Abstract

Waves typically propagate very differently through a homogeneous medium like free space than through an inhomogeneous medium like a complex dielectric structure. Here we present the surprising result that wave solutions in two-dimensional free space can be mapped to a solution inside a suitably designed non-Hermitian potential landscape such that both solutions share the same spatial distribution of their wave intensity. The mapping we introduce here is broadly applicable as a design protocol for a special class of non-Hermitian media across which specific incoming waves form scattering-free propagation channels. This protocol naturally enables the design of structures with a broadband unidirectional invisibility for which outgoing waves are indistinguishable from those of free space. We illustrate this concept through the example of a beam that maintains its Gaussian shape while passing through a randomly assembled distribution of scatterers with gain and loss.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

When waves propagate through an inhomogeneous medium, they typically diffract and produce complicated interference patterns due to multiple scattering. When the incoming wave is a strongly confined beam of light with a well-defined initial momentum, the inhomogeneities of the medium break up the beam’s profile such that the initial directionality will gradually get lost. In this sense, waves do behave very differently in an inhomogeneous medium than in homogeneous free space, where beams preserve their initial momentum and are often amenable to a fully analytical description.

One of the key concepts in modern optics is to bridge the divide between free space and inhomogeneous dielectric media through “transformation optics” [1,2]. The central idea here is that for the propagation of light, certain inhomogeneous dielectric media are equivalent to a curved geometry of space, provided that a conformal mapping between these two situations exists. In this way, dielectric structures can be designed that guide light in an effectively curved space around objects that can thereby be cloaked [37]. While the light then travels along curved trajectories, it appears to the external observer as if it had propagated along the straight-line paths of free space.

Here, we will present a new strategy to link homogeneous with inhomogeneous media by mapping the solutions of the wave equation in these two spaces onto each other. As a result of this mapping procedure, an incoming wave impinging onto a dielectric structure produces exactly the same wave intensity pattern as when propagating through free space—even inside the inhomogeneous structure itself. Rather than diverting the light around a complicated object, our approach allows us to guide light straight through this object without distortions. The key ingredient to suppress back-reflections and scattering by the object is a suitable distribution of gain and loss that is added to its dielectric function (no materials with negative index or epsilon-near-zero are required). In this way, our results connect to the new area of non-Hermitian photonics [814], where many exciting results such as reflection-less potentials [1523], constant-intensity waves (CI waves) [2429], asymmetric mode switches [30,31], coherent perfect absorbers [3236], cloaks based on parity-time (PT) symmetry [18,37,38], exceptional point sensors [3941], and topological lasers [4246] have been reported.

Our starting point is the two-dimensional Helmholtz equation describing the propagation of a scalar electric field $E$ in the presence of a non-uniform and potentially non-Hermitian dielectric function $\varepsilon(x,y)$:

$${\nabla ^2}E(x,y) + {k^2}\left[{n_{\rm ref}^2 + \varepsilon (x,y)} \right]E(x,y) = 0 ,$$
where ${\nabla ^2}$ is the 2D Laplacian, $k = 2\pi /\lambda$ is the wavenumber in vacuum, $\varepsilon (x,y)$ is the medium’s complex dielectric function $\varepsilon = {\varepsilon _R} + i{\varepsilon _I}$, varying in the $x{-}y$ plane, and ${n_{\rm ref}}$ is the refractive index of the homogeneous background medium.

We now wish to connect the above solution $E(x,y)$ (representing, e.g., a scattering state) to the solution $\phi (x,y)$ of the scalar Helmholtz wave equation in homogeneous space with dielectric constant $n_{\rm ref}^2$ :

$${\nabla ^2}\phi (x,y) + {k^2}n_{\rm ref}^2\,\phi (x,y) = 0.$$

Demanding that the two solutions have equal intensities, $|E(x,y{)|^2} = |\phi (x,y{)|^2}$, we connect them through the following simple transformation:

$$E(x,y) = \phi (x,y){e^{ik\theta (x,y)}} ,$$
where $\theta (x,y)$ is a smooth and real-valued generating function that can be chosen at will. Inserting this ansatz into Eq. (1), we obtain the following relation:
$$\varepsilon (x,y) = (\nabla \theta {)^2} - \frac{i}{k}\left({{\nabla ^2}\theta + 2\nabla \theta \cdot \frac{{\nabla \phi}}{\phi}} \right) .$$

This equation provides us the desired expression for the dielectric function $\varepsilon (x,y)$ of an inhomogeneous and typically non-Hermitian medium, with a refractive index $n(x,y) = \sqrt {n_\textit{ref}^2 + \varepsilon (x,y)} = {n_R}(x,y) + i{n_I}(x,y)$, which supports solutions $E(x,y)$ with the same spatial intensity distribution as the reference solution $\phi (x,y)$ in a homogeneous medium such as free space, where ${n_{\rm ref}} = 1$. Formally, the above mapping between $E (x,y)$ and $\phi (x,y)$ requires that $\phi (x,y)$ has no zeros in positions different from those where its derivative goes to zero.

We emphasize that our approach works not only when the functions $\theta (x,y)$ and $\phi (x,y)$ are known from the outset, but also, when the real parts of the dielectric functions ${\varepsilon _R}(x,y)$ and $\phi (x,y)$ are given, the required imaginary part ${\varepsilon _I}(x,y)$ of the non-Hermitian potential can be determined. For this purpose, we express the solution $\phi (x,y)$ as

$$\phi (x,y) = {\phi _A}(x,y){e^{i{\phi _P}(x,y)}},$$
with both ${\phi _A}(x,y)$ and ${\phi _P}(x,y)$ being real known functions. The corresponding $\theta (x,y)$ can be determined by solving the following differential equation:
$${(\nabla \theta)^2} + \frac{2}{k}\nabla \theta \cdot \nabla {\phi _P} = {\varepsilon _R}.$$
The calculated $\theta (x,y)$ can then be used to evaluate the ${\varepsilon _I}(x,y)$ by substituting into (4).

An important point associated with the mapping in Eq. (3) is the following: if the generating function $\theta (x,y)$ is localized and vanishes outside the scattering region of interest, the asymptotic wave solutions of $E(x,y)$ and $\phi (x,y)$ are not just identical in their intensities, but also in their amplitudes and phases. As such, the far field of the beam does not reveal the presence of the scattering potential it was designed for; the potential can thus be considered as being unidirectionally invisible [15,19,21,22,27]. In Supplement 1 (Section 2.B), we show explicitly that variations around the design frequency have a similar effect on the output fields $E(x,y)$ and $\phi (x,y)$, such that this invisibility is a broadband effect with exciting prospects for the undistorted propagation of pulses through such media.

As a first example of our methodology, we consider the scattering from a dipole-shaped refractive index $n(x,y)$ generated by a single Gaussian function $\theta (x,y)$ [see Figs. 1(a) and 1(b)]. The reference electric field ${\phi _G}(x,y)$ is chosen to be the paraxial Gaussian beam in a homogeneous medium with ${n_{\rm ref}} = 3$ (see Supplement 1). By solving the scattering problem numerically using NGSolve [4749], we find for the Hermitian case [$n(x,y) = {n_R}(x,y)$] that the beam exhibits a strong interference peak and breaks up into multiple parts [see Fig. 1(c)]. When including the appropriate imaginary refractive index distribution ${n_I}(x,y)$ based on Eq. (4), the intensity distribution is smooth and follows exactly the propagation pattern that corresponds to homogeneous space both inside and outside the scattering region [see Fig. 1(d)]. The values for gain and loss required to achieve such a behavior depend on the derivatives of the function $\theta (x,y)$, as well as on the wavelength of operation; smoother variations in ${n_R}(x,y)$ are typically associated with smaller amplitudes of ${n_I}(x,y)$.

 

Fig. 1. Scattering of a Gaussian beam of width ${w_0} = 5\lambda$ by a simple dipole-shaped potential. The real and imaginary parts of the non-Hermitian refractive index distribution $n(x,y) = {n_R}(x,y) + i{n_I}(x,y)$ are presented in (a), (b), respectively. The diffraction intensity patterns of an incident Gaussian beam are shown in (c) for the Hermitian $n(x,y) = {n_R}(x,y)$ and (d) for the corresponding non-Hermitian medium $n(x,y) = {n_R}(x,y) + i{n_I}(x,y)$. Panel (d) demonstrates how the non-Hermitian mapping of Eq. (4) entirely suppresses the scattering of the Gaussian beam due to the presence of gain and loss. The white dashed rectangles in (c), (d) denote the limits of the scattering region, depicted in (a), (b) (homogeneous space with ${n_{\rm ref}} = 3$ is assumed outside).

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An even more dramatic demonstration of our mapping is found when considering the same input beam propagating inside a disordered potential, depicted in Fig. 2. Here the dielectric function $\varepsilon (x,y)$ of the medium is generated from a function $\theta (x,y)$ consisting of a superposition of $N = 300$ Gaussians with random center positions, widths, and amplitudes (see Supplement 1). When the beam propagates through the Hermitian part of the disordered refractive index distribution [see Fig. 2(a)], the intensity distribution exhibits very strong fluctuations characteristic of multiple scattering [see Fig. 2(c)].

 

Fig. 2. Scattering of a Gaussian beam of width ${w_0} = 5\lambda$ through a disordered medium calculated through a generating function $\theta (x,y)$ given by a random superposition of $N = 300$ Gaussians (see Supplement 1). Real and imaginary parts of the medium’s refractive index $n(x,y)$ are shown in (a) and (b), respectively. In (c), we show how the intensity of the incoming beam produces a complicated interference pattern when the medium consists of only the real part of the refractive index ${n_R}(x,y)$ as shown in (a). In (d), the same incoming beam propagates through this medium with the imaginary part ${n_I}(x,y)$ included according to Eq. (4). Here the intensity pattern of the beam is the same as in a homogeneous medium. The white dashed rectangles in (c) and (d) denote the limits of the scattering region, depicted in (a) and (b) (homogeneous space with ${n_{\rm ref}} = 3$ outside).

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A remarkably different behavior is seen when the imaginary part of the refractive index [see Fig. 2(b)] is added to the real part. The beam now propagates through the medium without any noticeable distortions, exhibiting only a reduction in the central peak amplitude and an increase in width [see Fig. 1(d)], in perfect equivalence to the propagation of a Gaussian beam in homogeneous space. The multiple scattering and interference effects are thus completely cancelled by including the gain/loss distribution of Fig. 2(b) into the Hermitian medium shown in Fig. 2a. What makes this observation particularly striking is the fact that the width of the Gaussian beam is comparable to the length scale of the modulations of its refractive index—a situation that conventionally gives rise to highly complex scattering effects.

In contrast to the approach of transformation optics, our mapping thus produces wave states that have the same intensity distribution as in a homogeneous reference medium not just outside but also inside an inhomogeneous dielectric structure. The price to pay for this strong equivalence is, apparently, that according to our methodology, only a single solution can be mapped onto homogeneous space, whereas in transformation optics, the transformed medium exhibits the desired behavior for any propagation direction and beam profile. We will thus investigate to which degree our design strategy is robust to varying the properties of the incident beam. In order to systematically determine this robustness, we have varied the beam’s incidence angle $\alpha$ and detuned its wavevector $k$ in small steps while keeping the disordered dielectric function $\varepsilon (x,y)$ fixed as before. The detailed results of scanning the input parameters are presented in Supplement 1; here we will summarize the interesting physical findings that we draw from this analysis.

For the case in which the angle of the incoming Gaussian beam is detuned from $\alpha {= 0^ \circ}$ to the values $\alpha = \pm {0.5^ \circ}$, the detuning just leads to small modulations of the beam’s profile inside the scattering region (not shown). For higher detuning of the angle, $|\alpha | \ge {1^ \circ}$, the beam deformations become increasingly significant, including a breakup of the beam into two parts. An example of such a behavior is depicted in Fig. 3(a), where we plot the intensity distribution for a Gaussian beam incident at an angle $\alpha {= 3^ \circ}$ onto a medium that is designed for supporting a Gaussian beam at $\alpha {= 0^ \circ}$. To demonstrate the flexibility of our approach, we now compare this situation with the case where the Gaussian beam is incident with a tilt of $\alpha {= 3^ \circ}$ and penetrates a refractive index distribution that is constructed based on the same generating function $\theta (x,y)$ as before, but where the reference solution ${\phi _G}$ now has the tilt of $\alpha {= 3^ \circ}$ already built in. As shown in Fig. 3(b), this new design removes the distortions that the tilt gave rise to before; instead, the incoming tilted Gaussian maintains its initial direction and its intensity profile as in homogeneous space.

 

Fig. 3. A Gaussian beam of width ${w_0} = 5\lambda$ entering a disordered medium at a tilt angle of $\alpha {= 3^ \circ}$ with respect to the $x$ axis. (a) The real and imaginary parts of the refractive index $n(x,y)$ are those of Fig. 2, following the design using a Gaussian reference beam without tilt ($\alpha {= 0^ \circ}$). The angular mismatch of 3° leads to distortions that ultimately break up the beam during propagation. (b) Here we take into account the tilt of the incoming beam by using a Gaussian reference solution ${\phi _G}(x,y)$ in Eq. (4) with an adjusted incidence angle of $\alpha {= 3^ \circ}$. The incoming beam at this angle propagates through this medium without any distortions. As in Fig. 2, the white dashed rectangles denote the limits of the scattering region.

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Regarding the sensitivity with respect to variations of the wavenumber $k$ of the incident beam, our analysis shows a relatively higher degree of robustness (see Supplement 1). More specifically, we find that if the reference solution that determines the design of $\varepsilon (x,y)$ has a wavenumber ${k_0}$ detuned by less than about 5% from the $k$-value of the input beam, the intensity distribution of this beam remains very similar to the case of homogeneous space, whereas larger distortions are observed for $k$-values of a higher mismatch (see Supplement 1). This broadband frequency robustness also leaves the unidirectional invisible character of our potentials unaffected in a comparably large frequency region (see Supplement 1, Section 2.B), in agreement with previous studies of CI waves in 1D non-Hermitian scattering media [27].

These special CI wave states were shown to be able to penetrate through complex non-Hermitian potential landscapes without any back-reflections and without changing their initial intensity. While most of the literature on CI waves has focused so far on 1D scattering problems [24,25,27,28], recent work shows how to generalize this concept also to scattering problems with two spatial dimensions [29]. From the point of view of our present work, it now turns out that these 2D CI waves correspond to the special case where the reference solution $\phi (x,y)$ in Eq. (4) is a simple plane wave. For this case, the amplitude of the reference solution $\phi (x,y)$ is constant, and the term in Eq. (4) involving $\nabla \phi (x,y)$ simply drops out, if one considers a boundary value problem with perfectly transmitting boundaries (see e.g.,  [25] and Supplement 1). To check the relevance of this term when considering inputs that are different from that of a plane wave, we consider the same real function $\theta (x,y)$ as used in Fig. 2 to construct the dielectric function $\varepsilon (x,y)$ for an incoming Gaussian at incidence angle $\alpha {= 0^ \circ}$. When replacing this Gaussian reference solution by a plane wave to construct the corresponding CI potential one finds that the real part of the refractive index ${n_R}(x,y)$ stays very similar, but its imaginary part ${n_I}(x,y)$ changes significantly [compare Figs. 2(b) and 4(a)]. As a result, when injecting the Gaussian beam of width ${w_0} = 5\lambda$ onto this CI potential, ranging from $[- 13.3\lambda ,13.3\lambda]$ in $y$ direction, we find that the beam gets distorted during propagation, since the improperly configured values of gain and loss are unable to mitigate scattering entirely [see Fig. 4(b)]. When, however, injecting a super-Gaussian beam with a width ${w_0} = 36.35\lambda$ larger than the scattering region, the amplitude profile of this beam approximates sufficiently well that of a plane wave, such that the beam’s intensity does not get distorted by the CI potential [see Fig. 4(c)]. In Supplement 1, we examine in more detail the sensitivity of the CI potential on the width of the input beam, and find that indeed the profile becomes more distorted when the transverse dimension of the beam becomes smaller than the width of the scattering region. Additionally, we have performed a parametric study of the sensitivity on variations of $\alpha$ and $k$ also for the super-Gaussian beams in the CI case, and found behavior similar to the corresponding case of the Gaussian beam.

 

Fig. 4. Propagation through a disordered refractive index distribution designed to support a CI wave solution (see text). (a) Imaginary part of the refractive index distribution (the real part is similar to the one in Fig. 2a; see Supplement 1). (b) Intensity of Gaussian beam with a width ${w_0} = 5\lambda$ (smaller than the width of the scattering region), exhibiting sizeable distortions during propagation. (c) Intensity distribution for a super-Gaussian beam with a width larger than that of the scattering region $[E(x = 0,y) = {\exp(-y^8/w_0^8)}$, ${w_0} = 36.35\lambda]$. Here the shape is maintained, since the beam approximates well a CI wave inside the scattering region. The dashed rectangles in (b) and (c) denote the limits of the scattering region in (a).

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In conclusion, we present here a systematic method for mapping the solutions of the 2D Helmholtz equation in homogeneous space onto a scattering problem in inhomogeneous dielectric environments with spatially modulated gain and loss. The photonic structures that can be designed based on this approach are not only unidirectionally invisible to an outside observer, but also support scattering-free propagation of a beam inside the optical potential. This contrasts sharply with the strategy of transformation optics, which seeks to design media for which the light behaves as a free-space solution only in the asymptotic region. As such, we believe that our approach opens up a whole new avenue for the design of dielectric structures with novel properties that do not rely on special material parameters such as a negative refractive index [50] or a dielectric constant with epsilon-near-zero [51]. The key characteristic of our new method is, instead, a tailored gain–loss distribution, which could be added to the medium in a flexible post-fabrication procedure using adaptive pumping [52].

Funding

European Commission (MSCA-RISE 691209); Austrian Science Fund (FWF) (P32300).

Acknowledgment

The authors would like to thank Matthias Kühmayer for his assistance during the early stages of the project. A.B. acknowledges the Austrian Academy of Sciences for providing a DOC Fellowship at the Institute of Theoretical Physics of Vienna University of Technology (TU Wien). I.K. acknowledges a Postdoctoral Fellowship from the Scholarship Foundation of the Republic of Austria. The computational results presented in this paper were achieved using the Vienna Scientific Cluster (VSC).

Disclosures

The authors declare no competing financial interests or conflicts of interest.

 

See Supplement 1 for supporting content.

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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780 (2006).
    [Crossref]
  2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777 (2006).
    [Crossref]
  3. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387 (2010).
    [Crossref]
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2020 (3)

D. Zezyulin and V. V. Konotop, “A universal form of localized complex potentials with spectral singularities,” New J. Phys. 22, 013057 (2020).
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Y. Zeng, U. Chattopadhyay, B. Zhu, B. Qiang, J. Li, Y. Jin, L. Li, A. G. Davies, E. H. Linfield, B. Zhang, Y. Chong, and Q. J. Wang, “Electrically pumped topological laser with valley edge modes,” Nature 578, 246 (2020).
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Z. K. Shao, H. Z. Chen, S. Wang, X. R. Mao, Z. Q. Yang, S. L. Wang, X. X. Wang, X. Hu, and R. M. Ma, “A high-performance topological bulk laser based on band-inversion-induced reflection,” Nat. Nanotechnol. 15, 67 (2020).
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2019 (5)

K. Pichler, M. Kühmayer, J. Böhm, A. Brandstötter, P. Ambichl, U. Kuhl, and S. Rotter, “Random anti-lasing through coherent perfect absorption in a disordered medium,” Nature 567, 351 (2019).
[Crossref]

S. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Nat. Mater. 18, 783 (2019).
[Crossref]

M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

S. A. R. Horsley, “Indifferent electromagnetic modes: bound states and topology,” Phys. Rev. A 100, 053819 (2019).
[Crossref]

A. Brandstötter, K. G. Makris, and S. Rotter, “Scattering-free pulse propagation through invisible non-Hermitian media,” Phys. Rev. B 99, 115402 (2019).
[Crossref]

2018 (8)

E. Rivet, A. Brandstötter, K. G. Makris, H. Lissek, S. Rotter, and R. Fleury, “Constant-pressure sound waves in non-Hermitian disordered media,” Nat. Phys. 14, 942 (2018).
[Crossref]

S. Yu, X. Piao, and N. Park, “Bohmian photonics for independent control of the phase and amplitude of waves,” Phys. Rev. Lett. 120, 193902 (2018).
[Crossref]

J. W. Yoon, Y. Choi, C. Hahn, G. Kim, S. H. Song, K. Y. Yang, J. Y. Lee, Y. Kim, C. S. Lee, J. K. Shin, H. S. Lee, and P. Berini, “Time-asymmetric loop around an exceptional point over the full optical communications band,” Nature 562, 86 (2018).
[Crossref]

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nat. Phys. 14, 11 (2018).
[Crossref]

S. Longhi, “Parity-time symmetry meets photonics: a new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2018).
[Crossref]

A. Kord, D. L. Sounas, and A. Alù, “Active microwave cloaking using parity-time-symmetric satellites,” Phys. Rev. Appl. 10, 054040 (2018).
[Crossref]

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

2017 (9)

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187 (2017).
[Crossref]

W. Chen, S. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192 (2017).
[Crossref]

F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, Y. Ma, and S. He, “Transformation optics: from classic theory and applications to its new branches,” Laser Photon. Rev. 11, 1700034 (2017).
[Crossref]

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
[Crossref]

K. G. Makris, A. Brandstötter, P. Ambichl, Z. H. Musslimani, and S. Rotter, “Wave propagation through disordered media without backscattering and intensity variations,” Light Sci. Appl. 6, e17035 (2017).
[Crossref]

P. Sebbah, “A channel of perfect transmission,” Nat. Photonics 11, 337 (2017).
[Crossref]

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nat. Photonics 11, 752 (2017).
[Crossref]

F. Loran and A. Mostafazadeh, “Perfect broadband invisibility in isotropic media with gain and loss,” Opt. Lett. 42, 5250 (2017).
[Crossref]

2016 (4)

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A 93, 031802 (2016).
[Crossref]

S. Longhi, “Bidirectional invisibility in Kramers–Kronig optical media,” Opt. Lett. 41, 3727 (2016).
[Crossref]

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref]

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[Crossref]

2015 (4)

D. L. Sounas, R. Fleury, and A. Alù, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4, 014005 (2015).
[Crossref]

K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, and S. Rotter, “Constant-intensity waves and their modulation instability in non-Hermitian potentials,” Nat. Commun. 6, 7257 (2015).
[Crossref]

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9, 15 (2015).
[Crossref]

S. A. R. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramers–Kronig relations and the reflection of waves,” Nat. Photonics 9, 436 (2015).
[Crossref]

2014 (2)

V. V. Konotop and D. A. Zezyulin, “Families of stationary modes in complex potentials,” Opt. Lett. 39, 5535 (2014).
[Crossref]

N. Bachelard, S. Gigan, X. Noblin, and P. Sebbah, “Adaptive pumping for spectral control of random lasers,” Nat. Phys. 10, 426 (2014).
[Crossref]

2013 (2)

X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38, 2821 (2013).
[Crossref]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. B. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2013).
[Crossref]

2012 (2)

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337, 549 (2012).
[Crossref]

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4, 5277 (2012).
[Crossref]

2011 (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889 (2011).
[Crossref]

2010 (3)

Y. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref]

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

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010).
[Crossref]

2008 (1)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

2007 (1)

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[Crossref]

2006 (2)

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

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

2004 (1)

D. R. Smith, J. B. Pendry, and M. C. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788 (2004).
[Crossref]

1997 (1)

J. Schöberl, NETGEN: an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Vis. Sci. 1, 41 (1997).
[Crossref]

Almeida, V. R.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. B. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108 (2013).
[Crossref]

Alù, A.

M. A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

A. Kord, D. L. Sounas, and A. Alù, “Active microwave cloaking using parity-time-symmetric satellites,” Phys. Rev. Appl. 10, 054040 (2018).
[Crossref]

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
[Crossref]

D. L. Sounas, R. Fleury, and A. Alù, “Unidirectional cloaking based on metasurfaces with balanced loss and gain,” Phys. Rev. Appl. 4, 014005 (2015).
[Crossref]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[Crossref]

Ambichl, P.

K. Pichler, M. Kühmayer, J. Böhm, A. Brandstötter, P. Ambichl, U. Kuhl, and S. Rotter, “Random anti-lasing through coherent perfect absorption in a disordered medium,” Nature 567, 351 (2019).
[Crossref]

K. G. Makris, A. Brandstötter, P. Ambichl, Z. H. Musslimani, and S. Rotter, “Wave propagation through disordered media without backscattering and intensity variations,” Light Sci. Appl. 6, e17035 (2017).
[Crossref]

Artoni, M.

S. A. R. Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramers–Kronig relations and the reflection of waves,” Nat. Photonics 9, 436 (2015).
[Crossref]

Aubry, A.

J. B. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, “Transformation optics and subwavelength control of light,” Science 337, 549 (2012).
[Crossref]

Bachelard, N.

N. Bachelard, S. Gigan, X. Noblin, and P. Sebbah, “Adaptive pumping for spectral control of random lasers,” Nat. Phys. 10, 426 (2014).
[Crossref]

Bahari, B.

B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, “Nonreciprocal lasing in topological cavities of arbitrary geometries,” Science 358, 636 (2017).
[Crossref]

Bandres, M. A.

G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, “Topological insulator laser: theory,” Science 359, eaar4003 (2018).
[Crossref]

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Baranov, D. G.

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
[Crossref]

Berini, P.

J. W. Yoon, Y. Choi, C. Hahn, G. Kim, S. H. Song, K. Y. Yang, J. Y. Lee, Y. Kim, C. S. Lee, J. K. Shin, H. S. Lee, and P. Berini, “Time-asymmetric loop around an exceptional point over the full optical communications band,” Nature 562, 86 (2018).
[Crossref]

Böhm, J.

K. Pichler, M. Kühmayer, J. Böhm, A. Brandstötter, P. Ambichl, U. Kuhl, and S. Rotter, “Random anti-lasing through coherent perfect absorption in a disordered medium,” Nature 567, 351 (2019).
[Crossref]

J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76 (2016).
[Crossref]

Brandstötter, A.

A. Brandstötter, K. G. Makris, and S. Rotter, “Scattering-free pulse propagation through invisible non-Hermitian media,” Phys. Rev. B 99, 115402 (2019).
[Crossref]

K. Pichler, M. Kühmayer, J. Böhm, A. Brandstötter, P. Ambichl, U. Kuhl, and S. Rotter, “Random anti-lasing through coherent perfect absorption in a disordered medium,” Nature 567, 351 (2019).
[Crossref]

E. Rivet, A. Brandstötter, K. G. Makris, H. Lissek, S. Rotter, and R. Fleury, “Constant-pressure sound waves in non-Hermitian disordered media,” Nat. Phys. 14, 942 (2018).
[Crossref]

K. G. Makris, A. Brandstötter, P. Ambichl, Z. H. Musslimani, and S. Rotter, “Wave propagation through disordered media without backscattering and intensity variations,” Light Sci. Appl. 6, e17035 (2017).
[Crossref]

Cao, H.

W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science 331, 889 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Y. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref]

Chan, C. T.

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

Chattopadhyay, U.

Y. Zeng, U. Chattopadhyay, B. Zhu, B. Qiang, J. Li, Y. Jin, L. Li, A. G. Davies, E. H. Linfield, B. Zhang, Y. Chong, and Q. J. Wang, “Electrically pumped topological laser with valley edge modes,” Nature 578, 246 (2020).
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Supplementary Material (1)

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» Supplement 1       Supplemental document

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Figures (4)

Fig. 1.
Fig. 1. Scattering of a Gaussian beam of width ${w_0} = 5\lambda$ by a simple dipole-shaped potential. The real and imaginary parts of the non-Hermitian refractive index distribution $n(x,y) = {n_R}(x,y) + i{n_I}(x,y)$ are presented in (a), (b), respectively. The diffraction intensity patterns of an incident Gaussian beam are shown in (c) for the Hermitian $n(x,y) = {n_R}(x,y)$ and (d) for the corresponding non-Hermitian medium $n(x,y) = {n_R}(x,y) + i{n_I}(x,y)$. Panel (d) demonstrates how the non-Hermitian mapping of Eq. (4) entirely suppresses the scattering of the Gaussian beam due to the presence of gain and loss. The white dashed rectangles in (c), (d) denote the limits of the scattering region, depicted in (a), (b) (homogeneous space with ${n_{\rm ref}} = 3$ is assumed outside).
Fig. 2.
Fig. 2. Scattering of a Gaussian beam of width ${w_0} = 5\lambda$ through a disordered medium calculated through a generating function $\theta (x,y)$ given by a random superposition of $N = 300$ Gaussians (see Supplement 1). Real and imaginary parts of the medium’s refractive index $n(x,y)$ are shown in (a) and (b), respectively. In (c), we show how the intensity of the incoming beam produces a complicated interference pattern when the medium consists of only the real part of the refractive index ${n_R}(x,y)$ as shown in (a). In (d), the same incoming beam propagates through this medium with the imaginary part ${n_I}(x,y)$ included according to Eq. (4). Here the intensity pattern of the beam is the same as in a homogeneous medium. The white dashed rectangles in (c) and (d) denote the limits of the scattering region, depicted in (a) and (b) (homogeneous space with ${n_{\rm ref}} = 3$ outside).
Fig. 3.
Fig. 3. A Gaussian beam of width ${w_0} = 5\lambda$ entering a disordered medium at a tilt angle of $\alpha {= 3^ \circ}$ with respect to the $x$ axis. (a) The real and imaginary parts of the refractive index $n(x,y)$ are those of Fig. 2, following the design using a Gaussian reference beam without tilt ($\alpha {= 0^ \circ}$). The angular mismatch of 3° leads to distortions that ultimately break up the beam during propagation. (b) Here we take into account the tilt of the incoming beam by using a Gaussian reference solution ${\phi _G}(x,y)$ in Eq. (4) with an adjusted incidence angle of $\alpha {= 3^ \circ}$. The incoming beam at this angle propagates through this medium without any distortions. As in Fig. 2, the white dashed rectangles denote the limits of the scattering region.
Fig. 4.
Fig. 4. Propagation through a disordered refractive index distribution designed to support a CI wave solution (see text). (a) Imaginary part of the refractive index distribution (the real part is similar to the one in Fig. 2a; see Supplement 1). (b) Intensity of Gaussian beam with a width ${w_0} = 5\lambda$ (smaller than the width of the scattering region), exhibiting sizeable distortions during propagation. (c) Intensity distribution for a super-Gaussian beam with a width larger than that of the scattering region $[E(x = 0,y) = {\exp(-y^8/w_0^8)}$, ${w_0} = 36.35\lambda]$. Here the shape is maintained, since the beam approximates well a CI wave inside the scattering region. The dashed rectangles in (b) and (c) denote the limits of the scattering region in (a).

Equations (6)

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2 E ( x , y ) + k 2 [ n r e f 2 + ε ( x , y ) ] E ( x , y ) = 0 ,
2 ϕ ( x , y ) + k 2 n r e f 2 ϕ ( x , y ) = 0.
E ( x , y ) = ϕ ( x , y ) e i k θ ( x , y ) ,
ε ( x , y ) = ( θ ) 2 i k ( 2 θ + 2 θ ϕ ϕ ) .
ϕ ( x , y ) = ϕ A ( x , y ) e i ϕ P ( x , y ) ,
( θ ) 2 + 2 k θ ϕ P = ε R .

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