PTMMetamaterials via Complex-Coordinate Transformation Optics

We extend the transformation-optics paradigm to a complex spatial coordinate domain, in order to deal with electromagnetic metamaterials characterized by balanced loss and gain, giving special emphasis to parity-time (PT) symmetric metamaterials. We apply this general theory to complex-source-point radiation and anisotropic transmission resonances, illustrating the capability and potentials of our approach in terms of systematic design, analytical modeling, and physical insights into complex-coordinate wave objects and resonant states. Comments Castaldi, G., Savoia, S., Galdi, V., Alù, A., & Engheta, N. (2013). PT Metamaterials via Complex-Coordinate Transformation Optics. Physical Review Letters, 110(17), 173901. doi: 10.1103/PhysRevLett.110.173901 © 2013 American Physical Society This journal article is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/630 PT Metamaterials via Complex-Coordinate Transformation Optics Giuseppe Castaldi, Silvio Savoia, Vincenzo Galdi,* Andrea Alù, and Nader Engheta Waves Group, Department of Engineering, University of Sannio, I-82100 Benevento, Italy Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78712, USA Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA (Received 29 October 2012; revised manuscript received 18 January 2013; published 23 April 2013) We extend the transformation-optics paradigm to a complex spatial coordinate domain, in order to deal with electromagnetic metamaterials characterized by balanced loss and gain, giving special emphasis to parity-time (PT ) symmetric metamaterials. We apply this general theory to complex-source-point radiation and anisotropic transmission resonances, illustrating the capability and potentials of our approach in terms of systematic design, analytical modeling, and physical insights into complexcoordinate wave objects and resonant states. DOI: 10.1103/PhysRevLett.110.173901 PACS numbers: 42.25.Bs, 11.30.Er, 42.70. a Balanced loss-gain artificial materials have elicited growing attention in optics and photonics, mostly inspired by the emerging parity-time (PT ) symmetry concept, which was originally introduced in connection with quantum physics [1] (see Ref. [2] for a comprehensive review). Against the traditional axioms in quantum mechanics, Bender and co-workers [2] proved that even nonHermitian Hamiltonians may exhibit entirely real energy eigenspectra, as long as they commute with the combined PT operator and share the same eigenstates. This implies, as a necessary condition on the quantum potential, Vð rÞ 1⁄4 V ðrÞ, with r denoting a vector position and * indicating complex conjugation. In view of the formal analogies between Schrödinger and paraxial Helmholtz equations, the above concepts and conditions may be straightforwardly translated to scalar optics and photonics scenarios, with complex-valued refractive-index profiles nð rÞ 1⁄4 n ðrÞ playing the role of the quantum potential. Such symmetry condition cannot be found in natural materials, but it may be engineered within current metamaterial technology, with a judicious spatial modulation of optical gain and losses (either along or across the propagation direction). Besides providing convenient experimental test beds for PT -symmetryinduced quantum-field effects that are still a subject of debate, PT -symmetric metamaterials constitute per se a very intriguing paradigm, as the complex interplay between losses and gain may give rise to a wealth of anomalous, and otherwise unattainable, light-matter interaction effects that extend far beyond the rather intuitive loss (over)compensation effects [3]. These include, for instance, double refraction [4], power oscillations [4,5], spontaneous PT -symmetry breaking [6,7], beam switching [8], absorption-enhanced transmission [6], effectively nonreciprocal propagation [9–15], spectral singularities [16], and coherent perfect absorption [17,18], with perspective applications to new-generation optical components, switches, lasers, and absorbers. In this Letter, we show that the transformation optics (TO) framework [19,20] may be extended, via complex analytic continuation of the spatial coordinates, in order to deal with PT -symmetric metamaterials. This extension brings along the powerful TO ‘‘bag of tools,’’ already applied successfully to a wide variety of field-manipulating metamaterials [21], in terms of systematic design, analytical modeling, and valuable physical insights. Our approach may be related to recent efforts in applying the coordinatetransformation methods to quantum mechanics in order to generate classes of exactly solvable PT -symmetric potentials (see, e.g., Refs. [22,23] and references therein). For simplicity, we start by considering an auxiliary vacuum space with Cartesian coordinates r0 ðx0; y0; z0Þ, where time-harmonic [ expð i!tÞ] electric (J0) and magnetic (M0) sources radiate an electromagnetic (EM) field (E0, H0). We then consider a coordinate transformation r0 1⁄4 FðrÞ (1) into a new curved-coordinate space. By relying on the covariance properties of Maxwell’s equations, the TO framework [20] allows for a ‘‘material’’ interpretation of the field-manipulation effects induced by the coordinate transformation in (1), in terms of a new set of sources (J, M) and fields (E, H) residing in a flat physical space r ðx; y; zÞ filled up by an inhomogeneous, anisotropic ‘‘transformation medium’’ (characterized by relative permittivity and permeability tensors " and , respectively) that are related to the original quantities as follows fE;HgðrÞ1⁄4 TðrÞ fE0;H0g1⁄2FðrÞ ; (2a) fJ;MgðrÞ1⁄4det1⁄2 ðrÞ 1ðrÞ fJ0;M0g1⁄2FðrÞ ; (2b) "ðrÞ1⁄4 ðrÞ1⁄4det1⁄2 ðrÞ 1ðrÞ TðrÞ: (2c) In (2), @ðx0; y0; z0Þ=@ðx; y; zÞ indicates the Jacobian matrix of the transformation in (1), while the symbol det and the superscripts 1 and T denote the determinant, PRL 110, 173901 (2013) P HY S I CA L R EV I EW LE T T E R S week ending 26 APRIL 2013 0031-9007=13=110(17)=173901(5) 173901-1 2013 American Physical Society the inverse, and the inverse transpose, respectively. From (2c), it is evident that, in order for the resulting transformation medium to exhibit loss and/or gain, the coordinate transformation in (1) must be complex valued. Complex-coordinate extensions of TO have already been explored in connection with single-negative transformation media [24] and field-amplitude control [25]. In the present investigation, although the framework can deal in principle with general (asymmetrical, unbalanced) loss-gain configurations, we focus on transformation media characterized by balanced loss and gain obeying the PT -symmetry conditions. First, it can be shown (see Ref. [26] for details) that the necessary condition nð rÞ 1⁄4 n ðrÞ, usually considered in the scalar case to achieve PT symmetry, can be generalized to our vector scenario as: "ð rÞ 1⁄4 " ðrÞ [or, equivalently, ð rÞ 1⁄4 ðrÞ]. From (2c), we observe that such conditions are automatically fulfilled if the coordinate transformation in (1) is chosen so that