spin) which will be discussed later.

View Article: PubMed Central - PubMed

ABSTRACT

The relativistic trajectory of a charged particle driven by the Lorentz force is different from the classical one, by velocity-dependent relativistic acceleration term. Here we show that the evolution of optical polarization states near the polarization singularity can be described in analogy to the relativistic dynamics of charged particles. A phase transition in parity-time symmetric potentials is then interpreted in terms of the competition between electric and magnetic &lsquo;pseudo&rsquo;-fields applied to polarization states. Based on this Lorentz pseudo-force representation, we reveal that zero Lorentz pseudo-force is the origin of recently reported strong polarization convergence to the singular state at the exceptional point. We also demonstrate the deterministic design of achiral and directional eigenstates at the exceptional point, allowing an anomalous linear polarizer which operates orthogonal to forward and backward waves. Our results linking parity-time symmetry and relativistic electrodynamics show that previous PT-symmetric potentials for the polarization singularity with a chiral eigenstate are the subset of optical potentials for the E&times;B &ldquo;polarization&rdquo; drift.

No MeSH data available.


Related in: MedlinePlus