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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 2 — Jan. 27, 2014
  • pp: 1760–1767
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Demonstration of a large-scale optical exceptional point structure

Liang Feng, Xuefeng Zhu, Sui Yang, Hanyu Zhu, Peng Zhang, Xiaobo Yin, Yuan Wang, and Xiang Zhang  »View Author Affiliations


Optics Express, Vol. 22, Issue 2, pp. 1760-1767 (2014)
http://dx.doi.org/10.1364/OE.22.001760


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Abstract

We report a large-size (4-inch) optical exceptional point structure at visible frequencies by designing a multilayer structure of absorbing and non-absorbing dielectrics. The optical exceptional point was implemented as indicated by the realized unidirectional reflectionless light transport at a wafer scale. The associated abrupt phase transition is theoretically and experimentally confirmed when crossing over the exceptional point in wavelengths. The large scale demonstration of phase transition around exceptional points will open new possibilities in important applications in free space optical devices.

© 2013 Optical Society of America

1. Introduction

Optical losses are responsible for power attenuation, and thus typically degrade the performance of optical devices. Consequently, conventional optical devices are usually made of non-absorbing transparent dielectric materials. Most recently, it has been suggested that the use of highly-absorbing dielectrics could result in intriguing optical phenomena [1

1. M. A. Kats, R. Blanchard, P. Genevet, and F. Capasso, “Nanometre optical coatings based on strong interference effects in highly absorbing media,” Nat. Mater. 12(1), 20–24 (2013). [CrossRef] [PubMed]

]. If material losses are further designed intercorrelated with the index of refraction, unique optical functionalities can be expected. Judiciously designed photonic synthetic matters, which take advantage of optical losses by engineering the entire complex dielectric permittivity plane, may be investigated to mimic the complex non-Hermitian Hamiltonians in quantum mechanics [2

2. C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]

] and thus realize unique directional light transport and novel devices [3

3. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). [CrossRef] [PubMed]

9

9. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010). [CrossRef]

] at the parity-time phase transition point, a type of exceptional points. However, exceptional point is much more general concept, contains broader and richer physics, and should thus be expected in a larger non-Hermitian family: exceptional points are singularities in the complex eigen spectrum and inherently associated with degenerate eigen states, leading to a series of interesting physical effects including level repulsion and crossing, bifurcation and chaos, as well as phase transition in a number of open quantum system configurations [10

10. Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104(15), 153601 (2010). [CrossRef] [PubMed]

12

12. 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(5), 787–790 (2001). [CrossRef] [PubMed]

].

In this work, driven by the unique transport behaviors enabled by non-Hermiticity, we exploited absorbing dielectric materials such as silicon (Si) to achieve a large size planar optical structure with an exceptional point at visible frequencies. Instead of minimizing optical losses, we intentionally utilized the losses together with the modulated index of refraction. By engineering their interplay, we developed a 4-inch sized multilayer exceptional structure and demonstrated its associated accidental degeneracy at the exceptional point. The implemented exceptional point also accounted for an abrupt change in the generalized power spectrum observed experimentally, manifesting a phase transition at a wafer scale. Moreover, because of the intrinsic planar configuration, our exceptional point structure is highly scalable in fabrication and can be implemented in both micro- and macro-scale applications.

2. Exceptional point and phase transition

Similar to the Hamiltonian matrix in quantum mechanics, light propagation in a dielectric medium can be described using the corresponding optical scattering matrix, which relates the initial state and the final state of an optical system. Here, for simplicity in the design, we will consider a two-port optical system whose scattering matrix is described asS=(trbrft) [12

12. 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(5), 787–790 (2001). [CrossRef] [PubMed]

14

14. F. Cannata, J. P. Dedonder, and A. Ventura, “Scattering in PT-symmetric quantum mechanics,” Ann. Phys. 322(2), 397–433 (2007). [CrossRef]

], where t is the transmission coefficient, rf and rb are reflection coefficients in forward and backward directions, respectively. As quantum exceptional points are designed by engineering the Hamiltonian matrix [10

10. Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104(15), 153601 (2010). [CrossRef] [PubMed]

], optical exceptional points can be achieved by manipulating the elements in the corresponding optical scattering matrix. The corresponding eigen values of the studied two-port system are sn=t±rfrb. As stated above, the desired optical exceptional point require the realization of degeneracy of two eigen values of such an optical system. Apparently, two eigen values become degenerate to form exceptional points in the complex eigen spectrum when either rf or rb is zero, i.e., the system is unidirectional reflectionless.

Here, we will design and demonstrate a general optical exceptional point rather than the special parity-time phase transition point in a large-scale free space device. A multilayer structure consisted of absorbing and non-absorbing dielectrics at visible frequencies, i.e., amorphous Si and silica, respectively [Fig. 1(a)
Fig. 1 (a) Schematic of the optical exceptional point structure on a glass wafer designed at the wavelength of 532 nm, where nSi = 4.86 – j0.65 and nsilica = 1.46. (b) Complex eigen value solutions of the modified scattering matrix at 532 nm construct a multi-valued Riemann surface as a function of Im(nSi), where red color indicates optical gain and blue corresponds to absorption as we have in Si layers. (c) Simulations of light propagating through the exceptional point structure in the forward (upper panel) and backward (lower panel) directions, in which the distribution of the amplitude of the magnetic field is shown normalized to the incident magnetic field of 1. (d) and (e) are amplitude and phase of the magnetic field in the device as light propagate in the forward (red) and backward (black) directions, respectively.
], to tailor the asymmetric optical response. Each unit cell consisted of 4 alternating thin film layers of Si and silica (23 nm Si/ 26 nm silica/ 9 nm Si/43 nm silica).

In this multi-layer system, light propagation in each layer j can be described by the matrix
Mj=(coskjLj1kjsinkjLjkjsinkjLjcoskjLj),j=1,2,...,N
(1)
where kj is the corresponding wave number, Lj the thickness of the layer, and N is the total number of layers. It is worth noting that this transfer matrix method is valid for the 1-dimentsional system, so in the design we assume each layer is infinite in the transverse dimensions. Therefore the transfer matrix of the entire system can be written as Mf=MN...M2M1 for light propagating in the forward direction from layer 1 to layer N. The corresponding transmission and reflection coefficients can be derived:
t=2ik0eik0L[M11fM22fM12fM21fM21f+k02M21f+ik0(M11f+M22f)]
(2)
and
rf=[(M21f+k02M12f)+ik0(M22fM11f)(M21f+k02M12f)+ik0(M22f+M11f)],
(3)
where L is the total thickness of the multi-layer system and k0 is the wave vector in free space. When light is propagating in the backward direction, the transfer matrix then becomes Mb=M1...MN1MN. The corresponding reflection coefficient can be derived accordingly:
rb=[(M21b+k02M12b)+ik0(M22bM11b)(M21b+k02M12b)+ik0(M22b+M11b)].
(4)
The desirable unidirectional characteristics (rf=0) was accomplished at the wavelength of 532 nm by simultaneously engineering the thickness of Si and silica layers according to Eqs. (1)(4), such that the accumulated phase change in both layers is in-phase correlated with absorption due to the imaginary part of the refractive index of Si. The absorption of Si layers caused strong attenuation in transmission. Although transmissions are the same in both directions, the multilayer structure shows significant direction-dependent response in reflection. In the forward direction, light constructively interferes in the Si layers but destructively interferes at the top surface of the structure, leading to a zero reflection coefficient in the complex plane with strong absorption in the structure; while in the backward direction, light is mainly confined in the silica layers with a destructive interference at the bottom surface of the structure, resulting in high reflection. Although single unit cell is unidirectional reflectionless by itself, the structure in the design comprises 4 periodic unit cells to further enhance the backward reflection, while still keeping the forward reflection zero. Compared to the previously studied parity-time symmetric optical materials [5

5. 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(3), 192–195 (2010). [CrossRef]

,7

7. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef] [PubMed]

,8

8. 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,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef] [PubMed]

], the geometry of our current exceptional point structure is less complicated for implementing the optical exceptional point and can be performed using the well-developed thin-film techniques.

Because of the associated accidental degeneracy, exceptional points are typically responsible for bifurcation and phase transition in open quantum systems [10

10. Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104(15), 153601 (2010). [CrossRef] [PubMed]

,16

16. H. Cartarius, J. Main, and G. Wunner, “Exceptional points in atomic spectra,” Phys. Rev. Lett. 99(17), 173003 (2007). [CrossRef] [PubMed]

]. In our structure, the optical exceptional point also experiences a phase transition as a function of wavelengths according to the generalized power representation T+RfRb [17

17. L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012). [CrossRef]

], where T=|t|2 is transmittance, and Rf=|rf|2and Rb=|rb|2are reflectance in forward and backward directions, respectively. The generalized power relates all the elements in the scattering matrix and represents the power summation of superpositions of the two eigen states derived from the scattering matrix. According to Eqs. (2)(4), the generalized power spectrum and its differential spectrum can be obtained as shown in Fig. 2
Fig. 2 (a) and (b) are calculated spectra of generalized total power and its partial derivatives as a function of wavelength detuning. The red curves denote the spectra with the exact exceptional point design, while the blue dash curves are the case near the exceptional point.
, in which material dispersions are also considered in calculations. Here, wavelength detuning is defined asΔ=λλEP, where λEP corresponds to the designed exceptional point wavelength where reflection is forward reflectionlessness. It is evident that, as shown in Fig. 2(a), the exceptional point (Δ = 0) divides the generalized power spectrum into two phases: a power-decreasing phase at Δ < 0 and the other with increasing generalized power at Δ > 0. Moreover, the differential generalized power spectrum manifests an abrupt phase change at the exceptional point [Fig. 2(b)], which resembles the classical solid-liquid phase transition in which the derivative of the Gibbs free energy on temperature is discontinuous at the transition point. These results originate from the realized optical exceptional point as well as its associated level crossing. If crossing over the exceptional point in any parameter axis such as wavelengths in our case, optical phase of reflected light undergoes an abrupt π phase jump, similar to the phase transition from the parity-time symmetric phase to the parity-time broken phase in the parity-time symmetric systems, in which the chosen parameter axis is the amplitude of the imaginary part of the index modulation [5

5. 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(3), 192–195 (2010). [CrossRef]

8

8. 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,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef] [PubMed]

].

3. Experimental validation

The 4-inch exceptional point structure was fabricated by alternating thin film depositions of amorphous Si and silica layers on a cleaned glass wafer using plasma enhanced chemical vapor deposition (PECVD) [Fig. 3(a)
Fig. 3 (a) SEM pictures of the cross-section of the fabricated exceptional point structure. Scale bar, 50 nm. (b) and (c) are numerically calculated and experimentally measured reflection spectra of the exceptional point structure from 450 nm to 600 nm, respectively, for both forward (red) and backward (black) incidence.
]. Although the PECVD process is not able to realize atomically flat thin films as we assumed in analysis and simulations, the resulted non-uniformity is on the order of nanometers, which is so small that light still “sees” flat thin films when propagating through.

The reflection spectra of the structure for both forward and backward directions were numerically calculated and experimentally measured, as shown in Figs. 3(b) and 3(c), respectively. The results consistently showed that with normal incident light, reflection spectra in forward and backward directions are in significant contrast within the spectral range of interest. The unidirectional reflectionless light transport was clearly demonstrated around the wavelength of 520 nm in measurements, where reflection in the forward direction was diminished owing to the implemented exceptional point. The small discrepancy between the calculations and experiments may result from the imperfect control of the film thickness in the PECVD process. Additionally, the PECVD Si films may contain less absorption because of the hydrogenation in the process compared to the index used in the calculations.

With the data in Fig. 3(c), the experimentally measured generalized power spectrum and its differential one can be derived as shown in Fig. 4
Fig. 4 Experimentally measured generalized power spectrum (a) and its differential spectrum (b) around the exceptional point.
. Phase transition from decreasing power to increasing power can be clearly observed around the exceptional point [Fig. 4(a)], but the transition in the differential spectrum is not as abrupt as predicted in Fig. 2(b). The fact that the spectrum in Fig. 4(b) is still differential at Δ = 0 is because it is difficult to exactly reach the exceptional point in experiments owing to its inherent singularity. The designed exceptional point requires the implemented geometry parameters to exactly match the design, which is hard to achieve in experiments. For example, the absorption of the PECVD Si film is typically smaller and thin film thickness may slightly vary in the fabrication. However, even though the structure parameters are slightly off the exceptional point (see Fig. 2 for the case near the exceptional point in which we considered lower absorption for Si films with Im(nSi)=0.6), Δ = 0 still divides the generalized power spectrum into two phases and the drastic phase change in the differential power spectrum can be convincingly observed in both experiments and calculations, though the slope in the generalized power spectrum at Δ > 0 is different than that in calculations (Fig. 2) due to the imperfect PECVD fabrication. It is expected that the transition becomes sharper when it is even closer to the exceptional point.

4. Conclusion

Acknowledgments

This work was supported by the U.S. Department of Energy, Basic Energy Sciences Energy Frontier Research Center (DoE-LMI-EFRC) under award DOE DE-AC02-05CH11231.

References and links

1.

M. A. Kats, R. Blanchard, P. Genevet, and F. Capasso, “Nanometre optical coatings based on strong interference effects in highly absorbing media,” Nat. Mater. 12(1), 20–24 (2013). [CrossRef] [PubMed]

2.

C. M. Bender and S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]

3.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). [CrossRef] [PubMed]

4.

S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008). [CrossRef] [PubMed]

5.

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(3), 192–195 (2010). [CrossRef]

6.

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(21), 213901 (2011). [CrossRef] [PubMed]

7.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef] [PubMed]

8.

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,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef] [PubMed]

9.

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010). [CrossRef]

10.

Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, and K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104(15), 153601 (2010). [CrossRef] [PubMed]

11.

S. B. Lee, J. Yang, S. Moon, S. Y. Lee, J. B. Shim, S. W. Kim, J. H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103(13), 134101 (2009). [CrossRef] [PubMed]

12.

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(5), 787–790 (2001). [CrossRef] [PubMed]

13.

J. G. Muga, J. P. Palao, B. Navarroa, and I. L. Egusquiza, “Complex absorbing potentials,” Phys. Rep. 395(6), 357–426 (2004). [CrossRef]

14.

F. Cannata, J. P. Dedonder, and A. Ventura, “Scattering in PT-symmetric quantum mechanics,” Ann. Phys. 322(2), 397–433 (2007). [CrossRef]

15.

W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. Math. Gen. 37(6), 2455–2464 (2004). [CrossRef]

16.

H. Cartarius, J. Main, and G. Wunner, “Exceptional points in atomic spectra,” Phys. Rev. Lett. 99(17), 173003 (2007). [CrossRef] [PubMed]

17.

L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012). [CrossRef]

18.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010). [CrossRef] [PubMed]

19.

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012). [CrossRef] [PubMed]

OCIS Codes
(310.4165) Thin films : Multilayer design
(310.6845) Thin films : Thin film devices and applications

ToC Category:
Thin Films

History
Original Manuscript: November 4, 2013
Revised Manuscript: December 17, 2013
Manuscript Accepted: December 18, 2013
Published: January 17, 2013

Citation
Liang Feng, Xuefeng Zhu, Sui Yang, Hanyu Zhu, Peng Zhang, Xiaobo Yin, Yuan Wang, and Xiang Zhang, "Demonstration of a large-scale optical exceptional point structure," Opt. Express 22, 1760-1767 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1760


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References

  1. M. A. Kats, R. Blanchard, P. Genevet, F. Capasso, “Nanometre optical coatings based on strong interference effects in highly absorbing media,” Nat. Mater. 12(1), 20–24 (2013). [CrossRef] [PubMed]
  2. C. M. Bender, S. Böttcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]
  3. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). [CrossRef] [PubMed]
  4. S. Klaiman, U. Günther, N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008). [CrossRef] [PubMed]
  5. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]
  6. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011). [CrossRef] [PubMed]
  7. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef] [PubMed]
  8. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef] [PubMed]
  9. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010). [CrossRef]
  10. Y. Choi, S. Kang, S. Lim, W. Kim, J. R. Kim, J. H. Lee, K. An, “Quasieigenstate coalescence in an atom-cavity quantum composite,” Phys. Rev. Lett. 104(15), 153601 (2010). [CrossRef] [PubMed]
  11. S. B. Lee, J. Yang, S. Moon, S. Y. Lee, J. B. Shim, S. W. Kim, J. H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103(13), 134101 (2009). [CrossRef] [PubMed]
  12. C. Dembowski, H. D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, A. Richter, “Experimental observation of the topological structure of exceptional points,” Phys. Rev. Lett. 86(5), 787–790 (2001). [CrossRef] [PubMed]
  13. J. G. Muga, J. P. Palao, B. Navarroa, I. L. Egusquiza, “Complex absorbing potentials,” Phys. Rep. 395(6), 357–426 (2004). [CrossRef]
  14. F. Cannata, J. P. Dedonder, A. Ventura, “Scattering in PT-symmetric quantum mechanics,” Ann. Phys. 322(2), 397–433 (2007). [CrossRef]
  15. W. D. Heiss, “Exceptional points of non-Hermitian operators,” J. Phys. Math. Gen. 37(6), 2455–2464 (2004). [CrossRef]
  16. H. Cartarius, J. Main, G. Wunner, “Exceptional points in atomic spectra,” Phys. Rev. Lett. 99(17), 173003 (2007). [CrossRef] [PubMed]
  17. L. Ge, Y. D. Chong, A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures,” Phys. Rev. A 85(2), 023802 (2012). [CrossRef]
  18. N. Liu, M. Mesch, T. Weiss, M. Hentschel, H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010). [CrossRef] [PubMed]
  19. Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012). [CrossRef] [PubMed]

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