Abstract
The relativistic trajectory of a charged particle driven by the Lorentz force is different from the classical one, by velocitydependent 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 paritytime symmetric potentials is then interpreted in terms of the competition between electric and magnetic ‘pseudo’fields applied to polarization states. Based on this Lorentz pseudoforce representation, we reveal that zero Lorentz pseudoforce 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 paritytime symmetry and relativistic electrodynamics show that previous PTsymmetric potentials for the polarization singularity with a chiral eigenstate are the subset of optical potentials for the E×B “polarization” drift.
Introduction
With the universal existence of open systems^{1,2} of nonequilibrium and timedependent^{3,4,5} potential energy, the concept of paritytime (PT) symmetry^{6,7} has become a multidisciplinary topic^{8,9,10,11,12,13,14,15}. PT symmetry successfully offers the special form of potentials V(x) = V^{*}(−x), providing physical observables even for nonequilibrium systems. The existence of real observables in PTsymmetric complex potentials has opened the field of nonHermitian quantum mechanics^{6,8}, which exhibits the phase transition^{6,16,17} between real and complex eigenspectra in stark contrast to a purely real eigenspectrum observed in Hermitian potentials. By utilizing optical gain and loss materials in the refractive index form n(x) = n^{*}(−x), the physics of PTsymmetric potentials has been applied to the exotic control of light flows^{17,18}. The effective realization of PT symmetry has also been extended to acoustics^{9,13,19}, optomechanics^{10}, electronics^{11}, gyrotropic systems^{20,21}, and population genetics^{12}. For all of these fields, critical traits of PT symmetry, e.g. unidirectionality^{22,23,24}, nonHermitian degeneracy^{18}, and chirality^{25,26}, impose intriguing features on wave dynamics in terms of the ‘singularity’^{27,28}: the coalescence of eigenstates with a chiral form^{25,26,29,30,31,32} at the exceptional point (EP, or phase transition point)^{33}.
Meanwhile, it is known that relativistic electrodynamics^{34} for charged particles also exhibits the inherent feature of open systems. The famous relativistic energy expression^{35}, Ẽ = mc^{2}/[1 − (v/c)^{2}]^{1/2}, shows that observers in different frames will see different values of total energy for moving charged particles of velocity v. We note that this nonequilibrium condition results in nonHermitian form of Hamiltonians, the necessary condition for the achievement of PT symmetry. In the context of the multidisciplinary realization of PT symmetry^{8,9,10,11,12}, therefore, the link between relativistic behaviors of charged particles in electromagnetic fields and wave dynamics in PTsymmetric potentials could offer different viewpoints on the physics of EP singularity in PTsymmetric potentials.
Inspired by the polarization equation of motion from the Schrödingerlike form of Maxwell’s equations^{36}, here we interpret the evolution of optical states of polarizations (SOP) near the EP singularity in direct analogy to the relativistic E×B drift (the movement under orthogonal E and B fields) of charged particles^{34}, which we call the relativistic E×B “polarization” drift of light. The phase transition in PTsymmetric potentials^{6,16,17} is then understood in view of the competition between electric and magnetic ‘pseudo’fields, and we prove that strong chiral conversion of optical SOP at the EP^{15,16} corresponds to the accidental cancellation of the Lorentz pseudoforce on the Poincaré hemisphere. By employing this “Lorentzforce picture” in the analysis of the polarization singularity, we then extend the class of the polarization singularity in vector wave equations^{25,26,37,38,39}, revealing the existence of achiral and directional eigenstates at the EP. Our approach paves the way for the unconventional control of optical polarizations, such as anomalous directional polarizers.
Results
Lorentz pseudoforces for optical polarizations
Consider the planewave propagating along the zaxis of the electrically anisotropic material, with the unity permeability (μ = 1). For the later discussion, we express the arbitrary permittivity tensor in the xy plane, in terms of Pauli matrices^{26,31} σ_{1–3} as ε = (ε_{2}σ_{1} + ε_{3}σ_{2} + ε_{1}σ_{3}), or
where ε_{o}(z) and ε_{1−3}(z) have slowlyvarying complex values, σ_{1} = [0, 1; 1, 0], σ_{2} = [0, −i; i, 0], and σ_{3} = [1, 0; 0, −1]. By applying the spin (x ± iy)/2^{1/2} bases^{36}, Maxwell’s equations become the vector Schrödingerlike equation dψ_{e}/dz = H_{s}·ψ_{e} with the temporallike zaxis^{40}, where the spinor representation of ψ_{e} = [ψ_{e+}, ψ_{e−}]^{T} is the electric field amplitude of (±) optical spin waves ((+) for rightcircular polarization (RCP) (x + iy)/2^{1/2}, and (−) for leftcircular polarization (LCP) (x − iy)/2^{1/2}), and H_{s} is the traceless Hamiltonian expressed with Pauli matrices^{36,41} as H_{s} = (ε_{1}σ_{1} + ε_{2}σ_{2} + ε_{3}σ_{3})/(iλ) for λ = 2ε_{o}^{1/2}/k_{0} and the freespace wavenumber k_{0}. If we assign the symmetry axis between x and y axes, the condition of PTsymmetric potentials^{15,16} requires realvalued ε_{o}, ε_{2}, and ε_{3}, and imaginaryvalued ε_{1}. Note that imaginaryvalued ε_{1} corresponds to the linear dichroism^{42,43}, the different dissipation for each linear polarization, while realvalued ε_{2} represents the birefringence. ε_{3} represents the magnetooptical change of plasma permittivity induced by an external static magnetic field^{34}, and for now, we assume the nonmagnetic case of ε_{3} = 0.
In this representation, the SOP of light is described by the Stokes parameters^{44} S_{j} = ψ_{e}^{†}·σ_{j}·ψ_{e} (j = 0, 1, 2, 3). The change of the SOP can then be expressed in view of lightmatter interactions^{36} by applying the governing equation dψ_{e}/dz = H_{s}·ψ_{e} and its conjugate form dψ_{e}^{†}/dz = ψ_{e}^{†}·H_{s}^{†}, which leads to the Lorentz pseudoforce equation of motion for the SOP^{36} (also see Supplementary Note 1)
where S_{n} = [S_{1}, S_{2}, S_{3}]^{T}/S_{0} is the pseudovelocity of the ‘hypothetical’ charged particle corresponding to the SOP of light, and E = 2·Im[ε_{1}, ε_{2}, ε_{3}]^{T}/λ and B = −2·Re[ε_{1}, ε_{2}, ε_{3}]^{T}/λ are the electric and magnetic pseudofield in relation to imaginary and realparts of the permittivity, respectively. Note that Eq. (2) provides direct analogy to the relativistic dynamics of massless charged particles with the motion equation^{20,22} of ∂_{t}β = E + β×B −(β·E)β. In this representation of optical polarization states, the acceleration of optical SOP comes from the Lorentz pseudoforce, F ~ dS_{n}/dz (Fig. 1). The first two terms of Eq. (2) are the counterparts of the classical electromagnetic Lorentz force E + β×B, and the 3^{rd} term – (S_{n}·E)S_{n} corresponds to the Joule effect^{34} – (β·E)β in the relativistic equation of motion.
Figure 1a–c shows the effect of each component of the Lorentz pseudoforce on the SOP of propagating light, induced by nonmagnetic PTsymmetric materials (imaginary ε_{1}, real ε_{2}, and zero ε_{3}). While ε_{2} of the birefringence derives the circulating acceleration on the Poincaré sphere (B = −2ε_{2}·e_{2}/λ, Fig. 1a, Hermitian case), ε_{1} of the amplification or dissipation results in the linear drift of the SOP (E = 2·Im[ε_{1}]·e_{1}/λ, Fig. 1b, nonHermitian case). We also note that the energy variation from gain and loss materials S_{n}·E (Fig. 1c) provides the relativistic nonlinear acceleration of the SOP with respect to E. Consequently, with the orthogonality between pseudofields (E⊥B), PTsymmetric potentials naturally satisfy the ideal E×B drift^{34,45} condition to optical polarization states.
Lorentz force picture on PT symmetry
Based on the Lorentz pseudoforce equation of Eq. (2), a phase transition^{6,16,17} between real and complex eigenspectra in PTsymmetric potentials can be interpreted in terms of the E×B drift^{34,45}: the competition between electric and magnetic pseudoforces. Figure 2a,b shows the evolution of real and imaginary eigenvalues Δε_{eig} for the Hamiltonian equation dψ_{e}/dz = H_{s}·ψ_{e}, as a function of the imaginary potential (ε_{i} = Im[ε_{1}], ε_{c} = ε_{2}). First, before the EP where eigenvalues are real and nondegenerate (Fig. 2a, ε_{i} < ε_{c}), the magnetic pseudofield is larger than the electric pseudofield, resulting in the counterdirective acceleration of SOP (lower panels in Fig. 2c) to northern/southernhemispheres. At the EP with the coalescence (d point in Fig. 2a,b, ε_{i} = ε_{c}), the equal magnitude of E = 2·ε_{i}·e_{1}/λ and B = −2ε_{c}·e_{2}/λ fields derives the suppression of total Lorentz pseudoforce on the southern Poincaré sphere, especially with the zero net force at the south pole (S_{n} = −e_{3}, dS_{n}/dz = E + S_{n} × B − (S_{n}·E)S_{n} = 2·ε_{i}·e_{1}/λ + e_{3} × 2ε_{c}·e_{2}/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 chirality^{26}. After the EP with amplifying and dissipative states (Fig. 2b, ε_{i} > ε_{c}), the strong electric pseudofield dominates the motion equation of the SOP, with the codirective force (lower panels in Fig. 2e) to opposite hemispheres. In the context of electrodynamics analogy, the phase of eigenvalues in PTsymmetric potentials can thus be divided by the (i) Bdominant (before the EP), (ii) B = E (at the EP) and (iii) Edominant 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 ε_{c} (converting the fast and slow axes for the birefringence) or ε_{i} (converting the gain and loss axes for the linear dichroism), allowing perfect RCP chirality. In terms of this Lorentz pseudoforce representation of SOP, we also note that PTsymmetric potentials^{15,16} with realvalued ε_{2} and imaginaryvalued ε_{1} are the special case of the E×B drift with specific field vectors E = 2·ε_{i}·e_{1}/λ and B = −2ε_{c}·e_{2}/λ, implying the existence of unconventional polarization singularity at other SOPs (e.g. without optical spin) which will be discussed later.
E×B polarization drift in PTsymmetric potentials
We then investigate the “evolution” of the SOP, for different pseudoforces shown in Fig. 2. Figure 3a–c shows the change of the initial SOP of (+, RCP) and (−LCP) spins under different phases of PT symmetry, in relation to the charged particle movement at different phases^{45} of the relativistic E×B drift (Fig. 3d–f). Because the magnetic pseudofield dominates the dynamics of SOP before the EP (Fig. 3a,d, B > E), the SOP for each spin simply rotates around the B field following the S_{n}×B of Eq. (2). Yet, with different magnitudes of the forces in northern and southern hemispheres (Fig. 2c), the ‘speed’ of the SOP rotation near each pole is different, analogous to different magneticallygyrating arcs of charged particles in the E×B drift^{34,45}. The directional drift of relativistic particles along the E×B axis (Fig. 3d, toward −S_{3} axis) is thus reproduced by the slow evolution of SOPs near the S_{n} = −e_{3} on the Poincaré sphere (Fig. 3a).
The extraordinary case of the relativistic E×B polarization drift is achieved at the singular state of EP (Fig. 3b,e), for the case of B = E. Because of the force cancellation near the perfectly stable south pole (S_{n} = −e_{3} for dS_{n}/dz = 0), the (+) spin state converges to the (−) spin when the state approaches the south pole through the gyration by the magnetic pseudofield (orange line in Fig. 3b), similar to the convergence of the velocity in the motion of relativistic particles (orange line in Fig. 3e). Because the (−) spin state is stationary, we note that the entire SOP, which can be represented in terms of the linear combination of the LCP (−spin) and RCP (+spin), is thus converted to the LCP chiral wave. After the EP (B < E, Fig. 3c,f), the electric force is dominant, resulting in the linear acceleration mostly towards the direction of E. For all cases, it is noted that the relativistic correction from nonHermitian Hamiltonians retains the evolution of SOP S_{n} = [S_{1}, S_{2}, S_{3}]^{T}/S_{0} on the Poincaré sphere, in contrast to the classical evolutions (dotted lines in Fig. 3a–c) which do not include the third term of Eq. (2).
Realization of achiral and directional singularity
Extending the special case of the E×B polarization drift derived from E = 2·ε_{i}·e_{1}/λ and B = −2ε_{c}·e_{2}/λ, we now work on other types of E×B polarization drift which allow unconventional polarization singularity without optical spin, by manipulating the direction of electromagnetic pseudofields. The vector form of the Lorentz pseudoforce equation provides larger degrees of freedom for the intuitive control of the eigenstate at the singularity, in contrast to the fixed chiral form^{25,26,29,30,31,32} of scalar PTsymmetric equation. Although the magnetic transition of PT symmetry can also be utilized to achieve the achiral (spinless) eigenstate at the EP (ε_{3} ≠ 0, Supplementary Note 2), here we investigate the realization of achiral and directional singularity with the use of nonmagnetic chiral materials. From the constitutive relation including optical chirality^{46} (or biisotropy) D = εE − iχH and B = μH + iχE, the condition of μ = μ_{0} and diagonal ε with ε_{x} ≠ ε_{y} represents the nonmagnetic chiral material with general electrical anisotropy, including both birefringence (Re[ε_{x}] ≠ Re[ε_{y}] for different x and ywavevector) and linear dichroism^{42,43} (Im[ε_{x}] ≠ Im[ε_{y}] for different x and ydissipation). The spinbased Hamiltonian equation dψ_{e}/dz = H_{s}·ψ_{e}, for slowlyvarying ε_{x} = ε_{o} + Δε(z) and ε_{y} = ε_{o} − Δε(z) and constant χ = χ_{o} with realvalued ε_{o} and χ_{o}, derives the Hamiltonian H_{s} for general chiral materials, in the form of
and k^{2} = ω^{2}·(μ_{0}ε_{o} + χ_{o}^{2}) (see Supplementary Note 3 for the general case of spatiallyvarying χ(z)). The Pauli expression^{41} of H_{s} = a_{1}σ_{1} + a_{2}σ_{2} + a_{3}σ_{3} has the coefficients of a_{1} = −(ik/2)·(Δε/ε_{o}), a_{2} = −(ωχ/2)·(Δε/ε_{o}), and a_{3} = iωχ. The electric and magnetic pseudofields for Eq. (2) are then defined as
with the degree of electrical anisotropy ρ = Δε/ε_{o}.
Equation (4) proves that the pseudofield components E(ρ, χ) and B(ρ, χ) driving the SOP are strongly dependent on the type of the anisotropy: birefringence (real ρ) or linear dichroism (imaginary ρ) both satisfying the condition of the E×B drift (E⊥B). For the case of birefringence with E(ρ, χ) = −ωχ_{o}ρ·e_{2} and B(ρ, χ) = −kρ·e_{1} + 2ωχ_{o}·e_{3}, the pseudofield satisfies E < B in most cases and the condition of E ≥ B enforces ε_{o} < −4χ_{o}^{2}/(μ_{0}·ρ^{2}) and thus prohibits the existence of propagating waves at the EP. It is interesting to note that this restriction proves the necessity of complex potentials for obtaining the singularity; therefore we employ linear dichroism for achieving the EP for the propagating wave, by fulfilling the condition of B = E.
Figure 4 shows the case of linearlydichroic (ρ = i·ρ_{i}) chiral materials, which derive pseudofields of E(ρ, χ) = kρ_{i}·e_{1} and B(ρ, χ) = −ωχ_{o}ρ_{i}·e_{2} + 2ωχ_{o}·e_{3} with the EP condition of ρ_{i}^{2} = 4χ_{o}^{2}/(μ_{0}ε_{o}) for B = E. The PTsymmetrylike phase transition (Fig. 4b–d) around the EP (marked with red dots in Fig. 4c,f) occurs in linearlydichroic chiral materials from the competition between E and B (Fig. 4a), and the direction of the E×B drift is controlled by changing ρ_{i} for the pseudomagnetic field B, allowing the realization of the achiral singularity (S_{n} ~ −e_{2} in Fig. 4c); in sharp contrast to the case of PTsymmetric potentials. Furthermore, the obtained EP state has the directionality in its propagation due to the wavevectordependency of Eq. (4) (Fig. 4b–d vs Fig. 4e–g, S_{n} ~ e_{2} in Fig. 4f), which originates from the broken mirror symmetry of chiral materials for forward and backward waves.
This directionality with achiral designer eigenstate at the EP allows the implementation of unconventional polarizers based on the polarization convergence at the EP. Figure 4h shows an example of the anomalous linear polarizer, operating ‘orthogonal’ to forward and backward waves. While the SOP of forward waves are converged to the +45° linear polarization (Fig. 4c), the SOP of backward waves becomes −45° linear polarization. Moreover, in contrast to the case of classical linear polarizers which perfectly reflect the orthogonally polarized waves (e.g. the ypolarized incidence to the xpolarizer), the linearpolarizing functionality in the structure of Fig. 4h operates for the ‘entire’ SOP due to the nonorthogonality between eigenstates.
Discussion
In summary, we found the link between the seemingly unrelated fields of PT symmetry optics and relativistic electrodynamics. This reinterpretation of PT symmetry brings insight to the singularity in polarization space, broadening the class of paritytime symmetric Hamiltonians in vector wave equations^{39}. The counterintuitive achievement of the achiral and directional EP eigenstate is also demonstrated, which allows the realization of anomalous linear polarizers for randomly polarized incidences. The comprehensible understanding of the EP in terms of the dynamics of charged particles will provide a novel design methodology near the singularity: the generation of chiral waves^{47,48,49}, topological photonics which has focused only on chiral states^{38}, coherent wave dynamics^{50}, PTsymmetrylike potentials based on causality^{51,52} or supersymmetric optics^{53,54}, and optical analogy of spintronics. At the same time, this classical viewpoint on relativistic electrodynamics also enables the analogy of EP dynamics in charged particle movements, the linear E×B drift toward a single direction for every initial velocity vectors.
Additional Information
How to cite this article: Yu, S. et al. Acceleration toward polarization singularity inspired by relativistic E×B drift. Sci. Rep. 6, 37754; doi: 10.1038/srep37754 (2016).
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Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) through the Global Frontier Program (GFP, 2014M3A6B3063708) and the Global Research Laboratory Program (GRL, K20815000003), all funded by the Ministry of Science, ICT & Future Planning of the Korean government. S. Yu was also supported by the Basic Science Research Program (2016R1A6A3A04009723), and X. Piao and N. Park were also supported by the Korea Research Fellowship Program (KRF, 2016H1D3A1938069) through the NRF, all funded by the Ministry of Education of the Korean government.
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Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Korea
 Sunkyu Yu
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Contributions
S.Y. conceived the presented idea. S.Y. and X.P. developed the theory and performed the computations. N.P. encouraged S.Y. to investigate the link between relativistic electrodynamics and nonHermitian physics while supervising the findings of this work. All authors discussed the results and contributed to the final manuscript.
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The authors declare no competing financial interests.
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Correspondence to Namkyoo Park.
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