One-Way Optical Transition based on Causality in Momentum Space

The concept of parity-time (PT) symmetry has been used to identify a novel route to nonreciprocal dynamics in optical momentum space, imposing the directionality on the flow of light. Whereas PT-symmetric potentials have been implemented under the requirement of V(x) = V*(-x), this precondition has only been interpreted within the mathematical frame for the symmetry of Hamiltonians and has not been directly linked to nonreciprocity. Here, within the context of light-matter interactions, we develop an alternative route to nonreciprocity in momentum space by employing the concept of causality. We demonstrate that potentials with real and causal momentum spectra produce unidirectional transitions of optical states inside the k-continuum, which corresponds to an exceptional point on the degree of PT-symmetry. Our analysis reveals a critical link between non-Hermitian problems and spectral theory and enables the multi-dimensional manipulation of optical states, in contrast to one-dimensional control from the use of a Schrodinger-like equation in previous PT-symmetric optics.

Photons exhibit bosonic statistics: thus, the actual quantities of photons can only be altered by the interaction with materials, which follows from the linearity of the chargeless Maxwell's equations [1]. Consequently, lightmatter interaction is a prerequisite to manipulate the flow of light, which is not only a classical subject but also an emerging research topic involved in recent discoveries of optics (e.g., Snell's law in 1621 and its generalization in 2011 [2], respectively). In this regard, continuous efforts have been made to manipulate photon states using lightmatter interactions in either energy (E=ħω) or momentum (p=ħk) space [3][4][5][6][7][8][9][10][11]. The development of cutting-edge techniques to control the linear permittivity in time [12] or space [13] has resulted in the revisitation of the fundamental Fourier dynamics between the time-energy or spacemomentum domains to access unexplored regimes in k-ω space. For example, exotic phenomena in ω-space, such as dynamic slow-light [4], time-reversal symmetry breaking [5], or effective magnetic fields [6], have been achieved using a time-varying permittivity. Likewise, a spatially varying permittivity down to a deep-subwavelength scale and the ultimate control of the linear and angular momentum of light were recently demonstrated using lattices [7], disorder [8], chiral metamaterials [9], and phase-modulating surfaces [2]. These achievements are extremely encouraging but were obtained by controlling only the real optical potentials that correspond to double-sided spectra in Fourier space: thus, the considerable opportunities offered by manipulating the potential of generalized spectra have been overlooked.
From a mathematical perspective, continual efforts have been made to overcome the well-known restriction of Hermiticity in quantum mechanics [14,15]. Bender first proved the existence of real eigenvalues for complex potentials [14] when the potentials satisfy parity-time (PT) symmetry. This striking discovery has been adopted in various fields [15][16][17][18][19] to interpret the physics of complex potentials. In optics, the use of cleverly designed PTsymmetric potentials has resulted in inspiring achievements in the nonreciprocal dynamics of linear [16][17][18] and angular [19] optical momentum. Since early approaches in PTsymmetric optics were implemented to effectively model quantum-mechanical problems, optical potentials have been designed to naturally satisfy V(x) = V * (-x) from the commutative relation between PT and the Hamiltonian operators for a Schrödinger-like equation. However, the most interesting feature of the relation V(x) = V * (-x) has surprisingly been neglected: the potential momentum is realvalued, in contrast to non-PT-symmetric cases.
In this paper, we propose a novel pathway to nonreciprocal dynamics by employing the perspective of light-matter interactions. In this context, we consider the general problem of light excursions in k-space, which we define as a 'momentum transition' along the isofrequency contour (IFC). We show that the momentum transition in the weak coupling regime is mediated by the potential momentum. We then demonstrate that 'causality' in potential momentum space produces a one-way transition inside the kcontinuum, corresponding to an exceptional point (EP) on the degree of PT symmetry. Our results facilitate the deliberate control of optical momentums with complex potentials, such for collimated beam steering or excitations in the inaccessible regime of low-or high-k states, and provide a logical mechanism for understanding PT-symmetric potentials V(x) = V * (-x) and the EP from the general perspective of spectral analysis. Figure 1 shows examples of light excursions in k-space including beam steering ( Fig. 1a and 1b) and high-k (Fig. 1c) and low-k (Fig. 1d) excitations. To tailor the evolution of the optical state in k-space, we address the one-way optical transition (red arrows in Fig. 1) along the IFC, which suppresses the back transfer (grey arrows in Fig. 1) to the initial state. Notably, the one-way transition can be understood in the context of the relation between 'cause' (the initial state) and 'effects' (the directionally excited states) along the IFC, which is also known as causality (Fig. 1e). To prove this prediction, we first derive the coupled mode equation between the optical momentum states by generalizing the continuous coupled mode theory [20] to 2dimensional anisotropic materials. Without loss of generality, we consider a TM-polarized wave in a nonmagnetic anisotropic material (H z , E x , and E y with ε x,y ) that produces a k-continuum for an elliptic IFC ( Fig.  1a and 1b), a hyperbolic IFC ( [21], Fig. 1c), or a quasi-linear IFC with extreme anisotropy ( [22], Fig. 1d) in momentum space. Here, we apply two standard approximations to the time-harmonic wave equation at a frequency ω: a weak (|Δε x,y (x,y)| << |ε x0,y0 |, where ε x,y (x,y) = ε x0,y0 + Δε x,y (x,y)) and a slowly-varying modulated potential (|Δε y -1 •∂ x Δε y | << |k x | and |Δε x -1 •∂ y Δε x | << |k y |). We use the IFC relation of k 0 2 = k x 2 /ε y0 + k y 2 /ε x0 , where k 0 = ω/c is the free-space wavenumber, to derive the following expression [ where ψ [kx,ky] is the spatially varying envelope of the magnetic field for H z (x,y) = ∫∫ψ [kx,ky] )y is the εnormalized momentum vector, and σ k (x,y) = (k x •Δε y (x,y)/ε y0 )x + (k y •Δε x (x,y)/ε x0 )y is the local modulation vector. Equation (1) clearly shows the source of the momentum transitions β k •σ k (x,y) that induce the locally modulated envelope ∇ψ.
With the Fourier expansion (Δε pq (p,q)) of the modulated potential Δε(x,y) and the use of the divergence theorem, the 2-dimensional coupled mode equation between the momentum states k = (k x ,k y ) and (k x-p ,k y-q ) is obtained as Equation (2) defines the coupling along the IFC k 0 2 = k x 2 /ε y0 + k y 2 /ε x0 (including the multipath coupling through Δε pq (p,q) with a finite bandwidth) and can be used to derive the criterion for the directional coupling that prohibits back transfers (grey arrows in Fig. 1), thereby efficiently delivering optical energy into the targeted momentum state. Note that the potential momentum Δε pq in Eq. (2) mediates the coupling between states, whereby a highly efficient unidirectional momentum transition results from enforcing the causality condition in potential momentum space (p,q) (i.e., Δε pq ≠ 0 only for a single quadrant, which leads to a zero value for the integral of the back transfer). The selection of a nonzero quadrant is also clearly determined by the transition direction, e.g., the high-k excitation (the red arrow in Fig. 1c) is produced by restricting the potential momentum to the 1 st quadrant (p,q ≥ 0), whereas the low-k excitation (red arrow in Fig. 1d) is produced by selecting the 3 rd quadrant spectrum (p,q ≤ 0). The aforementioned conditions for both cases, the conditions in the momentum and spatial domains can be easily achieved by employing the multi-dimensional Hilbert transform for single orthant spectra [24], such as Δε pq In the spatial domain, the "one-way coupling potentials" for the low-k and high-k excitations then become or simply is the potential for the low-(or high-) k excitation with the upper (or lower) sign, and H T (or H p ) is the total (or partial) Hilbert transform [24]. We emphasize that Eq. (3) not only shows that complex potentials in the spatial domain are essential to produce one-way dynamics but also that the one-way complex potentials of Δε 0 •exp(-ip 0 x) that have been previously identified [17] are only a manifestation of a special case, i.e., pointwise coupling (Δε rp = Δε 0 •π[δ(p-p 0 ) + δ(p+p 0 )]) in a 1-dimensional problem. Note that this formalism, which is based on momentum causality, also allows for the deterministic composition of one-way designer potentials from the Δε rpq in momentum space. This condition can easily be extended to isofrequency 'surfaces' by employing a 3-dimensional Hilbert transform [24].
Most importantly, Eq. (3) offers critical physical insight into the link between PT symmetry [14][15][16][17][18][19] and causality in momentum space, which has not been previously elucidated, to the best of our knowledge. The one-way coupling potentials of Eq. (3) from causality satisfy the necessary condition of PT symmetry Δε L,H (x,y) = Δε L,H * (-x,-y) [14][15][16][17][18][19] and also guarantee real-valued spectra in momentum space (p,q). Because 'perfect' one-way dynamics in PT-symmetric potentials is achieved only at the EP [16,19] where PT symmetry breaking occurs, the causality potentials of Eq. (3) obviously correspond to the EP, and the regimes before and after the EP correspond to noncausal, real-valued spectra in momentum space.
We illustrate the aforementioned results by considering an arbitrary PT-symmetric potential Δε s (x,y) in space, where Re[Δε s ] (or Im[Δε s ]) is an even (or odd) real-valued function that satisfies the precondition Δε s (x,y) = Δε s * (-x,-y). The potential momentum Δε m (p,q) = F{Δε s (x,y)} is then expressed by the sum of real-valued functions as Δε m (p,q) = Δε m-even (p,q) + Δε m-odd (p,q), where Δε m-even = F{Re[Δε s ]} is an even function, and Δε m-odd = -Im[F{Im[Δε s ]}] is an odd function. To clarify the relation between the degree of PT symmetry and the potential momentum, we assume a potential for which the real and imaginary parts of Δε s (x,y) have identical distributions, such as a potential with a Gaussian envelope Δε s (x,y) = [Δε sr0 · cos(p 0 x + q 0 y) + iΔε si0 · sin(p 0 x + q 0 y)] · exp(-(x 2 +y 2 )/(2σ 2 )), where both Δε sr0 and Δε si0 are real values, and Δε sr0 = Δε si0 at the EP. Figure 2 shows the potential momentum in each regime with PT symmetry. Whereas the spectrum of the potential momentum satisfies causality at the EP (Fig. 2a), the potentials of the regimes before (Fig. 2b, with in-phase spectra) and after (Fig.  2c, with out-of-phase spectra) the EP break causality. Therefore, a critical result is that the concept of PT symmetry breaking is equivalent to a phase transition from an in-phase potential momentum spectrum to an out-of-phase potential momentum spectrum (Fig. 2b vs. 2c), which are separated by the causal phase (Fig. 2a). This interpretation provides an intuitive understanding of the degree of PT symmetry, which is not restricted to the relative magnitude between the real and imaginary parts of the potentials [16,19] but results from a direct spectral analysis of the 'degree of the causality' for the real-valued potential momentum. Fig. 2. Potential momentum spectra for degrees of PT symmetry: (a) at the EP (Δεsr0 = Δεsi0), (b) before the EP (Δεsr0 > Δεsi0), and (c) after the EP (Δεsr0 < Δεsi0). Lower figures illustrate the corresponding coupling between momentum states for each degree. Green (purple) solid line denotes the momentum state that corresponds to the 'cause' ('effect'). As shown, causality is only maintained at the EP. Gaussian spectra with σ = 0.25 and p 0 = q 0 = 1 are assumed, without loss of generality.
We now apply Eq. (2) to demonstrate the momentum transition using the one-way coupling potentials in Eq. (3). Without loss of generality, we investigate a case of high-k excitations along the hyperbolic IFC (p,q ≥ 0, Fig. 1c). We assume that a y-axis-invariant wave is incident on the oneway potential (x ≥ 0) from the left side (k x0 > 0), as illustrated in Fig. 3a for the spatial domain. We accommodate potentials of arbitrary shape by discretizing the potential in both the spatial (Fig. 3a) and momentum (Fig. 3b) domains. By setting the y-infinite unit volume V with a deepsubwavelength spatial discretization Δx, the surface integral of Eq. (2) is determined on the S L (x L ) and S R (x R ) surfaces, and the volume integral can be evaluated from the average of the values in S L and S R . The discretization for the momentum states also carries over to the IFC (circles in Fig. 3b) from the phase matching condition. The discretized form of Eq. (2) is then expressed as where the subscript m denotes the m-th momentum state of (k xm ,k ym ); n = 1 is the incident state; p = k xm -k xn ; q = k ymk yn ; Δp n = k x(n+1) -k xn ; and Δq n = k y(n+1) -k yn . Equation (4) can be used to perform a serial calculation for the integral of the envelope, starting from the left boundary (a more detailed procedure for the serial calculation is given in [23]). As a result of the causality condition that is imposed on the Δε pq , only the eigenstates on the bounded region of the IFC (blue circles in Fig. 3b) participate in the coupling to the (k x ,k y ) state. Fig. 3. Discretization of (a) spatial and (b) momentum domains for the derivation of Eq. (4). S L and S R present the left and right surfaces, respectively, of the unit volume V (colored in blue). A wave with a unit amplitude (at the (k x0 ,k y0 ) state, shown by red arrows in (a)) is incident on the left side of the spatial domain. Circles in (b) represent discretization in momentum space. Blue circles denote states that participate in the coupling to the calculated state (k x ,k y ).
The high-k excitation process is shown in Fig. 4. For general curvilinear IFCs, the transition through the multiple linear-path coupling is adopted (Fig. 4a). In this specific example, we assume a potential modulation for five realvalued momentum spectra (Fig. 4b). A finite bandwidth is used for each spectrum to accommodate quasi-phase matching. Figures 4c and 4d show the amplitude and phase of the complex potential given by Eq. (3) and present the confinement in space from the finite bandwidth and the mixed phase evolution from the multi-harmonics. Figures 4e and 4f show the results for high-k excitations in momentum space at the point x = 100λ 0 for different bandwidths of the potential momentum spectra. The variation in the effective index along the x-axis is illustrated in Fig. 4g, using n eff (x) = ∫∫n(k x ,k y )•|ψ [kx,ky] (x)| 2 dk x dk y / ∫∫|ψ [kx,ky] (x)| 2 dk x dk y and the excited envelopes at each x value. For all cases, successful multistage delivery of optical energy to the high-k regime is observed and is more efficient for larger modulation depths (Fig. 4g). Note that even the higherk states are excited above the targeted final (5 th ) state (which is shown as a black dotted line in Fig. 4g). This behavior results from the linear asymptotic behavior of the hyperbolic IFC (k y ~ (-ε x /ε y ) 1/2 •k x ), which alleviates the phase matching condition in the high-k regime. This result indicates that a perpetual transition to higher-k states becomes possible for the hyperbolic IFC provided that the minor phase-mismatch is compensated for by the bandwidth of the modulation spectra, as shown by the superior excitations in the high-k regime with broadband potentials (solid vs. dotted lines after the arrows in Fig. 4g).   (Fig. 4a, with the nonzero 4 th quadrant of (p,q) space), the transition is allowed only within the 1 st quadrant of the IFC, as can be clearlyobserved by comparing incidences for n y > 0 (red squares) with those for n y < 0 (grey squares). The beam trajectories in Fig. 4b are calculated from Eq. (4) and confirm that strong, selective beam steering only occurs in the cases with lateral positive momentum components k y0 , as predicted. In contrast to the high-k excitation example with asymptotic behavior (Figs. 3e-3g), in this case, we note the convergence toward the final k state that selectively facilitates collimating behavior (blue solid lines, angular bandwidth from 44° to 17°). Beam trajectories (solid lines) for different incidences of n y > 0 (blue dotted lines) and n y < 0 (orange). ε x (x,y)/ε x0 = ε y (x,y)/ε y0 = 0.2 and σ x,y = k 0 /200. All other parameters of the potential are the same as those given in Fig. 4.
In conclusion, in this study, we develop and analyze the unidirectional excursion of excited states along the kcontinuum to expand optical potentials into the complex domain of generalized spectra. We derive the condition for one-way k-transitions within the context of optical and potential momentum interactions and expand the interpretation of a singular PT-symmetric potential as the causal potential of real-valued momentum spectra. Our approach offers a fundamental understanding of the degree of PT symmetry in terms of causal momentum interactions and enables us to tailor optical evolution in k-space using oneway complex potentials that are directly designed in momentum space. We have demonstrated novel applications, such as excitations in the inaccessible k regime and nonreciprocal beam steering and collimation. A further application for complex potentials would be to apply causality to the frequency ω domain, i.e., time-varying complex potentials could be used to produce temporal non-Hermitian dynamics, and employing the relation between causality and a complex potential momentum could result in a novel research subject: the physical interpretation of non-PT-symmetric potentials [25] with real spectra. [1] Linearity holds in the classical limit of ħ0, neglecting vacuum polarization by QED effects. See, e.g., J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
We apply the spatial discretization of y-infinite unit cells (Fig. 3a) and the causality condition for the potential momentum (p,q ≥ 0) to the integral form of the coupled mode equations; thus (Eq. (2)) becomes We apply the subwavelength limit to evaluate the volume integral from the average of the values in S L and S R as (S6) For discretization in momentum space with sufficiently small Δk (Fig. 3b), Eq. (S5) can be approximated by the following equation for the m th momentum state: where p = k xm -k xn , q = k ym -k yn , Δp n = k x(n+1) -k xn , Δq n = k y(n+1) -k yn , and n denotes each momentum state before the m th state. Because the spatial boundary condition is applied to the left side of the structure, the calculation is performed from the left to the right side in space. Additionally, because of the causality condition, n has the lower limit of n = 1 which is defined by the momentum state of an incident wave (k x0 ,k y0 ), and the calculation in momentum space should be performed from n = 1 to n = m. Therefore, we separate the unknown and known integral terms in Eq. (S7) as We can now perform the serial calculation with the boundary condition ∫ψ 1 (x=0)dy. At the fixed point (x = x f ), all of the momentum states can be obtained from Eq. (S8) in the order ∫ψ 1 (x=x f )dy, ∫ψ 2 (x=x f )dy, …, ∫ψ m (x=x f )dy. These results are applied to calculate the states at the next position (x = x f + Δx). For a unity incidence wave on the boundary, the density of the envelope is directly proportional to the integral of the density of the envelope.