Open Advanced Search
Get 20M+ Full-Text Papers For Less Than $1.50/day.
Start a 14-Day Trial for You or Your Team.
Learn More →
One-Way Zero Reflection in an Insulator-Metal-Insulator Structure Using the Transfer Matrix Method
One-Way Zero Reflection in an Insulator-Metal-Insulator Structure Using the Transfer Matrix Method
Noh, Heeso;Choi, Jai-Min
hv photonics Letter One-Way Zero Reﬂection in an Insulator-Metal-Insulator Structure Using the Transfer Matrix Method 1 2, Heeso Noh and Jai-Min Choi * Department of Physics, Kookmin University, Seoul 02707, Korea; firstname.lastname@example.org Division of Science Education, Jeonbuk National University, Jeonju 54896, Korea * Correspondence: email@example.com; Tel.: +82-63-270-2777 Abstract: We numerically demonstrate one-way zero reﬂection using the transfer matrix method. Using simulations, we adjusted the thickness of SiO layers in a simple SiO -Au-SiO layer structure. 2 2 2 We found two solutions, 47 nm-10 nm-32 nm and 71 nm-10 nm-60 nm, which are the thicknesses for one-way zero reﬂection at a wavelength of 560 nm. We conﬁrmed it with reﬂection spectra, where reﬂectance is zero for forwardly incident light and 2.5% for backwardly incident light at the wavelength 560 nm, and thickness 47 nm-10 nm-32 nm. Keywords: reﬂection; absorption; PT-symmetry 1. Introduction Many physical systems have a Hamiltonian with Hermitian properties, these proper- ties lead to real and orthogonal eigenvalues. Recently Hamiltonians with parity-time (PT) symmetry have attracted a lot of interest because their eigenvalues are real below a certain threshold but are complex above that threshold . PT-symmetric systems have a potential V(r) = V ( r). Such a system can also be realized in optics [2,3]. If the permittivity, #, of the system has symmetry such that #(r) = # ( r), then PT-symmetry can be realized Citation: Noh, H.; Choi, J.-M. in an optical systems. Therefore, in order to satisfy PT-symmetry in optics, one should One-Way Zero Reflection in an consider optical gain and loss in the system which, in turn, means # should be represented Insulator-Metal-Insulator Structure as a complex number. This has been demonstrated many times in previous research [4–7]. Using the Transfer Matrix Method. One of the interesting phenomena in systems with PT-symmetry is the existence of an Photonics 2021, 8, 8. https://doi.org/ exceptional point. Exceptional point exists at the threshold where an eigenvalue changes 10.3390/photonics8010008 from a real number to a complex number or vice versa. This behavior seems to be similar to that seen with degenerate states. The eigen modes, however, coalesce at an exceptional Received: 11 December 2020 point. In other words, only one eigen mode can exist, which opens up the possibility of Accepted: 29 December 2020 one-way reﬂection. There has been lots of research that has used this exceptional point in a Published: 31 December 2020 variety of applications such as in single mode lasers or unidirectional reﬂectionlessness Publisher’s Note: MDPI stays neu- among others [8–13]. Recent progress in developing exceptional point-based research is tral with regard to jurisdictional clai- well introduced in Ref. [14,15] and references therein. ms in published maps and institutio- Even though many physical systems use Hamiltonians to describe the physics at play, nal afﬁliations. this is not the only way to describe PT-symmetry. Any kind of system can be considered, all that is required is that the PT-symmetry condition is satisﬁed. In optics, a scattering matrix (s-matrix) is often used to describe optical phenomena. Therefore, an s-matrix is also a candidate to describe PT-symmetric systems. One-way zero reﬂection has been Copyright: © 2020 by the authors. Li- obtained using an s-matrix in previous research [16–18]. Even though the theory for zero censee MDPI, Basel, Switzerland. reﬂection comes from the use of an s-matrix, the reﬂection coefﬁcient can be calculated with This article is an open access article other methods. For layered structures, it is convenient to use the transfer matrix method distributed under the terms and con- (TMM) . ditions of the Creative Commons At- In this report, we use a three-layered structure that is composed of a metal layer tribution (CC BY) license (https:// sandwiched between two dielectric layers to obtain one-way zero reﬂection. We selected creativecommons.org/licenses/by/ Au as a metal layer and SiO as dielectric layers which are favored materials in photonics 4.0/). Photonics 2021, 8, 8. https://doi.org/10.3390/photonics8010008 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 8 2 of 7 research. Although there are demonstrations using Si-based structure [16,17], one-way zero reﬂection using the metal-based layer structure may also ﬁnd its applications in various ﬁelds. By adjusting the thickness of the SiO layers, we obtain zero reﬂection for forwardly incident light while we get a nonzero reﬂection value for backwardly incident light. 2. Simulation Reﬂection and transmission were obtained using TMM for the layered structure. The electric ﬁeld relation between the mth layer and the (m + 1)th layer in the TMM is as follows. ! ! f f E E m+1 = M , (1) b b E E m+1 in kd m m e 0 1/t r /t m,m+1 m,m+1 m,m+1 M = , (2) in kd m m r /t 1/t 0 e m,m+1 m,m+1 m,m+1 where E (E ) is the electric ﬁeld of forwardly (backwardly) propagating light; n and d m m m m are the refractive index and thickness of the mth layer, respectly; k is the wave number in a vacuum; r and t are the reﬂection and transmission coefﬁcients when light m,m+1 m,m+1 propagates from the mth layer to the (m + 1)th layer. For normal incidence, the equations for r and t follow when relative permittivity is 1. m,m+1 m,m+1 n n 2n m m+1 m r = , t = , (3) m,m+1 m,m+1 n + n n + n m m m+1 m+1 where n is the refractive index of the (m + 1)th layer. m+1 If the total number of layers is N and the electric ﬁeld amplitude of incident light is 1, when the propagation direction of incident light is forward, we can write a matrix form of TMM as follows, 1 1/t r /t t 0,1 0,1 0,1 f = M M M , (4) 2 N r r /t 1/t 0 f 0,1 0,1 0,1 where r , t are the total reﬂection and transmission coefﬁcients when the propagation f f direction of incident light is forward. When the propagation direction of incident light is backward, the relation is as follows, 0 1/t r /t r 0,1 0,1 0,1 = M M M . (5) 1 2 N t r /t 1/t 1 b 0,1 0,1 0,1 t r In two port system, s-matrix can be expressed as s = . Because PT- r t symmetric system maintains reciprocity, t = t = t . Furthermore, corresponding eigneval- f b ues are s = t r r . When either r or r is zero, two eigenvalues coalesce to form f b f b exceptional points, i.e., the system has one-way zero reﬂection. The structure for one-way zero reﬂection is depicted in Figure 1a. The structure presented consists of a SiO – Au – SiO layers in vacuum. The thickness of the gold is 2 2 ﬁxed at 10 nm. The reason for this is that Au is highly reﬂective in the visible wavelength range. If the thickness of Au is too high, we cannot achieve coupling between the light in the top SiO layer and that in the bottom SiO layer. On the other hand, if the thickness 2 2 of the Au is too low, we will no longer be able to achieve a continuous ﬁlm with the Au. The Au of 10 nm thickness is right above the percolation threshold . The thicknesses of the top and bottom SiO layers are denoted by d and d , respectively. We adjusted d 2 1 2 1 and d to achieve one-way zero reﬂection. The incident light from the left is denoted as forwardly incident light; the incident light from the right, backwardly incident light. Photonics 2021, 8, 8 3 of 7 (a) (b) 400 450 500 550 600 650 700 750 800 wavelength (nm) Figure 1. (a) Schematic of simulated structure. The three layers seen here are SiO -Au-SiO . We ﬁxed the thickness of Au as 2 2 10 nm, and d and d are varied to achieve one-way zero reﬂection. Incident light coming from the left (right) is denoted as 1 2 forwardly (backwardly) incident light. (b) dispersion curve of Au. The data points marked with the black ﬁlled circles () and the blue open circles () are from Ref. . Solid and dashed curves are lines of best ﬁt. The black curves represents the real part of #, the blue curves represents the imaginary part of #. The relative permittivity, #, is deﬁned as # = n. The # of SiO is 2.13. As the permit- tivity of Au is strongly dispersive, its value changing with wavelength is not negligible so one ﬁxed value cannot be used for a range of wavelengths. We used # of Au from Ref. . Figure 1b shows the dispersion of # as a function of wavelength. Blue represents the real part of #; black represents the imaginary part of #. The data points marked with the ﬁlled circles () and the open circles () come from the reference, while the solid and dashed curves come from interpolation with the spline method. The real part of # is negative because Au is a metal. r and r are generally represented with complex numbers. Therefore, one can write f b r and r in terms of amplitude and phase. Using phase gives a great advantage when f b one is trying to ﬁnd a zero for r . As such, it is possible to ﬁnd where r = 0 by simply f f observing the phase of r because there will be a 2p phase change around the value zero . We continuously adjusted d and d from 20 nm to 100 nm to ﬁnd the phases of 1 2 r and r . f b Figures 2a,b show the false color images of the phases of r and r respectively, f b when the wavelength l of the incident light is 560 nm. The colors indicate the phases of r and r . One can see that the phase changes by 2p, i.e., the color changes from the f b darkest blue to the darkest red, if we move counterclockwise around the bottom-left point P = (47 nm, 32 nm) of the dark line and around the top-right point P = (71 nm, 60 nm) of 1 2 the dark line, where the ﬁrst and second numbers correspond to d and d , respectively. Therefore, we can see that r becomes zero at P and P . r , however, does not go to zero at f 1 2 b P and P because there is no 2p phase change around those points in false color image 1 2 in Figure 2b. This indicates that one-way zero reﬂection has been realized at these points. It is also possible to get more accurate values for d and d by increasing the resolution of 1 2 the false color image; this can be done by decreasing the scanning step, i.e., by reducing the interval we increment d and d while creating these false color image. Note that r 1 2 b T T T T goes to zero at P = (32 nm, 47 nm) and P = (60 nm, 71 nm), where P (P ) is transposed 1 2 1 2 of P (P ). This is because the Au layer is embedded with the two SiO layers. 1 2 2 Re( ) Im( ) Photonics 2021, 8, 8 4 of 7 (a) (b) Figure 2. The false color image showing the phases of r (a) and r (b). (a) r = 0 at P = (47 nm, 32 nm) and P = (71 nm, 1 2 f b f T T 60 nm) because a phase change of 2p occurs around these points. (b) r = 0 at P = (32 nm, 47 nm) and P = (60 nm, 71 nm), 1 2 where the ﬁrst and second numbers correspond to d and d , respectively. 1 2 3. Results and Discussion In order to conﬁrm one-way zero reﬂection, we calculated the reﬂectance of the simulated structures: SiO (32 nm)-Au(10 nm)-SiO (47 nm) and SiO (60 nm)-Au(10 nm)- 2 2 2 SiO (71 nm). Reﬂectance is deﬁned as the square of the absolute value of the reﬂection coefﬁcient. Therefore, with forwardly incident light the reﬂectance is jr j , for backwardly incident light it is jr j . Figure 3a shows the reﬂectance as a function of wavelength for d = 32 nm and d = 47 nm. The blue solid curve is for the reﬂectance of forwardly 1 2 incident light, the black dashed curve is for the reﬂectance of backwardly incident light. The numerically calculated reﬂectance is less than 10 for forwardly incident light while it is not zero but 2.5% for backwardly incident light at l = 560 nm. These results conﬁrm that we have one-way zero reﬂection at l = 560 nm, which agrees with our simulation results 2 2 presented in Figure2. Figure 3b shows the transmittances of jt j (jt j ) for forwardly f b (backwardly) incident light. The blue solid curve shows the transmittance of forwardly incident light; the black dashed curve shows the transmittance of backwardly incident light. The blue solid and black dashed curves overlap exactly, which means the transmittance for forwardly incident light is the same as for backwardly incident light. (a) (b) 0.95 0.14 forward forward back back 0.90 0.12 0.85 0.10 0.80 0.08 0.75 0.06 0.70 0.04 0.65 0.02 0.60 0.00 400 450 500 550 600 650 700 400 450 500 550 600 650 700 Wavelength (nm) Wavelength (nm) Figure 3. Reﬂectance and transmittance for a structure with d = 32 nm and d = 47 nm (a) Reﬂectance of forwardly incident light (blue solid curve) and backwardly incident light (black dashed curve) as a function of wavelength. Only reﬂectance of forwardly incident light becomes zero at l = 560 nm. (b) Transmittance of forwardly incident light (blue solid curve) and backwardly incident light (black dashed curve) as a function of wavelength. The transmittance is the same for both forwardly and backwardly incident light. Reflectance Transmittance Photonics 2021, 8, 8 5 of 7 Figure 4a shows the reﬂectance as a function of wavelength for d = 60 nm and d = 71 nm. The blue solid curve shows the reﬂectance of forwardly incident light; the black dashed curve shows the reﬂectance of backwardly incident light. The numerically calculated reﬂectance is less that 10 for forwardly incident light while it is not zero but 1.4% for backwardly incident light at l = 560 nm. These results conﬁrm that we achieve one-way zero reﬂection at l = 560 nm. Figure 4b shows the transmittance. The blue solid curve shows the transmittance of forwardly incident light, the black dashed curve shows the transmittance of backwardly incident light. The blue solid and black dashed curves overlaps exactly. The reﬂectance and transmittance results indicate that the absorption is different for forwardly and backwardly incident light because Re f lectance + Transmittance + Absorption = 1. Figure 5 shows the absorption spectra of forwardly incident light (blue solid curve) and backwardly incident light (black dashed curve) as a function of wave- length for d = 47 nm and d = 32 nm (a), d = 60 nm and d = 71 nm (b). In both cases, 1 2 1 2 absorption is higher for forwardly incident light compared to backwardly incident light. (a) (b) 0.95 0.14 forward forward back back 0.90 0.12 0.85 0.10 0.80 0.08 0.75 0.06 0.70 0.04 0.65 0.02 0.60 0.00 400 450 500 550 600 650 700 400 450 500 550 600 650 700 Wavelength (nm) Wavelength (nm) Figure 4. Reﬂectance and transmittance for a structure with d = 60 nm and d = 71 nm (a) Reﬂectance of forwardly 1 2 incident light (blue solid curve) and backwardly incident light (black dashed curve) as a function of wavelength. Only the reﬂectance of forwardly incident light becomes zero at l = 560 nm. (b) Transmittance of forwardly incident light (blue solid curve) and backwardly incident light (black dashed curve) as a function of wavelength. The transmittance is the same for both forwardly and backwardly incident light. (a) (b) 0.40 0.40 forward forward back back 0.35 0.35 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 400 450 500 550 600 650 700 400 450 500 550 600 650 700 Wavelength (nm) Wavelength (nm) Figure 5. Absorption spectra of forwardly incident light (blue solid curve) and backwardly incident light (black dashed curve) as a function of wavelength for a structure with d = 47 nm and d = 32 nm (a), d = 60 nm and d = 71 nm (b). 1 2 1 2 Absorption Reflectance Absorption Transmittance Photonics 2021, 8, 8 6 of 7 4. Conclusions We used TMM to simulate one-way zero reﬂection through an insulator-metal-insulator (IMI) structure. We used SiO for the two outer insulator layers and Au for the middle metal layer. By examining a range of thicknesses for both the SiO layers with the 10-nm Au layer at l = 560 nm, we found two pairs of thicknesses that produced one-way zero reﬂection: d = 47 nm (60 nm) for the top layer and d = 32 nm (71 nm) for the bottom 1 2 layer. We also conﬁrmed one-way zero reﬂection using reﬂectance spectra. The reﬂectance becomes zero for forwardly incident light, while it is not zero for backwardly incident light. The reﬂectance of backwardly incident light is higher when d = 47 nm and d = 32 nm 1 2 compared to when d = 60 nm and d = 71 nm. 1 2 Even though we used a metal as the absorption material, any material with absorption could be used to achieve one-way zero reﬂection, such as silicon demonstrated in prior research with various conﬁgurations [16,17]. IMI structure proposed in the context of one-way zero reﬂection may be adapted in optical ﬁber-based biosensors to investigate the possible increase of light coupling to the sensing region . One-way zero reﬂection has great potential when applied in systems where no reﬂection is important, such as in solar cells. Author Contributions: Conceptualization, H.N. and J.-M.C.; methodology, H.N.; investigation, H.N.; data curation, J.-M.C.; writing—original draft preparation, H.N.; writing—review and editing, J.-M.C.; visualization, J.-M.C.; supervision, H.N.; project administration, H.N.; funding acquisition, H.N. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the Basic Science Research Program through the National Re- search Foundation of Korea (NRF) funded by the Ministry of Education (No. 2017R1D1A1A02018515). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Bender, C.M.; Boettcher, S. Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry. Phys. Rev. Lett. 1998, 80, 5243–5246. [CrossRef] 2. Longhi, S.; Della Valle, G. Photonic Realization of PT-Symmetric Quantum Field Theories. Phys. Rev. A 2012, 85, 012112. [CrossRef] 3. Bender, C.M. Making Sense of Non-Hermitian Hamiltonians. Rep. Prog. Phys. 2007, 70, 947–1018. [CrossRef] 4. Musslimani, Z.H.; Makris, K.G.; El-Ganainy, R.; Christodoulides, D.N. Optical Solitons in PT Periodic Potentials. Phys. Rev. Lett. 2008, 100, 030402. [CrossRef] [PubMed] 5. Guo, A.; Salamo, G. J.; Duchesne, D.; Morandotti, R.; Volatier-Ravat, M.; Aimez, V.; Siviloglou, G.A.; Christodoulides, D.N. Observation of PT-Symmetry Breaking in Complex Optical Potentials. Phys. Rev. Lett. 2009, 103, 093902. [CrossRef] [PubMed] 6. Rüter, C.E.; Makris, K.G.; El-Ganainy, R.; Christodoulides, D.N.; Segev, M.; Kip, D. Observation of Parity-Time Symmetry in Optics. Nat. Phys. 2010, 6, 192–195. [CrossRef] 7. Miri, M.-A.; LiKamWa, P.; Christodoulides, D.N. Large Area Single-Mode Parity–Time-Symmetric Laser Ampliﬁers. Opt. Lett. 2012, 37, 764. [CrossRef] 8. Lin, Z.; Ramezani, H.; Eichelkraut, T.; Kottos, T.; Cao, H.; Christodoulides, D.N. Unidirectional Invisibility Induced by PT- Symmetric Periodic Structures. Phys. Rev. Lett. 2011, 106, 213901. [CrossRef] 9. Liertzer, M.; Ge, L.; Cerjan, A.; Stone, A.D.; Türeci, H.E.; Rotter, S. Pump-Induced Exceptional Points in Lasers. Phys. Rev. Lett. 2012, 108, 173901. [CrossRef] 10. Feng, L.; Xu, Y.L.; Fegadolli, W.S.; Lu, M.H.; Oliveira, J.E.B.; Almeida, V.R.; Chen, Y.F.; Scherer, A. Experimental Demonstration of a Unidirectional Reﬂectionless Parity-Time Metamaterial at Optical Frequencies. Nat. Mater. 2013, 12, 108–113. [CrossRef] 11. Brandstetter, M.; Liertzer, M.; Deutsch, C.; Klang, P.; Schöberl, J.; Türeci, H.E.; Strasser, G.; Unterrainer, K.; Rotter, S. Reversing the Pump Dependence of a Laser at an Exceptional Point. Nat. Commun. 2014, 5, 4034. [CrossRef] [PubMed] 12. Feng, L.; Wong, Z.J.; Ma, R.M.; Wang, Y.; Zhang, X. Single-Mode Laser by Parity-Time Symmetry Breaking. Science 2014, 346, 972–975. [CrossRef] [PubMed] 13. Hodaei, H.; Miri, M.A.; Hassan, A.U.; Hayenga, W.E.; Heinrich, M.; Christodoulides, D.N.; Khajavikhan, M. Parity-Time- Symmetric Coupled Microring Lasers Operating around an Exceptional Point. Opt. Lett. 2015, 40, 4955. [CrossRef] [PubMed] 14. Miri, M.-A.; Alú, A. Exceptional points in optics and photonics. Science 2019, 363, eaar7709. [CrossRef] 15. Wiersig, J. Review of exceptional point-based sensors. Photonics Res. 2020, 8, 1457–1467. [CrossRef] Photonics 2021, 8, 8 7 of 7 16. Shen, Y.; Deng, X.H.; Chen, L. Unidirectional Invisibility in a Two-Layer Non-PT-Symmetric Slab. Opt. Express 2014, 22, 19440. [CrossRef] 17. Feng, L.; Zhu, X.; Yang, S.; Zhu, H.; Zhang, P.; Yin, X.; Wang, Y.; Zhang, X. Demonstration of a Large-Scale Optical Exceptional Point Structure. Opt. Express 2014, 22, 1760. [CrossRef] 18. Huang, Y.; Veronis, G.; Min, C. Unidirectional Reﬂectionless Propagation in Plasmonic Waveguide-Cavity Systems at Exceptional Points. Opt. Express 2015, 23, 29882. [CrossRef] 19. Na, J.; Noh, H. Investigation of a Broadband Coherent Perfect Absorber in a Multi-Layer Structure by Using the Transfer Matrix Method. J. Korean Phys. Soc. 2018, 72, 66–70. [CrossRef] 20. Ducourtieux, S.; Podolskiy, V.A.; Grésillon, S.; Buil, S.; Berini, B.; Gadenne, P.; Boccara, A.C.; Rivoal, J.C.; Bragg, W.D.; Banerjee, K.; et al. Near-Field Optical Studies of Semicontinuous Metal Films. Phys. Rev. B 2001, 64, 1654031–16540314. [CrossRef] 21. Johnson, P.B.; Christy, R.W. Optical Constants of the Noble Metals. Phys. Rev. B 1972, 6, 4370–4379. [CrossRef] 22. Andreasen, J.; Vanneste, C.; Ge, L.; Cao, H. Effects of Spatially Nonuniform Gain on Lasing Modes in Weakly Scattering Random Systems. Phys. Rev. A 2010, 81, 043818. [CrossRef] 23. González-Vila, Á.; Debliquy, M.; Lahem, D.; Zhang C.; Mégret, P.; Caucheteur, C. Molecularly imprinted electropolymerization on a metal-coated optical ﬁber for gas sensing application. Sens. Actuators B Chem. 2017, 244, 1145–1151. [CrossRef]
Multidisciplinary Digital Publishing Institute
One-Way Zero Reflection in an Insulator-Metal-Insulator Structure Using the Transfer Matrix Method
, Volume 8 (1) –
Dec 31, 2020
Share Full Text for Free
Add to Folder
Web of Science