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

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[Crossref]

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[Crossref]

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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).
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[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]

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[Crossref]

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[Crossref]

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[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).
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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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