Abstract

Giant transmission and reflection of a finite bandwidth are demonstrated at the same wavelength when the electromagnetic wave is incident on a subwavelength array of parity-time PT symmetric dimers embedded in a metallic film. Remarkably, this phenomenon vanishes if the metallic substrate is lossless while keeping other parameters unchanged. Moreover super scattering can also occur when increasing the loss of the dimers while keeping the gain unchanged. When the metafilm is adjusted to the vicinity of an exceptional point, tuning either the substrate dissipation or the loss of the dimers can lead to supper scattering in stark contrast to what would be expected in conventional systems. In addition, increasing the gain of the dimers can increase the absorption near the exceptional point. These phenomena indicate that the PT -synthetic plasmonic metafilm can function as a thinfilm PT -plasmonic laser or absorber depending on the tuning parameter. One implication is that super radiation is possible from a cavity by tuning cavity dissipation or lossy element inside the cavity.

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References

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2015 (2)

J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
[Crossref]

S. Yu, X. Piao, J. Hong, and N. Park, “Progress toward high-Q perfect absorption: A Fano antilaser,” Phys. Rev. A 92, 011802 (2015).
[Crossref]

2014 (14)

Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental Demonstration of a Coherent Perfect Absorber with PT Phase Transition,” Phys. Rev. Lett. 112, 143903 (2014).
[Crossref] [PubMed]

T. S. Luk, S. Campione, I. Kim, S. Feng, Y. C. Jun, S. Liu, J. B. Wright, I. Brener, P. B. Catrysse, S. Fan, and M. B. Sinclair, “Directional perfect absorption using deep subwavelength low-permittivity films,” Phys. Rev. B 90, 085411 (2014).
[Crossref]

R. Fleury, D. L. Sounas, and A. Alú, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113, 023903 (2014).
[Crossref] [PubMed]

S. Savoia, G. Castaldi, Vincenzo Galdi, Andrea Alú, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nature Commun. 54034 (2014).
[Crossref]

L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760 (2014).
[Crossref] [PubMed]

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT Symmetry Breaking in Polarization Space with Terahertz Metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref] [PubMed]

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).
[Crossref]

S. Bittner, B. Dietz, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schäfer, “Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance,” Phys. Rev. E 89032909 (2014).
[Crossref]

B. Peng, S. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328 (2014).
[Crossref] [PubMed]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346, 972 (2014).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time symmetric microring lasers,” Science 346, 975 (2014).
[Crossref] [PubMed]

M. Kang, H.-X. Cui, T.-F. Li, J. Chen, W. Zhu, and M. Premaratne, “Unidirectional phase singularity in ultrathin metamaterials at exceptional points,” Phys. Rev. A 89, 065801 (2014).
[Crossref]

R. Aalipour, “Optical spectral singularities as zero-width resonance frequencies of a Fabry-Perot resonator,” Phys. Rev. A 90, 013820 (2014).
[Crossref]

2013 (7)

F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. Hooft, and M. P. van Exter, “Surface Plasmon Lasing Observed in Metal Hole Arrays,” Phys. Rev. Lett. 110, 206802 (2013).
[Crossref] [PubMed]

V. Y. Fedorov and T. Nakajima, “All-angle collimation of incident light in μ-near-zero metamaterials,” Opt. Express 21, 27789 (2013).
[Crossref]

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

S. Yu, D. R. Mason, X. Piao, and N. Park, “Phase-dependent reversible nonreciprocity in complex metamolecules,” Phys. Rev. B 87, 125143 (2013).
[Crossref]

X. Yin and X. Zhang, “Unidirectional light propagation at exceptional points,” Nature Mater. 12, 175 (2013).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alú, and N. Engheta, “PT Metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
[Crossref] [PubMed]

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express 21, 21651 (2013).
[Crossref] [PubMed]

2012 (6)

S. Feng and K. Halterman, “Coherent perfect absorption in epsilon-near-zero metamaterials,” Phys. Rev. B 86, 165103 (2012).
[Crossref]

H. F. Jones, “Analytic results for a PT-symmetric optical structure,” J. Phys. A 45, 135306 (2012).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Paritytime synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

S. Feng, “Loss-Induced Omnidirectional Bending to the Normal in ε-Near-Zero Metamaterials,” Phys. Rev. Lett. 108, 193904 (2012).
[Crossref]

L. Sun, S. Feng, and X. Yang, “Loss enhanced transmission and collimation in anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 101, 241101 (2012).
[Crossref]

W. D. Heiss, “The physics of exceptional points,” J. Phys. A 45, 444016 (2012).
[Crossref]

2011 (6)

A. Mostafazadeh, “Optical spectral singularities as threshold resonances,” Phys. Rev. A 83, 045801 (2011).
[Crossref]

H. Benisty, A. Degiron, A. Lupu, A. D. Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Léronde, “Implementation of PT symmetric devices using plasmonics: principle and applications,” Opt. Express 19, 18004 (2011).
[Crossref] [PubMed]

S. Longhi, “Invisibility in PT-symmetric complex crystals,” J. Phys. A 44, 485302 (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] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333729 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref] [PubMed]

2010 (7)

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43, 265305 (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,” Nature Phys. 6, 192 (2010).
[Crossref]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent Perfect Absorbers: Time-Reversed Lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref] [PubMed]

S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (2010).
[Crossref]

S. Longhi, “Optical Realization of Relativistic Non-Hermitian Quantum Mechanics,” Phys. Rev. Lett. 105, 013903 (2010).
[Crossref] [PubMed]

2009 (5)

E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17, 8548 (2009).
[Crossref] [PubMed]

A. Mostafazadeh, “Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at Real Energies,” Phys. Rev. Lett. 102, 220402 (2009).
[Crossref] [PubMed]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref] [PubMed]

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A 42, 153001 (2009).
[Crossref]

2008 (2)

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nature Photon. 2, 351 (2008).
[Crossref]

2003 (1)

D. J. Bergman and M.I. Stockman, “Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[Crossref] [PubMed]

2002 (1)

I. Rotter, “Branch points in the complex plane and geometric phases,” Phys. Rev. E 65, 026217 (2002).
[Crossref]

2001 (1)

C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental Observation of the Topological Structure of Exceptional Points,” Phys. Rev. Lett. 86, 787 (2001).
[Crossref] [PubMed]

2000 (1)

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929 (2000).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 805243 (1998).
[Crossref]

1981 (1)

Aalipour, R.

R. Aalipour, “Optical spectral singularities as zero-width resonance frequencies of a Fabry-Perot resonator,” Phys. Rev. A 90, 013820 (2014).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Alaeian, H.

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).
[Crossref]

Almeida, V. R.

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

Alú, A.

R. Fleury, D. L. Sounas, and A. Alú, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113, 023903 (2014).
[Crossref] [PubMed]

G. Castaldi, S. Savoia, V. Galdi, A. Alú, and N. Engheta, “PT Metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
[Crossref] [PubMed]

Alú, Andrea

S. Savoia, G. Castaldi, Vincenzo Galdi, Andrea Alú, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

Ayache, M.

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M. Kang, H.-X. Cui, T.-F. Li, J. Chen, W. Zhu, and M. Premaratne, “Unidirectional phase singularity in ultrathin metamaterials at exceptional points,” Phys. Rev. A 89, 065801 (2014).
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H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time symmetric microring lasers,” Science 346, 975 (2014).
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T. S. Luk, S. Campione, I. Kim, S. Feng, Y. C. Jun, S. Liu, J. B. Wright, I. Brener, P. B. Catrysse, S. Fan, and M. B. Sinclair, “Directional perfect absorption using deep subwavelength low-permittivity films,” Phys. Rev. B 90, 085411 (2014).
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M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT Symmetry Breaking in Polarization Space with Terahertz Metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
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Li, H.-Q.

Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental Demonstration of a Coherent Perfect Absorber with PT Phase Transition,” Phys. Rev. Lett. 112, 143903 (2014).
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J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
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Y. Sun, W. Tan, H.-Q. Li, J. Li, and H. Chen, “Experimental Demonstration of a Coherent Perfect Absorber with PT Phase Transition,” Phys. Rev. Lett. 112, 143903 (2014).
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M. Kang, H.-X. Cui, T.-F. Li, J. Chen, W. Zhu, and M. Premaratne, “Unidirectional phase singularity in ultrathin metamaterials at exceptional points,” Phys. Rev. A 89, 065801 (2014).
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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).
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J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
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T. S. Luk, S. Campione, I. Kim, S. Feng, Y. C. Jun, S. Liu, J. B. Wright, I. Brener, P. B. Catrysse, S. Fan, and M. B. Sinclair, “Directional perfect absorption using deep subwavelength low-permittivity films,” Phys. Rev. B 90, 085411 (2014).
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M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT Symmetry Breaking in Polarization Space with Terahertz Metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
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M. Kang, H.-X. Cui, T.-F. Li, J. Chen, W. Zhu, and M. Premaratne, “Unidirectional phase singularity in ultrathin metamaterials at exceptional points,” Phys. Rev. A 89, 065801 (2014).
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Appl. Phys. Lett. (1)

L. Sun, S. Feng, and X. Yang, “Loss enhanced transmission and collimation in anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 101, 241101 (2012).
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J. Opt. Soc. Am. (1)

J. Phys. A (5)

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Nature (1)

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Paritytime synthetic photonic lattices,” Nature 488, 167 (2012).
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Nature Commun. (1)

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nature Commun. 54034 (2014).
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Nature Mater. (2)

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

X. Yin and X. Zhang, “Unidirectional light propagation at exceptional points,” Nature Mater. 12, 175 (2013).
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Nature Photon. (1)

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nature Photon. 2, 351 (2008).
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Nature Phys. (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192 (2010).
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Opt. Express (5)

Phys. Rev. A (8)

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M. Kang, H.-X. Cui, T.-F. Li, J. Chen, W. Zhu, and M. Premaratne, “Unidirectional phase singularity in ultrathin metamaterials at exceptional points,” Phys. Rev. A 89, 065801 (2014).
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J. Gear, F. Liu, S. T. Chu, S. Rotter, and J. Li, “Parity-time symmetry from stacking purely dielectric and magnetic slabs,” Phys. Rev. A 91, 033825 (2015).
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S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
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A. A. Sukhorukov, Z. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
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S. Yu, X. Piao, J. Hong, and N. Park, “Progress toward high-Q perfect absorption: A Fano antilaser,” Phys. Rev. A 92, 011802 (2015).
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Phys. Rev. B (5)

S. Savoia, G. Castaldi, Vincenzo Galdi, Andrea Alú, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89, 085105 (2014).
[Crossref]

T. S. Luk, S. Campione, I. Kim, S. Feng, Y. C. Jun, S. Liu, J. B. Wright, I. Brener, P. B. Catrysse, S. Fan, and M. B. Sinclair, “Directional perfect absorption using deep subwavelength low-permittivity films,” Phys. Rev. B 90, 085411 (2014).
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S. Yu, D. R. Mason, X. Piao, and N. Park, “Phase-dependent reversible nonreciprocity in complex metamolecules,” Phys. Rev. B 87, 125143 (2013).
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H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89, 075136 (2014).
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S. Feng and K. Halterman, “Coherent perfect absorption in epsilon-near-zero metamaterials,” Phys. Rev. B 86, 165103 (2012).
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Phys. Rev. E (3)

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Figures (12)

Fig. 1
Fig. 1 A schematic showing (a) a unit cell composed of P T -symmetric loss (blue) and gain (red) subwavelength elements embedded in an aluminum substrate and (b) a two-dimensional array of the unit cells in the x-y plane with the same period in both directions. The dimers and the aluminum film have the same thickness, i.e. the metallic mesh is filled with the gain-loss elements. The real part of the relative permittivity of the loss and gain elements is fixed at 3.6 through out this work. The imaginary part varies, but satisfies ε g a i n = ε l o s s to ensure the P T symmetry. The permeability is unit for all the materials. Period p = 3.5 μm, the dimer length a = 2.5 μm, the width b = 1.0 μm, and the separation between the loss and gain Δx = 0.5 μm are fixed throughout the paper. The incidence wave is p-polarized with the electric field parallel to the x-z plane.
Fig. 2
Fig. 2 Transmittance (blue solid curves) and reflectance (red dashed curves) of the normal incident wave on a lossless metafilm versus wavelength (upper panel) and thickness (lower panel) with the electric field parallel to the shorter edge of the dimers. The thickness of the metafilm d = 1.5 μm for the upper panel and the wavelength λ = 6 μm for the lower panel. The relative permittivity of the dimers is real and given by εr = 3.6.
Fig. 3
Fig. 3 Simulation without (left panels) and with (right panels) metallic substrate dissipation. Upper panels: Transmittance (blue solid curves) and reflectance (red dashed curves) of the normal incident beam with the electric field parallel to the shorter edge of the dimers. Lower panels: The magnitude of M11 (blue solid curves) and M21 (red dashed curves). The thickness of the mesh d = 1.5 μm. The relative permittivity of the gain/loss elements is given by ε = 3.6(1 ± i0.06).
Fig. 4
Fig. 4 Eigenvalues and associated eigenfunctions versus wavelength near the exceptional point. (a) Magnitude of the eigenvalues of the transfer matrix and (b) the corresponding phase. (c) Real and (d) imaginary parts of the left eigenfunctions. The mesh thickness d = 1.5 μm. The relative permittivity of the dimers is given by ε = 3.6(1 ± i0.06).
Fig. 5
Fig. 5 Transmission (solid curves) and reflection (dashed curves) vs. wavelength near the exceptional point for different losses of the dimers with a fixed gain εg = 3.6(1 − i0.06) in the absence of the substrate dissipation. The relative permittivity of the loss elements is εl = 3.6(1 + i0.06 + l). The blue curves δl = 0 correspond to the balanced loss and gain where the lasing does not occur. The transmission and reflection increase with the increase of δl, reach a maximum at δl = 0.43, and then reduce. The mesh thickness d = 1.5 μm.
Fig. 6
Fig. 6 Movement of the pole of the scattering matrix in a complex wavelength plane (x-axis: real λ; y-axis: imaginary λ) when increasing the loss of the dimers with a fixed gain. The arrow indicates the direction of the increase: δl = 0.1,0.15,0.2,0.25,0.3,0.35,0.4, and 0.43. The mesh thickness d = 1.5 μm. When δl = 0.43, ℑ(λ) = 0 (lasing occurs), corresponding to the highest peak in Fig. 5.
Fig. 7
Fig. 7 Gain controlled infrared absorption in the vicinity of the exceptional point with a fixed loss εl = 3.6(1 + i0.06) of the dimers in the absence of the substrate dissipation. The permittivity of the gain is given by εg = 3.6(1 − i0.06 − g). The mesh thickness d = 1.5 μm. Both absorption peak and bandwidth increase when increasing the gain. The maximum absorption is about 50% at the δg = 0.5. Further increasing δg, the absorption peak reduces but the bandwidth continues to increase. The negative absorption represents the regions of amplification.
Fig. 8
Fig. 8 Mode property of the metafilm. (a) Real and (b) imaginary parts of the normalized propagation constant versus wavelength without (blue dashed curves) and with (red solid curves) the substrate dissipation. The golden starts in (a) correlate to the free space excitation in Fig. 9. The corresponding imaginary part is zero. The arrow indicates the direction of increasing (polar) angle. The mesh thickness d = 1.5 μm. The relative permittivity of the dimers ε = 3.6(1 ± i0.06).
Fig. 9
Fig. 9 (a) Transmittance of a p-polarized wave and (b) the corresponding phase of the electric field for different (polar) angles of incidence from 0° to 40°. The parameters are the same as those in Fig. 3. The transmission maximums correspond to the excitation of the radiative modes, which are correlated to the golden stars in Fig. 8(a).
Fig. 10
Fig. 10 Transmittance (blue solid curves) and reflectance (red dashed curves) vs. wavelength when the beam is normally incident on the metafilm of thickness (a) d = 1.91 μm, (b) d = 1.96 μm. The lasing frequency or the resonant scattering frequency can be tuned by varying the thickness of the metafilm. The relative permittivity of the dimers is given by ε = 3.6(1 ± i0.056).
Fig. 11
Fig. 11 Transmittance (blue solid curves) and reflectance (red dashed curves) vs. wavelength for different diffraction orders. (a) Zero-order of p-polarized component (main diffraction). (b) Zero-order of s-polarized component (main cross-polarization). (c) The (11)-order scattering including both polarizations. (d) All the scattering orders including both polarizations minus the main diffraction (p00-component). Simulation parameters are the same as those in Fig. 2.
Fig. 12
Fig. 12 Numerical validations of the mathematical conditions. (a) Bi-orthogonality Eq. 12. (b) Identity det(M) = 1 (blue solid line) and | η + t η t | = 1 (red dashed line). Relationships (c) S21 = S12 or |S21S12| = 0, and (d) M12 = −M21 or |M12 + M21| = 0. Simulation parameters are the same as those in Fig. 4.

Equations (13)

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ε m = 1 ω p 2 ω 2 + i γ ω ,
i z ( E t z ^ × H t ) = H ˜ ( E t z ^ × H t ) ,
H ˜ = ( 0 k 0 μ t I ^ t + 1 k 0 t 1 ε z t k 0 ε t I ^ t + 1 k 0 z ^ × t 1 μ z z ^ × t 0 ) ,
t x ^ x + y ^ y .
| Ψ i = ( M 11 M 12 M 21 M 22 ) | Ψ o ,
| Ψ ( E t z ^ × H t ) .
S 11 = M 21 M 11 1 , S 21 = M 11 1 , S 22 = M 11 1 M 12 , S 12 = M 22 M 21 M 11 1 M 12 ,
η ± t = M 11 + M 22 2 ± ( M 11 + M 22 2 ) 2 1 .
η ± s = M 21 ± 1 M 11 .
| Ψ ± r = ( η ± t M 22 M 21 1 ) ,
| Ψ ± l = ( η ± t M 22 M 12 1 )
ψ + l | ψ r = ψ l | ψ + r = 0.
η + s , η s M 22 2 .

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