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Bohmian Photonics for Independent Control of the Phase and Amplitude of Waves

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Bohmian Photonics for Independent Control of the Phase and Amplitude of Waves PHYSICAL REVIEW LETTERS 120, 193902 (2018) Bohmian Photonics for Independent Control of the Phase and Amplitude of Waves Sunkyu Yu, Xianji Piao, and Namkyoo Park* Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Korea (Received 6 December 2017; published 9 May 2018) The de Broglie–Bohm theory is one of the nonstandard interpretations of quantum phenomena that focuses on reintroducing definite positions of particles, in contrast to the indeterminism of the Copenhagen interpretation. In spite of intense debate on its measurement and nonlocality, the de Broglie–Bohm theory based on the reformulation of the Schrödinger equation allows for the description of quantum phenomena as deterministic trajectories embodied in the modified Hamilton-Jacobi mechanics. Here, we apply the Bohmian reformulation to Maxwell’s equations to achieve the independent manipulation of optical phase evolution and energy confinement. After establishing the deterministic design method based on the Bohmian approach, we investigate the condition of optical materials enabling scattering-free light with bounded or random phase evolutions. We also demonstrate a unique form of optical confinement and annihilation that preserves the phase information of incident light. Our separate tailoring of wave information extends the notion and range of artificial materials. DOI: 10.1103/PhysRevLett.120.193902 The de Broglie–Bohm theory [1], also called Bohmian an individual particle trajectory in the form of Newton’s mechanics, suggests an alternative interpretation of second law, in which the exerted force is represented by quantum mechanics for the description of individual events two parts: the classical force −∇V and the quantum force in statistical quantum phenomena. Compared to the ðℏ2=2mÞ∇ð∇2R=RÞ. The annihilation of the quantum force Copenhagen interpretation, which emphasizes the indeter- from ℏ → 0 then approximates quantum phenomena by ministic nature of quantum mechanics [2], the de Broglie– Newtonian mechanics. Bohm approach, constructed upon the reformulation of the In this Letter, we show that the Bohmian formulation of Schrödinger equation [1,3], attempts to reintroduce a Maxwell’s wave equation provides a new perspective on definite position of each particle that is guided by a pilot optical materials for the independent tailoring of amplitude wave [4]. This causal view for “point particles” enables the and phase information of light, which has not been reported understanding of quantum phenomena in a classical-like so far. By deriving the optical counterpart of the QHJ sense [3], though there has been considerable debate [5–10] equation, we demonstrate that optical potentials can be regarding the measurement and nonlocality of the Bohmian understood as the sum of the classical-like part for the approach, such as surreal trajectories in a Welcher-Weg phase evolution and the quantumlike part for the energy measurement [5,10]. To our knowledge, at this stage, the confinement. In this vein, as a first example, we realize a de Broglie–Bohm theory does not contradict the orthodox quantum-mechanically-free potential [3] in an optical plat- Copenhagen interpretation and quantum-mechanical form, to derive a new class of constant-intensity waves experiments. Recent achievements are discussed in [11]. [12,13] that have freely designed phase distributions with The heart of the Bohmian formulation is in the application phase trapping or randomization functions. Anomalous of the polar-form wave function ψ ¼ ReiðS=ℏÞ to the confinement or annihilation of energy that perfectly 2 2 Schrödinger equation iℏ∂tψ ¼ −ðℏ =2mÞ∇ ψ þ Vψ to preserves the phase information of light will also be separate the real and imaginary parts of this complex-valued demonstrated with the construction of a scattering-free equation [3]. The resulting quantum Hamilton-Jacobi (QHJ) quantumlike potential. These results not only pave the way 2 2 2 equation ∂tS þð∇SÞ =2m − ðℏ =2mÞ∇ R=R þ V ¼ 0 and for the independent manipulation of wave quantities but 2 2 the continuity equation ∂tR þ ∇ · ðR ∇S=mÞ¼0 identify also deepen the understanding of artificial materials, such as the near-zero regime [14] in optical material parameters. To explore the consequences of the Bohmian approach in optics, we consider a monochromatic wave of frequency ω Published by the American Physical Society under the terms of propagating in the x-y plane within a medium that has a the Creative Commons Attribution 4.0 International license. ε x; y ε x; y iε x; y Further distribution of this work must maintain attribution to permittivity ð Þ¼ rð Þþ ið Þ. For the trans- the author(s) and the published article’s title, journal citation, verse electric mode (electric field along the z axis), and DOI. the field evolution obeys the Helmholtz wave equation 0031-9007=18=120(19)=193902(6) 193902-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 120, 193902 (2018) 2 2 ∇ Ez þVðx;yÞEz ¼0, where V ¼ k0εðx; yÞ and k0 ¼ ω=c. We then express the field profile in polar form [3] as iSðx;yÞ Ezðx; yÞ¼Rðx; yÞe , where R is a real amplitude function that is by definition non-negative, and S is a real phase function that determines the instantaneous wave vector kðx; yÞ¼∇S, which corresponds to the guidance equation [3] of light. Applying this polar-form electric field to the Helmholtz equation allows the separation of the real and imaginary parts of the wave equation to yield the following representations of the complex optical potential V ¼ Vrðx; yÞþiViðx; yÞ (Supplemental Material Note S1 in [15]) ∇2R V x; y ∇S 2 − ; rð Þ¼j j R ð1Þ ∇R V x; y −∇2S − 2 ∇S: ið Þ¼ R · ð2Þ It is noted that Eq. (1) corresponds to the optical counterpart of the stationary (∂t → 0) QHJ equation [3]. Similar to the QHJ equation, the real part of the optical 2 potential Vr ¼ k0εr [Fig. 1(a)] can always be decomposed into two parts, for each wave information S and R: “ ” V ∇S 2 the classical Hamilton-Jacobi term classical ¼j j “ ” [Fig. 1(b)] and the quantum potential counterpart FIG. 1. Bohmian interpretation of an optical potential by V −∇2R=R [Fig. 1(c)] quantum ¼ . Each potential separately separating the classical and quantum potential, each for phase governs the spatial phase evolution with Sðx; yÞ [Fig. 1(d)] evolution and energy confinement. (a) The real part of the optical 2 and the optical confinement with Rðx; yÞ [Fig. 1(e)], thus potential Vr ¼ k0εr. Vr is the sum of the (b) classical Hamilton- V ∇S 2 constructing the full spatial information of the wave as Jacobi potential classical ¼j j and (c) quantum potential iS V −∇2R=R Ez ¼ Re [Fig. 1(f)]. Meanwhile, Eq. (2), rewritten as quantum ¼ , each governing the (d) phase evolution 2 2 2 eiS R E ReiS ∇ · ðR ∇SÞþk0εiðx; yÞR ¼ 0, is the stationary continuity and (e) amplitude of the (f) entire light wave z ¼ . equation describing the equilibrium between optical sources (or sinks) represented by εiðx; yÞ and the diver- This expression allows for the alternative classification of 2 “ ” gence of the power flow R ∇S. conventional homogeneous materials to include the Critically, applying the Bohmian interpretation of the following: (i) ordinary material for the linearly varying S x; y k x k y ε x;y k2 k2 =k2 >0 optical potential allows the independent control of the phase ð Þ¼ x þ y with ð Þ¼ð x þ yÞ 0 phase and amplitude distributions of light, analogous to and (ii) epsilon-near-zero (ENZ) material for the constant the separation of classical and quantum phenomena in phase Sðx; yÞ¼S0 with εðx; yÞ¼0. Note that this CI-wave iS Bohmian mechanics [3]. For the given wave Ez ¼ Re class does not include the case of negative real permittivity 2 2 with targeted spatial functions of Sðx; yÞ and Rðx; yÞ that because Re½ε¼j∇Sj =k0 ≥ 0. satisfy the electromagnetic boundary conditions, the real Depending on the form of Sðx; yÞ, the CI-wave class can optical potential Vr is directly obtained as the sum of include “inhomogeneous” materials with nonconstant first- V ∇S 2 V −∇2R=R S x; y classical ¼j j and quantum ¼ . The imaginary order partial derivatives of ð Þ. Among the many possible optical potential Vi is also determined by the continuity designs, we examine spatial phase trapping phenomena Eq. (2), accomplishing the inverse design of the complex (Fig. 2). While conventional materials result in full range iS 2π potential V for the given wave Ez ¼ Re . phase evolution between 0 and , the Bohmian design with In light of this Bohmian separation of light-matter an appropriate function of Sðx; yÞ enables the “trapping” of interactions in terms of phase and amplitude, we explore the spatial phase in a certain range. For example, taking the a particular example: quantum-mechanically-free [3] one-dimensional (1D) phase evolution as SðyÞ¼Sb ·sinðqyÞ, V −∇2R=R 0 potentials of quantum ¼ ¼ . Among the general the designed permittivity becomes (Note S2 in [15]) 2 solutions of Laplace’s equation ∇ R ¼ 0, we consider the R x; y R q 2 constant-intensity (CI) wave [12,13] with ð Þ¼ 0 to ε y S2 2 qy ; focus on the design of unconventional phase evolution. The rð Þ¼ b k cos ð Þ 0 complex optical potential for the CI wave then becomes 2 2 2 q Vrðx; yÞ¼j∇Sj and Viðx; yÞ¼−∇ S, which lead to the εiðyÞ¼Sb sinðqyÞ: ð3Þ 2 2 2 k0 complex permittivity profile of εðx;yÞ¼ðj∇Sj −i∇ SÞ=k0. 193902-2 PHYSICAL REVIEW LETTERS 120, 193902 (2018) FIG. 3. Bohmian design of the disordered potential for CI-wave propagation.
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