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Abstract
Field solutions for a conventional Veselago-Pendry (VP) flat lens with $\epsilon =-\epsilon _0$ and $\mu =-\mu _0$ can be derived based on transformation optics (TO) principles. The TO viewpoint makes it clear that perfect imaging by a VP lens is a consequence of multivalued nature of the particular coordinate transformation involved. This transformation is equivalent to a “space folding” whereby one point in the transformed domain (source point) is mapped to three different points in the physical domain (the original source point plus two focal points). In theory, a VP lens would enable the recovery of the entire range of spectral components, i.e. both propagating and evanescent fields, thus characterizing a “perfect lens”. Such lens, if lossess, is indeed “perfect” for monochromatic waves; however, for any realistic wave packet the space folding interpretation provided by TO makes it clear that a VP lens violates primitive causality constraints, which precludes any practical realization. Here, we utilize complex transformation optics (CTO) to derive generalized Veselago-Pendry (GVP) lenses without requiring a multivalued transformation. Unlike the conventional VP lens, the proposed lenses can fully recover the evanescent spectra under more general conditions that include the presence of (anisotropic) material loss/gain.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Double-negative materials (or left-handed materials) [1] with $\epsilon =-\epsilon _0$ and $\mu =-\mu _0$ have attracted much attention since their experimental verification [2,3]. In particular, they motivated the concept of a “perfect” of flat lens (Veselago-Pendry lens) [4] with the ability to retrieve both the propagating and evanescent wave spectra (the latter through compensatory amplification) and overcome the conventional diffraction limit. However, perfect lensing is only possible in the limit of no losses [5–7]. In practice, the Veselago-Pendry lens must be dispersive and hence lossy in order to to satisfy Kramers-Kronig relations and preserve causality [8–10]. When loss and dispersion are considered, the performance of the Veselago-Pendry lens is significantly degraded [8,11]. Nevertheless, the quest to obtain a “superlens”, has provided much impetus to experimental efforts [12–15].
In parallel with these developments, transformation optics (TO) arose as a tool for deriving metamaterials with unprecedented functionalities [16–22]. TO explores the fact that properly chosen permitivity and permeability tensors can mimic the effect of (hypothetical) deformations of space in field solutions of Maxwell’s equations [23,24]. Interestingly, the VP lens can be derived via TO and interpreted as a space “folding” [9,25–27]. Not only this provides a simple interpretation of perfect lensing but it also makes it clear that the ideal Veselago-Pendry lens implies a multivalued transformation and hence violates causality for any signal with nonzero frequency bandwidth [9,28], thus rendering “perfect lensing” an elusive quest. So the question naturally arises as to whether it is possible to remove the multivalued nature of the TO mapping without affecting the necessary compensatory amplification of the evanescent spectrum.
TO has traditionally involved real-valued spatial coordinate transformations. More recently, there has been an increased interest in complex transformation optics (CTO) [22,29–32], whereby additional degrees of freedom enabled by complex-valued coordinate transformations [23,24] (for field solutions expressed in the Fourier domain, $e^{-i\omega t}$) are used to provide functionalities beyond the reach of conventional TO [22,30–32]. One early example of CTO-derived media are perfectly matched layers used to truncate numerical simulation domains [33–35]. Although perfectly matched layers have been primarily used for specific computational needs, they can provide metamaterial blueprints for reflectionless absorbers as well [24,34,36]. A recent application of CTO can be found in [30–32], where parity-time ($\mathcal {PT}$) and gain media were designed to produce, among other functionalities, directional beams from (complex) point sources [37–39].
In this work, we describe a CTO-based approach to obtain the constitutive properties of a generalized class of Veselago-Pendry (GVP) flat lenses (slabs) which can recover the evanescent spectrum under more general conditions and without implying a multivalued transformation. Similarly to what has been discussed in [25,26], negative refraction emerges naturally by choosing the real part of the coordinate transformation Jacobian to be negative. Differently from [25,26] though, the presence of a nonzero imaginary part in the coordinate transformation considered here leads to GVP slabs exhibiting anisotropic gain or loss, according to the sign choice (positive or negative) of the imaginary part. Akin to other types of transformation media, the obtained GVP slabs are impedance matched to free space and exhibit doubly anisotropic (uniaxial) constitutive parameters but no birefringence. Unlike conventional VP lens, where (isotropic) losses have a severe effect on performance (see Fig. 2(b)), a GVP lens perfectly retrieves the evanescent wave spectra (see Fig. 2(a)) irrespective of the induced anisotropic loss/gain level. On the other hand, the propagating spectra is only partially recovered in the presence of anisotropic losses. Thus “perfect” lensing is still not achievable in this case, albeit the effect of (anisotropic) losses on performance on GVP slabs than the effect of (isotropic) losses in VP slabs. In a gain media GVP slab, the propagating modes are amplified while the evanescent modes are again perfectly recovered (see Fig. 2(a)). Because of the complex nature of the coordinate transformation, there are actually two focal points inside the slab and two outside the slab, which correspond to branch points in the complex radial distance $\rho '$ [32], as discussed further below.
2. Generalized Veselago-Pendry slabs from CTO solutions
We denote a point in transformed space (with constitutive properties of vacuum) as $\left (x',\;y,\;z\right )$ and the associated point in physical space (flat space with constitutive properties of a CTO-derived transformation medium, see below) as $\left (x,\;y,\;z\right )$. Note that, our definitions of transformed space (“complex-space”) and flat-space coordinates follow the PML literature convention and are reversed with respect to the TO literature. The final material tensors are, of course, independent of convention. We assume the following coordinate transformation $\left (x,\;y,\;z\right ) \rightarrow \left (x',\;y,\;z\right )$:
The parameter $a_x$ controls the phase velocity in such transformation medium. In particular, the choice $a_x<0$ implies negative refraction and produces a GVP slab. On the other hand, $\sigma _x$ controls the wave amplitude. In general, $\sigma _x>0$ implies an absorptive GVP slab and $\sigma _x<0$ a gain GVP slab [23]. The only choices that yield an isotropic material are $a_x = \pm 1$ with $\sigma _x=0$; otherwise, the resulting material parameters from Eq. (2) correspond to doubly anisotropic (doubly uniaxial) media with induced loss (or gain) present along the $x$ direction only. It is immediate to see that the choice $a_x =1$ and $\sigma _x = 0$ corresponds to vacuum and $a_x =-1$ and $\sigma _x = 0$ to the conventional Veselago-Pendry lens.
In what follows, we let
in which case the transformation media corresponds to a lossy media if $b>0$ and to a gain media if $b<0$. The corresponding behavior of $\Im m [x']$ is shown by the dashed red and dash-dotted green curves in Fig. 1, respectively. For $a_x=-1$, one obtains $\Re e[x']$ as shown by the blue curve in Fig. 1 (implying negative refraction, as noted).
Fig. 1. The shaded map indicated the real part of the complex distance $\rho '$ for the transformation given in Eqs. (1) and (3) with $a_x =-1$ and $\sigma _x = 0.2$, with $d=2\lambda$. Superimposed on this $\rho '$ plot, we also show a representation of the complex-valued coordinate $x'$ as a function of $x$ entailed by Eqs. (1) and (3). In particular, the blue line shows $\Re e[x']$ for $a_x=-1$, the red dashed line shows $\Im m[x']$ for $b=0.4 \lambda$, and the green dash-dotted line shows $\Im m[x']$ for $b=-0.4 \lambda$. The source point gives rise to two focal points both inside and outside the transformation region in accordance with the $\sigma _x$ function. Note that while the mapping from $x$ to $\Re e[x']$ is still multivalued, the mapping from $x$ to $x$ is not multivalued anymore.
Under the CTO paradigm, the field solution for a point source in the transformed space can be found by first effecting an analytic continuation in the Green’s function. Assuming a 2D scenario for simplicity, the field produced by a current source $\vec {J'}=\hat {z} \delta \left (x'-x'_s\right ) \delta \left (y'-y'_s\right )$ can be written as [42]:
where $H_0^{(1)}$ is the Hankel function of first kind and zeroth order and $\rho '$ is the complex distance in the transformed space given by: where $(x'_s,\;y'_s)$ is the (generally complex) source position in the transformed coordinates associated to a source point at $(x_s,\;y_s)$ after the transformation given by Eq. (1). Fig. 1 shows the contour plot of $\Re e \left [\rho '\right ]$ assuming $a_x=-1$ and $\sigma _x=0.2$. The source is assumed to be placed on the left of the slab, at $\left (x_s,\;y_s\right )=\left (-\lambda ,0\right )$. The field behavior on the opposite side of the slab mimics that of a field produced by a complex source point (CSP), as discussed in [32]. The actual fields ${\vec E}$ and sources ${\vec J}$ in physical space can be simply written as [9,16,23] where in order to obtain the correct field solution, the Riemann sheet associated with $\Re e[\rho ']>0$ (the so-called “source-type” solution) is selected [37–39,43,44]. For the GVP lenses, the focal points do not always reside on the same cut ($y=0$) as the source [10]. In order to find the focal points of the GVP lens, we require: From this condition and by assuming $a_x=-1$, we can determine the focal points inside the lens as where $d_1$ is the source distance from the lens boundary and the focal points outside the lens as These focal point locations also coincide with the branch points in the field solutions (see Fig. 1).Assuming an incident spectral component (plane wave) with unit amplitude and ${\vec k} = (k_x,\;k_y,0)$ where $k_y=(\omega ^{2} \mu _0 \epsilon _0 - k_x^{2})^{1/2}$, we can express the transfer function $\tau (k_x; x_{f_2},\;y_{f_2})$ as the resulting spectral component amplitude at the focal point [5,45]. The GVP slab problem considered here constitutes a canonical planarly layered geometry with three layers (two interfaces) that can be easily solved [5,45–47] to find $\tau (k_x; x_{f_2},\;y_{f_2})$ . The solution is further facilitated by the fact that, as noted before, the two interfaces are reflectionless [34].
3. Results and discussion
Figure 2(a) plots the transfer function of GVP slabs with $\sigma _x > 0$ (gain) and $\sigma _x < 0$ (loss), while Fig. 2(b) plots the transfer function of conventional VP slabs with and without losses. It is seen that the evanescent wave spectra is fully recovered at the focal point for both the loss and gain GVP lenses. On the other hand, the propagating spectra is recovered with different amplitudes for loss and gain GVP lenses. As expected, propagating modes decay in loss media and grow in gain media but otherwise these results are quite interesting for two key reasons. First, since the CTO derived media is impedance matched to free space regardless of the (anisotropic) loss/gain levels, i.e. the presence of (anisotropic) loss/gain does not affect the (full) transmission of waves into the GVP lenses. In contrast, the presence of (isotropic) losses in the conventional VP lens (even for small values), distorts the transmission spectra even as the (transmitted) evanescent fields are eventually amplified inside the VP lens. Second, the coordinate transformation implied by the GVP lenses is not multivalued (one to one mapping between $x$ and $x'$) anymore. This means that, at least in principle, GVP lenses can be designed over a non-zero bandwidth.
Fig. 2. Transfer function of a (a) GVP with $\left [\epsilon \right ]= \epsilon _0 \left [\Lambda \right ]$ and $\left [\mu \right ] = \mu _0 \left [\Lambda \right ]$ where $\left [\Lambda \right ]$ is given by Eq. (2) for different $\sigma _x$ values given by Eq. (3), (b) Veselago-Pendry lens with $\epsilon _r=-1+i \sigma$, $\mu _r=-1+i \sigma$, and different loss parameters.
Fig. 3 presents results from the finite element based COMSOL$^{\textrm {TM}}$ software implementing Eq. (2) for a VP and GVP lens with thickness equal to $2\lambda$. Figure 3(a) shows the $\Re e[E_z]$ field distribution for a line source placed at $(x_s,\;y_s)=(-\lambda ,0)$ for a VP lens with $\epsilon _r=-1+i10^{-8}$ and $\mu _r=-1+i10^{-8}$. Figure 3(b) shows the intensity of the electric field cut at $x=x_{f_2}$. The transverse resolution as measured by the full width at half maximum (FWHM) is $0.36\lambda$ in this case. Figure 3(c, d) shows results for $\epsilon _r=-1+i5\times 10 ^{-2}$ and $\mu _r=-1+i5\times 10^{-2}$. In this case, the FWHM is $0.45\lambda$ and the effect of loss is clearly visible. Figure 3(e, f) show results for a loss GVP lenses with $[\epsilon ]=\epsilon _0 [\Lambda ]$ and $[\mu ]=\mu _0 [\Lambda ]$ and $[\Lambda ]$ is given by Eq. (2) where $a_x=-1$ and $\sigma _x=5\times 10 ^{-2}$ (lossy GVP slab). Figure 3(g, h) shows results for a gain GVP slab with $a_x=-1$ and $\sigma _x=-5\times 10 ^{-2}$. Note that, although we have used equivalent level of loss/gain parameter in the VP and loss/gain GVP lenses, both loss and gain GVP lens exhibits cross-range resolution similar to the ideal (lossless) VP lens, showing that a more effective recovery of the evanescent spectrum is possible. It should be noted that the fields do not decay monotonically inside the lossy GVP slab due to the amplification of evanescent waves. While propagating waves are mapped to decaying waves, evanescent waves are amplified to some extent [24]). On the contrary for gain GVP lens both propagating and evanescent waves are amplified as traveling inside the GVP slab. These conclusions can be observed in the field plots of Fig. 3(e, g). More quantitatively, Fig. 4 shows the comparison of four above cases in the horizontal cut at $y=0$ confirming our conclusion. The ripples (side lobes) observed in the numerical field solution are due to transition boundary and lateral truncation effects (finite size slab). Moreover, it is difficult for the finite element solution to precisely replicate the very sharp peaks present in the exact solution due to the spectral cutoff caused by the finite element discretization [5,45].
Fig. 3. $\Re e[E_z]$ field distribution due to a point source placed next to different slabs and the corresponding field distribution along the $x=x_{f_2}$ cut with FWHM value. (a, b) VP slab with $\epsilon _r=-1+i10^{-6}$ and $\mu _r=-1+i10^{-6}$. (c, d) VP slab with $\epsilon _r=-1+i5\times 10^{-2}$ and $\mu _r=-1+i5\times 10^{-2}$. (e, f) GVP slab based on Eq. (3) with $a_x=-1$ and $\sigma _x=b/d$ with $b=0.1\lambda$ (g, h) GVP slab based on Eq. (3) with $a_x=-1$ and $\sigma _x=b/d$ with $b=-0.1\lambda$.
Fig. 4. Horizontal cut ($y=0$) for the cases given in Fig. 3. Note the field levels both inside and outside of the lens region. As expected for gain GVP (dotted magenta) the field intensity is higher than lossless (approximately) VP case (solid blue) whereas the loss GVP (dash-dotted green) is lower than the same case. As the loss level of VP lens is increased, the field intensity inside and outside (dashed red) the slab quickly decays.
4. Concluding remarks
In summary, we have shown that generalized Veselago-Pendry (GVP) lenses can be derived based on CTO. The proposed GVP lenses bear additional parameter choices for controlling the field distribution both inside and outside the slab compared to a conventional VP slab. GVP lenses can, in fact, perfectly recover the evanescent spectrum of waves while enabling near perfect recovery of the propagating spectrum. In addition, the GVP lenses in general do not imply a multivalued coordinate transformation, which opens the possibility of realization over a non-zero frequency bandwidth. The realization of the proposed GVP lens should not be fundamentally more difficult than that of other anisotropic metamaterials with losses but a more challenging aspect would be the realization of the proposed GVP lens with anisotropic loss/gain. An enlarged parameter space should also facilitate efforts towards the eventual fabrication of such negative index structures by making use, for example, of proper dispersion engineering of the both real and imaginary parts of the constitutive tensors [48–51].
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