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 [57]. In practice, the Veselago-Pendry lens must be dispersive and hence lossy in order to to satisfy Kramers-Kronig relations and preserve causality [810]. 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 [1215].

In parallel with these developments, transformation optics (TO) arose as a tool for deriving metamaterials with unprecedented functionalities [1622]. 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” [9,2527]. 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) [22,2932], 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 [22,3032]. One early example of CTO-derived media are perfectly matched layers used to truncate numerical simulation domains [3335]. 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 [3032], where parity-time ($\mathcal {PT}$) and gain media were designed to produce, among other functionalities, directional beams from (complex) point sources [3739].

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 $\left (x,\;y,\;z\right ) \rightarrow \left (x',\;y,\;z\right )$:

$$x'(x)= \left\{ \begin{array}{lll} x & \textrm{if} & x \leq 0 \\ s_x x & \textrm{if} & 0 \leq x \leq d \\ x+(s_x-1)d & \textrm{if} & x \geq d \end{array} \right.$$
where $s_x = a_x + i \sigma _x$, and $d$ is the thickness of the transformation media slab. For simplicity, both $a_x$ and $\sigma _x$ are assumed to be uniform within the slab. Based on TO principles, the effect of such coordinate transformation can be mimicked in physical space by material tensors (transformation medium) $\left [\epsilon \right ] = \epsilon _0 \left [\Lambda \right ]$ and $\left [\mu \right ] = \mu _0 \left [\Lambda \right ]$ with
$$\left[\Lambda\right]=\textrm{diag}\left\{1/s_x,\;s_x,\;s_x\right\}$$
where $\left [\Lambda \right ] = \textrm {det}\left ([S]^{-1}\right ) \left [S\right ] \cdot \left [S\right ]^{T}$ and $[S]$ is the Jacobian of the coordinate transformation [16,17,22]. The actual fields in physical space can likewise be easily written in terms of those in the transformed space, see [18,19,22]. Under a continuous mapping $x \rightarrow x'$, the resulting transformation medium is reflectionless (i.e. impedance-matched to free space for all angles of incidence) [24,34]. The above transformation assumes a problem stated in the Fourier domain or, equivalently, to a time-harmonic steady-state problem. Frequency dependency could be embedded into the transformation by appropriate functional choices for $a_x(\omega )$ and $\sigma _x(\omega )$ where ensuing implications for causality would come into play [22,36,40,41].

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

$$\sigma_x= b/d \hspace{0.4cm} \textrm{for} \hspace{0.2cm} 0\leq x \leq d,$$
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.

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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]:

$$\vec{E}' ={-}{\hat z}\frac{I_0 k_0 \eta_0}{4} H_0^{(1)}\left(k_\rho \rho' \right)$$
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:
$$\rho' = \sqrt{(x'-x'_s)^{2}+(y'-y'_s)^{2}}$$
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]
$${\vec E} = \left[S^{{-}1}\right]^{T} \cdot {\vec E}'$$
$${\vec J} = \textrm{det}(\left[S\right])^{{-}1} \left[S\right] \cdot {\vec J}'$$
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 [3739,43,44]. 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:
$$\rho'= 0$$
From this condition and by assuming $a_x=-1$, we can determine the focal points inside the lens as
$$x_{f_1} = x_s + 2 d_1, \hspace{0.2cm} y_{f_1} ={\mp} \Im \textrm{m}[x'(x_{f_1})]$$
where $d_1$ is the source distance from the lens boundary and the focal points outside the lens as
$$x_{f_2} = x_s +2 d, \hspace{0.2cm} y_{f_2} ={\mp} \Im \textrm{m} [x'(x_{f_2})]$$
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 [5,4547] 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.

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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 $b=-0.1\lambda$.

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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.

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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 [4851].

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References

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  1. V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of $\epsilon$ϵ and $\mu$μ,” Phys.-Usp. 10(4), 509–514 (1968).
    [Crossref]
  2. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
    [Crossref]
  3. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
    [Crossref]
  4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [Crossref]
  5. D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
    [Crossref]
  6. A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92(11), 117403 (2004).
    [Crossref]
  7. V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30(1), 75–77 (2005).
    [Crossref]
  8. W. C. Chew, “Some reflections on double negative materials,” Prog. Electromagn. Res. 51, 1–26 (2005).
    [Crossref]
  9. U. Leonhardt and T. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).
  10. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001).
    [Crossref]
  11. R. E. Collin, “Frequency dispersion limits resolution in veselago lens,” Prog. Electromagn. Res. 19, 233–261 (2010).
    [Crossref]
  12. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
    [Crossref]
  13. A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32(23), 3432–3434 (2007).
    [Crossref]
  14. I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
    [Crossref]
  15. M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
    [Crossref]
  16. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [Crossref]
  17. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
    [Crossref]
  18. W. X. Jiang, J. Y. Chin, and T. J. Cui, “Anisotropic metamaterial devices,” Mater. Today 12(12), 26–33 (2009).
    [Crossref]
  19. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
    [Crossref]
  20. E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
    [Crossref]
  21. H. Chen, R.-X. Miao, and M. Li, “Transformation optics that mimics the system outside a schwarzschild black hole,” Opt. Express 18(14), 15183–15188 (2010).
    [Crossref]
  22. H. Odabasi, F. L. Teixeira, and W. C. Chew, “Impedance-matched absorbers and optical pseudo black holes,” J. Opt. Soc. Am. B 28(5), 1317–1323 (2011).
    [Crossref]
  23. F. Teixeira and W. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” J. Electromagn. Waves Appl. 13(5), 665–686 (1999).
    [Crossref]
  24. F. L. Teixeira and W. C. Chew, “Complex space approach to perfectly matched layers: a review and some new developments,” Int. J. Numer. Model. 13(5), 441–455 (2000).
    [Crossref]
  25. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006).
    [Crossref]
  26. M. Kuzuoglu, “Analysis of perfectly matched double negative layers via complex coordinate transformations,” IEEE Trans. Antennas Propag. 54(12), 3695–3699 (2006).
    [Crossref]
  27. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
    [Crossref]
  28. M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
    [Crossref]
  29. B.-I. Popa and S. A. Cummer, “Complex coordinates in transformation optics,” Phys. Rev. A 84(6), 063837 (2011).
    [Crossref]
  30. G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “$\mathcal {PT}$PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
    [Crossref]
  31. S. Savoia, G. Castaldi, and V. Galdi, “Complex-coordinate non-hermitian transformation optics,” J. Opt. 18(4), 044027 (2016).
    [Crossref]
  32. H. Odabasi, K. Sainath, and F. L. Teixeira, “Launching and controlling gaussian beams from point sources via planar transformation media,” Phys. Rev. B 97(7), 075105 (2018).
    [Crossref]
  33. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
    [Crossref]
  34. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43(12), 1460–1463 (1995).
    [Crossref]
  35. W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
    [Crossref]
  36. F. L. Teixeira, “On aspects of the physical realizability of perfectly matched absorbers for electromagnetic waves,” Radio Sci. 38(2), 15 (2003). 8014.
    [Crossref]
  37. J. B. Keller and W. Streifer, “Complex rays with an application to gaussian beams,” J. Opt. Soc. Am. 61(1), 40–43 (1971).
    [Crossref]
  38. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7(23), 684–685 (1971).
    [Crossref]
  39. L. B. Felsen, “Complex source point solution of the eld equations and their relation to the propagation and scattering of gaussian beams,” Symp. Matematica 18, 40–56 (1976).
  40. F. L. Teixeira and W. C. Chew, “On causality and dynamic stability of perfectly matched layers for fdtd simulations,” IEEE Trans. Microwave Theory Tech. 47(6), 775–785 (1999).
    [Crossref]
  41. K. Sainath and F. L. Teixeira, “Perfectly reflectionless omnidirectional absorbers and electromagnetic horizons,” J. Opt. Soc. Am. B 32(8), 1645–1650 (2015).
    [Crossref]
  42. R. F. Harrington, Time-Haarmonic Electromagnetic Fields (IEE Press, 2001).
  43. E. Heyman and L. B. Felsen, “Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18(7), 1588–1611 (2001).
    [Crossref]
  44. K. Tap, “Complex source point beam expansions for some electromagnetic radiation and scattering problems,” Ph.D. thesis, Ohio State University, The Ohio State University, Columbus, Ohio, USA (2007).
  45. T. Koschny, R. Moussa, and C. M. Soukoulis, “Limits on the amplification of evanescent waves of left-handed materials,” J. Opt. Soc. Am. B 23(3), 485–489 (2006).
    [Crossref]
  46. W. C. Chew, Waves and Fields in Inhomogenous Media (Wiley IEEE Press, 1999).
  47. T. M. Grzegorczyk, X. Chen, J. Pacheco, B.-I. Wu, and J. A. Kong, “Reflection coefficients and goos-hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
    [Crossref]
  48. S. Tretyakov, “Uniaxial omega medium as a physically realizable alternative for the perfectly matched layer (pml),” J. Electromagn. Waves Appl. 12(6), 821–837 (1998).
    [Crossref]
  49. L. Sun, X. Yang, and J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103(20), 201109 (2013).
    [Crossref]
  50. D. Ye, Z. Wang, K. Xu, H. Li, J. Huangfu, Z. Wang, and L. Ran, “Ultrawideband dispersion control of a metamaterial surface for perfectly-matched-layer-like absorption,” Phys. Rev. Lett. 111(18), 187402 (2013).
    [Crossref]
  51. D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5(1), 5841 (2014).
    [Crossref]

2018 (1)

H. Odabasi, K. Sainath, and F. L. Teixeira, “Launching and controlling gaussian beams from point sources via planar transformation media,” Phys. Rev. B 97(7), 075105 (2018).
[Crossref]

2016 (1)

S. Savoia, G. Castaldi, and V. Galdi, “Complex-coordinate non-hermitian transformation optics,” J. Opt. 18(4), 044027 (2016).
[Crossref]

2015 (1)

2014 (1)

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5(1), 5841 (2014).
[Crossref]

2013 (3)

L. Sun, X. Yang, and J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103(20), 201109 (2013).
[Crossref]

D. Ye, Z. Wang, K. Xu, H. Li, J. Huangfu, Z. Wang, and L. Ran, “Ultrawideband dispersion control of a metamaterial surface for perfectly-matched-layer-like absorption,” Phys. Rev. Lett. 111(18), 187402 (2013).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “$\mathcal {PT}$PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref]

2011 (2)

B.-I. Popa and S. A. Cummer, “Complex coordinates in transformation optics,” Phys. Rev. A 84(6), 063837 (2011).
[Crossref]

H. Odabasi, F. L. Teixeira, and W. C. Chew, “Impedance-matched absorbers and optical pseudo black holes,” J. Opt. Soc. Am. B 28(5), 1317–1323 (2011).
[Crossref]

2010 (3)

H. Chen, R.-X. Miao, and M. Li, “Transformation optics that mimics the system outside a schwarzschild black hole,” Opt. Express 18(14), 15183–15188 (2010).
[Crossref]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref]

R. E. Collin, “Frequency dispersion limits resolution in veselago lens,” Prog. Electromagn. Res. 19, 233–261 (2010).
[Crossref]

2009 (2)

W. X. Jiang, J. Y. Chin, and T. J. Cui, “Anisotropic metamaterial devices,” Mater. Today 12(12), 26–33 (2009).
[Crossref]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

2008 (1)

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[Crossref]

2007 (3)

A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32(23), 3432–3434 (2007).
[Crossref]

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[Crossref]

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[Crossref]

2006 (7)

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006).
[Crossref]

M. Kuzuoglu, “Analysis of perfectly matched double negative layers via complex coordinate transformations,” IEEE Trans. Antennas Propag. 54(12), 3695–3699 (2006).
[Crossref]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref]

T. Koschny, R. Moussa, and C. M. Soukoulis, “Limits on the amplification of evanescent waves of left-handed materials,” J. Opt. Soc. Am. B 23(3), 485–489 (2006).
[Crossref]

2005 (3)

T. M. Grzegorczyk, X. Chen, J. Pacheco, B.-I. Wu, and J. A. Kong, “Reflection coefficients and goos-hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[Crossref]

V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett. 30(1), 75–77 (2005).
[Crossref]

W. C. Chew, “Some reflections on double negative materials,” Prog. Electromagn. Res. 51, 1–26 (2005).
[Crossref]

2004 (1)

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92(11), 117403 (2004).
[Crossref]

2003 (2)

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

F. L. Teixeira, “On aspects of the physical realizability of perfectly matched absorbers for electromagnetic waves,” Radio Sci. 38(2), 15 (2003). 8014.
[Crossref]

2001 (3)

E. Heyman and L. B. Felsen, “Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18(7), 1588–1611 (2001).
[Crossref]

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001).
[Crossref]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[Crossref]

2000 (3)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

F. L. Teixeira and W. C. Chew, “Complex space approach to perfectly matched layers: a review and some new developments,” Int. J. Numer. Model. 13(5), 441–455 (2000).
[Crossref]

1999 (2)

F. Teixeira and W. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” J. Electromagn. Waves Appl. 13(5), 665–686 (1999).
[Crossref]

F. L. Teixeira and W. C. Chew, “On causality and dynamic stability of perfectly matched layers for fdtd simulations,” IEEE Trans. Microwave Theory Tech. 47(6), 775–785 (1999).
[Crossref]

1998 (1)

S. Tretyakov, “Uniaxial omega medium as a physically realizable alternative for the perfectly matched layer (pml),” J. Electromagn. Waves Appl. 12(6), 821–837 (1998).
[Crossref]

1995 (1)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43(12), 1460–1463 (1995).
[Crossref]

1994 (2)

W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

1976 (1)

L. B. Felsen, “Complex source point solution of the eld equations and their relation to the propagation and scattering of gaussian beams,” Symp. Matematica 18, 40–56 (1976).

1971 (2)

J. B. Keller and W. Streifer, “Complex rays with an application to gaussian beams,” J. Opt. Soc. Am. 61(1), 40–43 (1971).
[Crossref]

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7(23), 684–685 (1971).
[Crossref]

1968 (1)

V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of $\epsilon$ϵ and $\mu$μ,” Phys.-Usp. 10(4), 509–514 (1968).
[Crossref]

Alekseyev, L. V.

Alù, A.

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “$\mathcal {PT}$PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

Castaldi, G.

S. Savoia, G. Castaldi, and V. Galdi, “Complex-coordinate non-hermitian transformation optics,” J. Opt. 18(4), 044027 (2016).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “$\mathcal {PT}$PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref]

Chang, K.

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5(1), 5841 (2014).
[Crossref]

Chen, H.

Chen, X.

T. M. Grzegorczyk, X. Chen, J. Pacheco, B.-I. Wu, and J. A. Kong, “Reflection coefficients and goos-hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[Crossref]

Chew, W.

F. Teixeira and W. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” J. Electromagn. Waves Appl. 13(5), 665–686 (1999).
[Crossref]

Chew, W. C.

H. Odabasi, F. L. Teixeira, and W. C. Chew, “Impedance-matched absorbers and optical pseudo black holes,” J. Opt. Soc. Am. B 28(5), 1317–1323 (2011).
[Crossref]

W. C. Chew, “Some reflections on double negative materials,” Prog. Electromagn. Res. 51, 1–26 (2005).
[Crossref]

F. L. Teixeira and W. C. Chew, “Complex space approach to perfectly matched layers: a review and some new developments,” Int. J. Numer. Model. 13(5), 441–455 (2000).
[Crossref]

F. L. Teixeira and W. C. Chew, “On causality and dynamic stability of perfectly matched layers for fdtd simulations,” IEEE Trans. Microwave Theory Tech. 47(6), 775–785 (1999).
[Crossref]

W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

W. C. Chew, Waves and Fields in Inhomogenous Media (Wiley IEEE Press, 1999).

Chin, J. Y.

W. X. Jiang, J. Y. Chin, and T. J. Cui, “Anisotropic metamaterial devices,” Mater. Today 12(12), 26–33 (2009).
[Crossref]

Collin, R. E.

R. E. Collin, “Frequency dispersion limits resolution in veselago lens,” Prog. Electromagn. Res. 19, 233–261 (2010).
[Crossref]

Cui, T. J.

W. X. Jiang, J. Y. Chin, and T. J. Cui, “Anisotropic metamaterial devices,” Mater. Today 12(12), 26–33 (2009).
[Crossref]

Cummer, S. A.

B.-I. Popa and S. A. Cummer, “Complex coordinates in transformation optics,” Phys. Rev. A 84(6), 063837 (2011).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref]

Davis, C. C.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[Crossref]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7(23), 684–685 (1971).
[Crossref]

Eleftheriades, G. V.

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92(11), 117403 (2004).
[Crossref]

Engheta, N.

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “$\mathcal {PT}$PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref]

Felsen, L. B.

E. Heyman and L. B. Felsen, “Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18(7), 1588–1611 (2001).
[Crossref]

L. B. Felsen, “Complex source point solution of the eld equations and their relation to the propagation and scattering of gaussian beams,” Symp. Matematica 18, 40–56 (1976).

Galdi, V.

S. Savoia, G. Castaldi, and V. Galdi, “Complex-coordinate non-hermitian transformation optics,” J. Opt. 18(4), 044027 (2016).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “$\mathcal {PT}$PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref]

Gao, J.

L. Sun, X. Yang, and J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103(20), 201109 (2013).
[Crossref]

Grbic, A.

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92(11), 117403 (2004).
[Crossref]

Grzegorczyk, T. M.

T. M. Grzegorczyk, X. Chen, J. Pacheco, B.-I. Wu, and J. A. Kong, “Reflection coefficients and goos-hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[Crossref]

Harrington, R. F.

R. F. Harrington, Time-Haarmonic Electromagnetic Fields (IEE Press, 2001).

Heyman, E.

Huangfu, J.

D. Ye, Z. Wang, K. Xu, H. Li, J. Huangfu, Z. Wang, and L. Ran, “Ultrawideband dispersion control of a metamaterial surface for perfectly-matched-layer-like absorption,” Phys. Rev. Lett. 111(18), 187402 (2013).
[Crossref]

Hung, Y.-J.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[Crossref]

Jacob, Z.

Jiang, W. X.

W. X. Jiang, J. Y. Chin, and T. J. Cui, “Anisotropic metamaterial devices,” Mater. Today 12(12), 26–33 (2009).
[Crossref]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref]

Keller, J. B.

Kildishev, A. V.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32(23), 3432–3434 (2007).
[Crossref]

Kingsland, D. M.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43(12), 1460–1463 (1995).
[Crossref]

Kong, J. A.

T. M. Grzegorczyk, X. Chen, J. Pacheco, B.-I. Wu, and J. A. Kong, “Reflection coefficients and goos-hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[Crossref]

Koschny, T.

Kuzuoglu, M.

M. Kuzuoglu, “Analysis of perfectly matched double negative layers via complex coordinate transformations,” IEEE Trans. Antennas Propag. 54(12), 3695–3699 (2006).
[Crossref]

Lee, J.-F.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43(12), 1460–1463 (1995).
[Crossref]

Lee, R.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43(12), 1460–1463 (1995).
[Crossref]

Leonhardt, U.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref]

U. Leonhardt and T. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

Li, H.

D. Ye, Z. Wang, K. Xu, H. Li, J. Huangfu, Z. Wang, and L. Ran, “Ultrawideband dispersion control of a metamaterial surface for perfectly-matched-layer-like absorption,” Phys. Rev. Lett. 111(18), 187402 (2013).
[Crossref]

Li, M.

Miao, R.-X.

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref]

Moussa, R.

Narimanov, E.

Narimanov, E. E.

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Odabasi, H.

H. Odabasi, K. Sainath, and F. L. Teixeira, “Launching and controlling gaussian beams from point sources via planar transformation media,” Phys. Rev. B 97(7), 075105 (2018).
[Crossref]

H. Odabasi, F. L. Teixeira, and W. C. Chew, “Impedance-matched absorbers and optical pseudo black holes,” J. Opt. Soc. Am. B 28(5), 1317–1323 (2011).
[Crossref]

Pacheco, J.

T. M. Grzegorczyk, X. Chen, J. Pacheco, B.-I. Wu, and J. A. Kong, “Reflection coefficients and goos-hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[Crossref]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref]

Philbin, T.

U. Leonhardt and T. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006).
[Crossref]

Podolskiy, V. A.

Popa, B.-I.

B.-I. Popa and S. A. Cummer, “Complex coordinates in transformation optics,” Phys. Rev. A 84(6), 063837 (2011).
[Crossref]

Psaltis, D.

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[Crossref]

Ramakrishna, S. A.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

Ran, L.

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5(1), 5841 (2014).
[Crossref]

D. Ye, Z. Wang, K. Xu, H. Li, J. Huangfu, Z. Wang, and L. Ran, “Ultrawideband dispersion control of a metamaterial surface for perfectly-matched-layer-like absorption,” Phys. Rev. Lett. 111(18), 187402 (2013).
[Crossref]

Rosenbluth, M.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

Sacks, Z. S.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43(12), 1460–1463 (1995).
[Crossref]

Sainath, K.

H. Odabasi, K. Sainath, and F. L. Teixeira, “Launching and controlling gaussian beams from point sources via planar transformation media,” Phys. Rev. B 97(7), 075105 (2018).
[Crossref]

K. Sainath and F. L. Teixeira, “Perfectly reflectionless omnidirectional absorbers and electromagnetic horizons,” J. Opt. Soc. Am. B 32(8), 1645–1650 (2015).
[Crossref]

Savoia, S.

S. Savoia, G. Castaldi, and V. Galdi, “Complex-coordinate non-hermitian transformation optics,” J. Opt. 18(4), 044027 (2016).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “$\mathcal {PT}$PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref]

Schultz, S.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[Crossref]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[Crossref]

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[Crossref]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Smolyaninov, I. I.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[Crossref]

Soukoulis, C. M.

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref]

Stockman, M. I.

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[Crossref]

Streifer, W.

Sun, L.

L. Sun, X. Yang, and J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103(20), 201109 (2013).
[Crossref]

Tap, K.

K. Tap, “Complex source point beam expansions for some electromagnetic radiation and scattering problems,” Ph.D. thesis, Ohio State University, The Ohio State University, Columbus, Ohio, USA (2007).

Teixeira, F.

F. Teixeira and W. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” J. Electromagn. Waves Appl. 13(5), 665–686 (1999).
[Crossref]

Teixeira, F. L.

H. Odabasi, K. Sainath, and F. L. Teixeira, “Launching and controlling gaussian beams from point sources via planar transformation media,” Phys. Rev. B 97(7), 075105 (2018).
[Crossref]

K. Sainath and F. L. Teixeira, “Perfectly reflectionless omnidirectional absorbers and electromagnetic horizons,” J. Opt. Soc. Am. B 32(8), 1645–1650 (2015).
[Crossref]

H. Odabasi, F. L. Teixeira, and W. C. Chew, “Impedance-matched absorbers and optical pseudo black holes,” J. Opt. Soc. Am. B 28(5), 1317–1323 (2011).
[Crossref]

F. L. Teixeira, “On aspects of the physical realizability of perfectly matched absorbers for electromagnetic waves,” Radio Sci. 38(2), 15 (2003). 8014.
[Crossref]

F. L. Teixeira and W. C. Chew, “Complex space approach to perfectly matched layers: a review and some new developments,” Int. J. Numer. Model. 13(5), 441–455 (2000).
[Crossref]

F. L. Teixeira and W. C. Chew, “On causality and dynamic stability of perfectly matched layers for fdtd simulations,” IEEE Trans. Microwave Theory Tech. 47(6), 775–785 (1999).
[Crossref]

Tretyakov, S.

S. Tretyakov, “Uniaxial omega medium as a physically realizable alternative for the perfectly matched layer (pml),” J. Electromagn. Waves Appl. 12(6), 821–837 (1998).
[Crossref]

Tsang, M.

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[Crossref]

Veselago, V. G.

V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of $\epsilon$ϵ and $\mu$μ,” Phys.-Usp. 10(4), 509–514 (1968).
[Crossref]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Wang, Z.

D. Ye, Z. Wang, K. Xu, H. Li, J. Huangfu, Z. Wang, and L. Ran, “Ultrawideband dispersion control of a metamaterial surface for perfectly-matched-layer-like absorption,” Phys. Rev. Lett. 111(18), 187402 (2013).
[Crossref]

D. Ye, Z. Wang, K. Xu, H. Li, J. Huangfu, Z. Wang, and L. Ran, “Ultrawideband dispersion control of a metamaterial surface for perfectly-matched-layer-like absorption,” Phys. Rev. Lett. 111(18), 187402 (2013).
[Crossref]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

Wu, B.-I.

T. M. Grzegorczyk, X. Chen, J. Pacheco, B.-I. Wu, and J. A. Kong, “Reflection coefficients and goos-hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[Crossref]

Xin, H.

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5(1), 5841 (2014).
[Crossref]

Xu, K.

D. Ye, Z. Wang, K. Xu, H. Li, J. Huangfu, Z. Wang, and L. Ran, “Ultrawideband dispersion control of a metamaterial surface for perfectly-matched-layer-like absorption,” Phys. Rev. Lett. 111(18), 187402 (2013).
[Crossref]

Yang, X.

L. Sun, X. Yang, and J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103(20), 201109 (2013).
[Crossref]

Ye, D.

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5(1), 5841 (2014).
[Crossref]

D. Ye, Z. Wang, K. Xu, H. Li, J. Huangfu, Z. Wang, and L. Ran, “Ultrawideband dispersion control of a metamaterial surface for perfectly-matched-layer-like absorption,” Phys. Rev. Lett. 111(18), 187402 (2013).
[Crossref]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001).
[Crossref]

Appl. Phys. Lett. (3)

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett. 82(10), 1506–1508 (2003).
[Crossref]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: Broadband omnidirectional light absorber,” Appl. Phys. Lett. 95(4), 041106 (2009).
[Crossref]

L. Sun, X. Yang, and J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103(20), 201109 (2013).
[Crossref]

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7(23), 684–685 (1971).
[Crossref]

IEEE Trans. Antennas Propag. (2)

M. Kuzuoglu, “Analysis of perfectly matched double negative layers via complex coordinate transformations,” IEEE Trans. Antennas Propag. 54(12), 3695–3699 (2006).
[Crossref]

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43(12), 1460–1463 (1995).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

F. L. Teixeira and W. C. Chew, “On causality and dynamic stability of perfectly matched layers for fdtd simulations,” IEEE Trans. Microwave Theory Tech. 47(6), 775–785 (1999).
[Crossref]

Int. J. Numer. Model. (1)

F. L. Teixeira and W. C. Chew, “Complex space approach to perfectly matched layers: a review and some new developments,” Int. J. Numer. Model. 13(5), 441–455 (2000).
[Crossref]

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

J. Electromagn. Waves Appl. (2)

F. Teixeira and W. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” J. Electromagn. Waves Appl. 13(5), 665–686 (1999).
[Crossref]

S. Tretyakov, “Uniaxial omega medium as a physically realizable alternative for the perfectly matched layer (pml),” J. Electromagn. Waves Appl. 12(6), 821–837 (1998).
[Crossref]

J. Opt. (1)

S. Savoia, G. Castaldi, and V. Galdi, “Complex-coordinate non-hermitian transformation optics,” J. Opt. 18(4), 044027 (2016).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

Mater. Today (1)

W. X. Jiang, J. Y. Chin, and T. J. Cui, “Anisotropic metamaterial devices,” Mater. Today 12(12), 26–33 (2009).
[Crossref]

Microw. Opt. Technol. Lett. (1)

W. C. Chew and W. H. Weedon, “A 3d perfectly matched medium from modified maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7(13), 599–604 (1994).
[Crossref]

Nat. Commun. (1)

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5(1), 5841 (2014).
[Crossref]

Nat. Mater. (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref]

New J. Phys. (1)

U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (1)

B.-I. Popa and S. A. Cummer, “Complex coordinates in transformation optics,” Phys. Rev. A 84(6), 063837 (2011).
[Crossref]

Phys. Rev. B (2)

H. Odabasi, K. Sainath, and F. L. Teixeira, “Launching and controlling gaussian beams from point sources via planar transformation media,” Phys. Rev. B 97(7), 075105 (2018).
[Crossref]

M. Tsang and D. Psaltis, “Magnifying perfect lens and superlens design by coordinate transformation,” Phys. Rev. B 77(3), 035122 (2008).
[Crossref]

Phys. Rev. E (1)

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64(5), 056625 (2001).
[Crossref]

Phys. Rev. Lett. (6)

A. Grbic and G. V. Eleftheriades, “Overcoming the diffraction limit with a planar left-handed transmission-line lens,” Phys. Rev. Lett. 92(11), 117403 (2004).
[Crossref]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “$\mathcal {PT}$PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref]

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[Crossref]

D. Ye, Z. Wang, K. Xu, H. Li, J. Huangfu, Z. Wang, and L. Ran, “Ultrawideband dispersion control of a metamaterial surface for perfectly-matched-layer-like absorption,” Phys. Rev. Lett. 111(18), 187402 (2013).
[Crossref]

Phys.-Usp. (1)

V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of $\epsilon$ϵ and $\mu$μ,” Phys.-Usp. 10(4), 509–514 (1968).
[Crossref]

Prog. Electromagn. Res. (3)

W. C. Chew, “Some reflections on double negative materials,” Prog. Electromagn. Res. 51, 1–26 (2005).
[Crossref]

R. E. Collin, “Frequency dispersion limits resolution in veselago lens,” Prog. Electromagn. Res. 19, 233–261 (2010).
[Crossref]

T. M. Grzegorczyk, X. Chen, J. Pacheco, B.-I. Wu, and J. A. Kong, “Reflection coefficients and goos-hanchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Prog. Electromagn. Res. 51, 83–113 (2005).
[Crossref]

Radio Sci. (1)

F. L. Teixeira, “On aspects of the physical realizability of perfectly matched absorbers for electromagnetic waves,” Radio Sci. 38(2), 15 (2003). 8014.
[Crossref]

Science (5)

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[Crossref]

Symp. Matematica (1)

L. B. Felsen, “Complex source point solution of the eld equations and their relation to the propagation and scattering of gaussian beams,” Symp. Matematica 18, 40–56 (1976).

Other (4)

R. F. Harrington, Time-Haarmonic Electromagnetic Fields (IEE Press, 2001).

W. C. Chew, Waves and Fields in Inhomogenous Media (Wiley IEEE Press, 1999).

K. Tap, “Complex source point beam expansions for some electromagnetic radiation and scattering problems,” Ph.D. thesis, Ohio State University, The Ohio State University, Columbus, Ohio, USA (2007).

U. Leonhardt and T. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

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Figures (4)

Fig. 1.
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.
Fig. 2.
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.
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.
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.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

x ( x ) = { x if x 0 s x x if 0 x d x + ( s x 1 ) d if x d
[ Λ ] = diag { 1 / s x , s x , s x }
σ x = b / d for 0 x d ,
E = z ^ I 0 k 0 η 0 4 H 0 ( 1 ) ( k ρ ρ )
ρ = ( x x s ) 2 + ( y y s ) 2
E = [ S 1 ] T E
J = det ( [ S ] ) 1 [ S ] J
ρ = 0
x f 1 = x s + 2 d 1 , y f 1 = m [ x ( x f 1 ) ]
x f 2 = x s + 2 d , y f 2 = m [ x ( x f 2 ) ]

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