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 Re½ε ¼j∇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.
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FIG. 3. Bohmian design of the disordered potential for CI-wave propagation. (a) Real and (b) imaginary permittivity distributions and (c) the obtained field evolution. (d) The distribution of the Poynting vector overlaid on the amplitude of the electric field Ez. The background index is n0 ¼ 1.5. All of the field evolutions are FIG. 2. Bohmian design of spatial phase trapping materials. calculated using COMSOL Multiphysics. (a)–(c) Electric field Ez evolutions for different values of Sb: (a) S 0 2π S 0 5π S π – b ¼ . ,(b) b ¼ . , and (c) b ¼ . Arrows in (a) (c) “ ” denote the propagation direction of light. (d) Real and (e) example of CI-wave propagation in 2D disordered imaginary permittivity landscapes and (f) obtained phase evolu- complex potentials. We design the disordered phase S x; y tions for Sb ¼ 0.1π–0.4π with a 0.1π interval. The regions function ð Þ from the plane-wavelike linear phase between dashed lines denote the designed range according to evolution, with Sðx; yÞ¼n0k0y þ 2πWðx; yÞΔSðx; yÞ= Eq. (3). The materials in the input and output regions are maxfWðx; yÞΔSðx; yÞg for the background refractive index ε y ε y q determined to maintain the continuity of rð Þ and ið Þ. ¼ n0, where ΔSðx; yÞ is the random perturbation defined in k λ 0 for all cases. 0 is the free-space wavelength. All of the field the spectral domain as evolutions are calculated by the finite element method using Z k COMSOL Multiphysics. 0 ΔSðx; yÞ¼u½−1; 1 sinðqxx þ u½−π; π Þdqx 0Z k This inhomogeneous complex permittivity allows the 0 þ u½−1; 1 sinðqyy þ u½−π; π Þdqy; ð4Þ “scattering-free” CI wave in which the oscillating phase is 0 bounded in [−Sb, Sb][Figs.2(a)–2(c) for different values of where u½a; b is a uniform random number between a and Sb]. Because the stringent condition of Sb 0 directly leads ¼ b. Wðx; yÞ is the weighting function for the electromagnetic to the ENZ material (εr ¼ εi ¼ 0), the CI-wave class for S x; y S continuity condition of ð Þ given by b-restricted phase trapping materials corresponds to the extension of the phase halt in the ENZ medium. Because the 2πx 2πy W ¼ 1 − cos 1 − cos ; ð5Þ designed phase trapping material can possess a nonzero real Lx Ly part and a zero imaginary part of permittivity at the boundaries [Figs. 2(d) and 2(e)], achieving impedance where Lx and Ly are the lengths of each side of the matching to conventional homogeneous materials while rectangle-shaped design area (the regions between the nearly maintaining the properties of ENZ media, i.e., the dashed lines in Fig. 3: 0 ≤ x ≤ 20λ0 and 0 ≤ y ≤ 10λ0). constant intensity and nearly conserved optical phase, is From the designed Sðx; yÞ, the necessary permittivity 2 2 2 possible. It is also noted that an exotic phase evolution [here, distribution is obtained as εðx; yÞ¼ðj∇Sj − i∇ SÞ=k0 sinusoidal from the SðyÞ function, the region between the [Figs. 3(a) and 3(b)]. Figure 3(c) shows the spatial field dashed lines in Fig. 2(f)], in contrast to a linear phase evolution in this disordered medium, showing a random evolution, can be designed, without any scattering due to phase in the design area and a perfect plane wave at the the CI-wave condition. Although the designed permittivity input and output regions without any scattering. As shown profiles in Eq. (3) satisfy the parity-time (PT) symmetry [16] in the Poynting vector distribution [Fig. 3(d)], this CI-wave with the odd function SðyÞ, it is not a necessary condition for propagation in the disordered complex potential is obtained the phase trapping, as shown in Note S3 and Fig. S1 in [15] from the modification of the magnetic field while preserv- for the random evolution of the bounded SðyÞ. ing the unity intensity of the electric field. It is also worth The condition of the CI wave can also be extended mentioning that this Bohmian random potential supporting to two-dimensional (2D) problems. Figure 3 shows an the CI wave with tunable phase evolution belongs to the
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FIG. 4. Phase-conserved control of energy confinement and cancellation, designed by the Bohmian approach. The light energy can be (a)–(d) confined or (e)–(h) annihilated in the designed regions. (a),(e) Real and (b),(f) imaginary permittivity distributions, and (c),(g) obtained field evolutions. (d),(h) The distributions of the Poynting vector overlaid on the amplitude of the electric field Ez. The confinement and cancellation of optical energy are obtained with (a)–(d) R0 ¼ 2 and (e)–(h) R0 ¼ −0.6, respectively. n0 ¼ 1.5, σ ¼ 5λ0, and b ¼ 2 for defining the 2D bump function. The center of the design area is ðx0;y0Þ¼ð10λ0; 10λ0Þ. All of the field evolutions are calculated using COMSOL Multiphysics. class of artificial disordered potentials with designed wave requiring the R function to be of the C2 differentiability profiles [17]. class. To fully guarantee the continuity condition, we Having demonstrated the phase manipulation with the employ a “bump” function that is smooth, having continu- V 0 ∞ quantum-mechanically-free case of quantum ¼ ,wenow ous derivatives of all orders, i.e., of C class investigate the utilization of the nonzero quantum potential ( 2 2 2 b λ0 V ¼ −∇ R=R ≠ 0, which allows for the realization 1 R − 2 2 r ≤ σ quantum R x; y þ 0 exp σ −r for ; of exotic energy confinement. The confinement of light is ð Þ¼ ð7Þ 1 otherwise achieved with systems that hinder outgoing waves [18–20], in general, except for a few, such as the use of bound states where R0 determines the strength of energy confinement, σ in the continuum [21,22]. Because all of these methods are defines the confinement area in the polar coordinate inherently based on the scattering of waves, the confine- 2 2 1=2 representation of r ¼½ðx − x0Þ þðy − y0Þ , and b ment of light, or, more generally, the modulation of the represents the sharpness of the confinement function. R spatial amplitude distribution , leads to a subsequent, With the calculated permittivity profiles in Figs. 4(a) S inevitable disturbance to optical phase information . and 4(b), we show in Figs. 4(c) and 4(d) the clear energy Instead, we apply the Bohmian approach to achieve concentration of the light wave inside the confinement scattering-free energy confinement, completely preserving region, with perfect conservation of the incident phase the spatial phase information. information. The negative value of R0 in Eq. (7) allows We assume a plane-wavelike linear phase evolution scattering-free energy annihilation inside the designed S x; y n k y ð Þ¼ 0 0 in the entire space. The required potential area, along with an unperturbed phase distribution then becomes [Figs. 4(e)–4(h)]. The extension of the result to arbitrary geometries is also shown in Note S4 and Fig. S2 in [15],by ∇2R 2 2 utilizing the superposition of bump functions with random Vrðx; yÞ¼n0k0 − ; R distributions. Notably, the spatial profiles of energy con- ∇R V x; y −2n k y:ˆ finement and cancellation can be freely designed without ið Þ¼ 0 0 R · ð6Þ any phase distortion. We have shown that the application of the Bohmian To satisfy the scattering-free condition for the plane formulation in optics enables the separate understanding of wave excitation, ∇R=R and ∇2R=R should be continuous, light-matter interactions for optical phase and amplitude,
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