Abstract
In optical experiments onesided reflectionless (ORL) and coherent perfect absorption (CPA) are unusual scattering properties yet fascinating for their fundamental aspects and for their practical interest. Although these two concepts have so far remained separated from each other, we prove that the two phenomena are indeed strictly connected. We show that a CPA–ORL connection exists between pairs of points lying along lines close to each other in the 3D spaceparameters of a realistic lossy atomic photonic crystal. The connection is expected to be a generic feature of wave scattering in nonHermitian optical media encompassing, as a particular case, wave scattering in paritytime (PT) symmetric media.
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Introduction
Scattering from complex potentials and the associated nonHermitian Hamiltonians^{1} are usually introduced to describe dissipation or decay processes in open systems. Likewise light wave propagation phenomena through media with complex susceptibilities are genuine realizations of scattering from localized nonHermitian potentials and provide a clear illustration of how Hermitian and nonHermitian processes differ from one another. The optical scattering matrix S fully governs the propagation of light and, in particular, onesided reflectionless (ORL) scattering of light waves impinging from “one” direction^{2,3,4,5,6} can be associated with a nonHermitian degeneracy^{7} of the scattering matrix (also known as an exceptional point^{8}). More intriguing phenomena appear, however, when coherent waves impinge on “both” sides of a complex potential^{9}. Among them, coherent perfect absorption (CPA)^{10}, which refers to complete absorption of both incident waves, is being extensively investigated^{11,12,13,14,15,16,17,18}. The interest in CPA stems not only for fundamental reasons^{10,11,13}, since it can be interpreted as the timereversed counterpart of lasing and related to paritytime (PT) symmetry^{19}, but also in view of its potential applications. Such efforts have spurred investigations and experiments in various areas that span, among others, absorption enhancement^{20}, perfect energy feeding into nanoscale systems^{21}, intersubband polaritons^{22}, slow light waveguides^{23}, graphenebased perfect absorbers^{24,25,26,27}, and Fano resonant plasmonic metasurfaces^{28}.
The concepts of onesided reflectionless and coherent perfect absorption have remained so far separated from each other, probably because of the lack of suitable physical systems in which both features would be accessible. Here, we show how a lossy medium that exhibits ORL can in general also exhibit CPA. The connection is general, not restricted to PT symmetric media and could be easily observed in a realistic 1D lossy medium through smooth deformations of the system’s externally tunable parameters. We further argue how this connection, intrinsic to the structure of nonHermitian degeneracies of scattering matrix S, can actually be extended to all points of a CPAline. Such a line is a novel topological structure of nonHermitian optical media predicted to occur next to a ORLline. Although there has been a number of recent advances in each of these areas of research, particularly restricted to the case of PT symmetric media requiring a balance of loss and gain^{29,30,31,32,33,34}, onesided reflectionless and coherent perfect absorption – taken together – may lead to a more complete understanding of nonHermitian optics in a large class of materials where absorption plays a key role for applications. Photodetectors, photovoltaics and nonreciprocal optical devices just to mention a few instances. The connection we present here is fairly general, hinges on nonHermitian scattering degeneracies with common notions from quantum mechanics and, though clearly relevant to optics in view of oneway mirrors, cloaks of invisibility and coherent laser absorbers, may well be relevant to unusual phenomena recently observed for acoustic waves^{35,36,37,38,39,40,41}.
ORL and CPA
The scattering properties of a 1Dmedium are fully determined by the complex amplitudes t = t_{L} = t_{R}, r_{L} and r_{R} respectively for (reciprocal) transmission and reflection upon incidence from the left (L) or from the right (R). ORL means that r_{L} = 0 with r_{R} ≠ 0 (or vice versa). The CPA condition corresponds, instead, to a specific configuration of input beams, incident at the same time one from the left and one from the right with a definite phase relationship, which are completely absorbed by the sample. Thus, for this configuration of input beams, the output beams to the right and to the left are both vanishing. This means that the CPA input beams represent an eigenvector of the scattering matrix S with eigenvalue zero. As discussed below, the CPA condition can finally be stated as t^{2} = r_{R}r_{L}, i.e. det S = 0^{10} (see Eq. (3)).
Thus, the main focus of the work is how to connect in general the two conditions r_{L} = 0 (ORL) and t^{2} = r_{R}r_{L} (CPA) upon smooth deformations of medium’s external driving parameters. More specifically, for a lossy 1Dphotonic crystal, the scattering properties near Bragg reflection can be described^{5,4} by the following model susceptibility
with χ_{0}, , and w being non negative real parameters, a the crystal period and the phases {α, β} defined within the interval [0, π]. The real part of the spatially independent background susceptibility is ignored for simplicity as it plays no significant role, while its imaginary part χ_{0} should be large enough with respect to to have everywhere a lossy medium, i.e., . In this rather generic model, the ORL condition (r_{L} = 0) is simply attained when w = 0^{5,4}, in which case the real and imaginary parts of the susceptibility modulation χ(z) − iχ_{0} are spatially shifted by π/2 and satisfy the spatial KramersKronig relations^{6}. The reflection and transmission of a light beam with a wavevector can be described on the basis of a minimal coupledmode model accounting for Bragg scattering in a sample of length L ≫ a, as usual. Then, the CPA condition is attained when
where (with Re[η] > 0 due to losses). The last term in Eq. (2) holds when e^{ηL} ≫ 1 (t ≪ 1) and w ≪ 1, and this is precisely the regime we are interested in as it can occur near a ORL point in a lossy medium. It thus appears that, while the parameter α is immaterial, the CPA condition can in general be satisfied only if β can be tuned at will within the whole interval [0, π], regardless of the value of w. In fact, although , need not be small at the CPA point as kL ≫ 1.
Though solidstate photonic structures may be considered^{4}, coherentlyprepared multilevel atoms^{5,42} are attractive for exploring nonHermitian optics, because of the easy reconfiguration of the scattering process through well established control techniques enabled by electromagnetically induced transparency (EIT)^{43}. In fact, the realization of atomic platforms to investigate nonHermitian models is currently a very active experimental endeavor^{44,45}. We consider the realistic atomic system of Fig. 1, which provides an implementation of the model of Eq.(1). The photonic crystal consists of cold atoms coherently driven by a nearresonant probe beam (Ω_{p}, Δ_{p} ≈ 0), a resonant coupling beam (Ω_{c}, Δ_{c} = 0) and an fardetuned dressing field (Ω_{d},Δ_{d} ≫ 0). The latter has both a travelingwave (TW) and a standingwave (SW) components with opposite detunings and induces on level 2〉 a dynamic shift , where and the phase shift 2ϕ_{d} is relative to the optical lattice modulating the atomic density. As a matter of fact, by adjusting only three of the above independent control parameters, namely {Δ_{p}, δ_{d0}, ϕ_{d}}, it is possible to identify scattering processes for which the existence of the CPA–ORL connection can be proven. More specifically, this is done by solving the density matrix equations for the atomic level configuration of Fig. 1 whose matrix elements will depend, among other parameters kept fixed here as in Fig. 6 of ref. 45, on the three parameters (Δ_{p}, δ_{d0}, ϕ_{d}) (See sect. II of ref. 45). For each choice of these three experimentally tunable parameters, we numerically compute the full susceptibility χ(z), which can be cast in the form of Eq. (1) when its higher order Fourier components are disregarded. From χ(z) we then directly obtain through transfer matrix calculations^{46} the scattering amplitudes t, r_{L} and r_{R} that identify a specific scattering process.
A relevant sets of ORL points (r_{L} = 0) and the associated CPApoints (t^{2} = r_{R}r_{L}) are reported in the 3D parameter space {Δ_{p}, δ_{d0}, ϕ_{d}} of Fig. 2. A CPAline lying roughly parallel to an ORLline is shown there. Hence, we can access a CPApoint starting from a ORLpoint essentially by adjusting the parameter δ_{d0}. The reason is simply that (i) the transmission amplitude t is always small in our lossy atomic medium and (ii) the reflection amplitudes r_{L} and r_{R} are more sensitive to δ_{d0} than Δ_{p} at a fixed value of ϕ_{d}. A range of ϕ_{d} values centered at ϕ_{d} = π/4 is shown, being our system periodic in ϕ_{d} with period π, while varying ϕ_{d} from ϕ_{d} = π/4 to 3π/4 (or to −π/4) simply changes the reflectionless behavior from the “left” into reflectionless from the “right”. Notice also that the CPAlines and ORLlines are symmetric under the simultaneous changes ϕ_{d} → π/2−ϕ_{d} and Δ_{p} → −Δ_{p}. We can always find an isolated CPApoint associated to a nearby isolated ORLpoint through cuts along {Δ_{p}, δ_{d0}}planes as shown in Fig. 3. Figure 4 illustrates further examples of how ORLpoints and the associated CPApoints are computed. ORLpoints are characterized by r_{L} = 0 and are here obtained by solving the two real equations Re[r_{L}] = 0 and Im[r_{L}] = 0. In the neighborhood of a solution both Re[r_{L}] and Im[r_{L}] change sign and their product changes sign in four alternating sections (i.e., deformed quadrants) of the {Δ_{p}, δ_{d0}}plane as shown in Fig. 4(a–c,e–g). This corresponds to the fact that the phase of r_{L} varies by 2π when a ORLpoint is encircled in the {Δ_{p}, δ_{d0}}plane, which embodies the freedom of choice of β in Eq. (2), and is a key point as discussed below. CPApoints, characterized by t^{2} = r_{L}r_{R}, are illustrated instead in Fig. 4(b–d,f–h) as minima of the function t^{2} − r_{L}r_{R}.
Discussion
The CPA – ORL connection can also be assessed in more general terms starting from the twoports scattering process,
where the S matrix relates the outgoing (electric) field amplitudes and to the incoming (electric) field amplitudes and (see Fig. 1a). The eigenvalues and eigenvectors of S are obtained through the last term in Eq. (3). It is here worth noting that we have chosen one of the most common representations of the S matrix, the other one having instead r_{L} and r_{R} on the diagonal. While the scattering is solely determined by the measurable complex amplitudes t, r_{L} and r_{R} and all physical results are independent of which S matrix representation is used, the specific choice of S in Eq. (3) is appropriate to prove the CPA – ORL connection, where the ORL condition is in this case directly related to a nonHermitian degeneracy (or exceptional point) of S, as we illustrate in the following.
In general, S is nonHermitian, its eigenvalues
are complex and the (unnormalized) eigenvectors are not orthogonal. NonHermitian degeneracies of S occur when the eigenvalues merge into one another [Fig. 5(a–d)] and the eigenvectors coalesce into a single state^{7}, being the S matrix no longer diagonalizable. The two coalescing eigenvalues are analytically connected by a squareroot branchpoint, with associated Riemann sheets, and are physically associated with unidirectional reflectionless scattering states occurring when r_{L} = 0 (or r_{R} = 0)^{4,5}. For a nonHermitian matrix, degeneracies are of codimension two, that is points in a twoparameter space (NHDpoint) and curves in a threeparameter space (NHDline). Meanwhile, CPA occurs when either one of the two eigenvalues or vanishes [Fig. 5(a–d)] along with the determinant of S (this condition is independent of the specific choice of S matrix representation). The corresponding eigenvector describes a perfect absorption state^{10} with amplitudes and phases of the incoming fields from the left and from the right precisely chosen so that no outgoing light intensity can be observed^{13,18}.
We start providing an intuitive illustration of how CPA and ORL are connected with one another in the particular, but important, case for which (i) the reflection phases are such that ϕ_{L} + ϕ_{R} = {0, π} and (ii) the transmission amplitude t is real. The corresponding eigenvalues are either real or complex conjugate in pairs depending on whether the two phases add up to 0 or to π [Fig. 5(c,d)]. Thus (half) sum of the two eigenvalues represents t and can be depicted, as we move in the parameter space toward degeneracy, by a vector whose magnitude decreases along the real axis of Fig. 5(e) for decreasing transmission. So does (half) difference of the two eigenvalues representing the geometric mean of r_{L} and r_{R}, which can be depicted by a vector parallel to the imaginary axis. As we move through degeneracy, the eigenvalues sum will keep decreasing but their difference will increase after moving away from zero (degeneracy) [Fig. 5(f)] owing to the intrinsic bifurcation (topological) structure of the branchpoint. Hence there will always be a point where sum and difference will be equal (to each other), i.e., [Fig. 5(g)]. It is worth noting that under the conditions (i.) and (ii.) an Hermitian invertible transformation η exists indeed for which the adjoint of the (nonHermitian) scattering matrix S satisfies S^{†} = ηSη^{−1}, i.e., S is pseudoHermitian^{48}. The reverse is also true and hence the pseudoHermiticity of S is the basic mathematical structure responsible for the direct connection between the ORL and the CPA point, at least for the specific spectrum of S shown in Fig. 5(c,d). Note that this particular case – realized in the alloptically tunable atomic system of Fig. 1 simply setting Δ_{p} = 0 – is essentially analogous to a PT symmetric one, even though our system is always lossy, both before and after the NHD point.
Yet, a CPApoint can be typically found in the vicinity of a ORLpoint under more general conditions and, in particular, without restricting ourselves to pseudoHermiticity. For definiteness we take the NHDpoint at assuming, without loss of generality, that around this point r_{R} and t are nonvanishing. For lossy media we may further take t ≪ 1, with r_{R} being in general on the order of unity^{5}. The perfect absorption condition is satisfied when r_{L}r_{R} = t^{2}, i.e. when
are both satisfied, implying that r_{L} and arg(r_{L}) should be independently adjusted (just as the phase β in Eq. (2) should be tuned at will, regardless of the value of w). Note that the CPA conditions in Eq. (5) generalize those given above for the pseudoHermitian case, and are only restricted by the requirement that r_{L} be small at the CPApoint, which occurs when this point is associated to a nearby ORLpoint. In general, we do expect t^{2}/r_{R} to be smoothly varying in the vicinity of this point while arg(r_{L}) can be varied at will when the parameters defining the system are smoothly changed so to encircle the ORLpoint, i.e. the NHD of S^{5}. A simple geometric illustration of this property similar to that provided in Fig. 5(e–h) is not so viable in the general, non pseudoHermitian case (such as that of Fig. 5(a,b)); yet, a direct analytical argument shows that r_{L} and arg(r_{L}) can be independently adjusted when encircling the ORLpoint.
In a typical scattering process, r_{L} depends smoothly on several experimental parameters. We consider here how the real (u) and the imaginary (v) parts of r_{L} vary near the ORLpoint as a function of only two of these parameters, keeping all other ones fixed. In terms of these two parameters, say x and y, one has
where the partial derivatives u_{x} = ∂u/∂x, u_{y} = ∂u/∂y, v_{x} = ∂v/∂x, and v_{y} = ∂v/∂y are evaluated at the ORLpoint taken at (x, y) = (0, 0). Note that it is not needed to combine x and y into a single complex parameter x + iy as r_{L} is not assumed to be holomorphic here. When u_{x}v_{y} − v_{x}u_{y} ≠ 0, it is always possible to select x and y to obtain any required values of arg(r_{L}) and of r_{L}, provided the latter is small enough that higher order terms in Eq. (6) are indeed negligible. Thus, under typical circumstances we expect a CPA and a ORL points to be close to each other in a scattering process from lossy media with t small. For example, in Fig. 3 the case ϕ_{d} = 0.25 × π (pinkarrow) represents changes in the scattering matrix as one moves from its NHDpoint () to its CPAcompanion (), namely for a pseudoHermitian matrix (Δ_{p} = 0). Similarly, the case ϕ_{d} = 0.15 × π (bluearrow) represents changes as one moves from the NHDpoint () to its CPAcompanion (), namely for the general nonHermitian case. Actually, the case in which u_{x}v_{y} − v_{x}u_{y} = 0 cannot be excluded. Assuming that (u_{y}, v_{y}) ≠ (0, 0) and writing (u_{x}, v_{x}) = μ(u_{y}, v_{y}) with μ real, one then has
which implies that, while r_{L} = Δr_{L} can be varied, arg(r_{L}) = arg(Δr_{L}) is fixed because . In this case, we expect to find no CPApoint in the vicinity of a ORLpoint when all other parameters are kept constant. Clearly, also when higher order terms in the above expansion of r_{L} become important as for instance in the peculiar case where all partial derivatives in Eq. (6) are vanishingly small, the occurrence of the CPA point is not granted.
Defining ρe^{iθ} ≡ −r_{L}/t, the scattering matrix eigenvector at the CPApoint, where the corresponding eigenvalue vanishes, can be eventually written as,
The complex quantity ρe^{iθ} is examined in Fig. 6 both for the pseudoHermitian and nonHermitian cases. At the CPApoint, the eigenvector’s components scale as , with ρ ≪ 1 according to Eq. (5). Both modulus (ρ) and phase (θ) of the (small) incoming field from the right, with respect to the incoming field from the left (i.e, the nearly reflectionless side), should be properly chosen to observe the typical perfect absorption behavior. Since the CPApoint considered here is associated to a ORL point, in general, perfect absorption requires very unbalanced incoming fields. As a matter of fact, the characteristic destructive interference conditions leading to perfect absorption for light scattering in both directions occur here for very unbalanced right and left reflectivities r_{R} ≫ r_{L}. In turn, a tiny input field from the right is sufficient to ensure that the outgoing field to the left vanishes, while a large input field from the left is necessary to destructively interfere with the reflected field from the right side. This CPA configuration provides, in particular, a highcontrast reflectivity control of a test beam incident from the right via a pump beam incident from the left.
Conclusions
A new insight into the nonHermitian optics of a familiar class of lossy photonic crystals is here discussed. Through continuous deformations of the scattering matrix S around a onesided reflectionless (ORL) point, a CPA point can be typically attained. Nearby pairs of ORL and CPA “points” or even “lines” appear, respectively, through controlling the crystal 2D or 3D parameter space. In such cases, the CPA scattering states associated to ORL points turn out to be significantly unbalanced, indicating a dynamically reversible highcontrast reflectivity control of the input beams. Finally, while the results here presented refer to realistic atomic structures^{44,45}, our general discussion can be easily adapted to atomiclike multilevel centers^{49} in solids, such as NV diamond or rareearthdoped crystals, also allowing for EIT control of light scattering^{50,51}. Hence the optics of photonic crystals is poised to have a privileged place in assessing that not only standard Hermitian models but also a broad set of nonHermitian ones are bound to have physical interpretations.
Additional Information
How to cite this article: Wu, J.H. et al. Coherent perfect absorption in onesided reflectionless media. Sci. Rep. 6, 35356; doi: 10.1038/srep35356 (2016).
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Acknowledgements
Insightful discussions with S. Horsley and M. Berry at the early stage of the work are kindly acknowledged. J.H.W is grateful for the hospitality at Scuola Normale Superiore in Pisa and the support from National Natural Science Foundation of China (No. 61378094, 11534002, and 11674049).
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J.H.W. developed theoretical frameworks and implemented numerical calculations. M.A. and G.C.L.R. conceived the mechanism, analyzed the results, and wrote the manuscript.
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Wu, JH., Artoni, M. & La Rocca, G. Coherent perfect absorption in onesided reflectionless media. Sci Rep 6, 35356 (2016). https://doi.org/10.1038/srep35356
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