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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 16 — Aug. 11, 2014
  • pp: 19440–19447
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Unidirectional invisibility in a two-layer non-PT-symmetric slab

Yun Shen, Xiao Hua Deng, and Lin Chen  »View Author Affiliations


Optics Express, Vol. 22, Issue 16, pp. 19440-19447 (2014)
http://dx.doi.org/10.1364/OE.22.019440


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Abstract

Recently, unidirectional invisibility has been demonstrated in parity-time (PT) symmetric periodic structures and has attracted great attention. Nevertheless, fabrication of a complex periodic structure may not be practically easy. In this paper, a simple two-layer non-PT-symmetric slab structure is proposed to realize unidirectional invisibility. We numerically show that in such conventional structure consisting of two slabs with different real parts of refractive indices, unidirectional invisibility can be achieved as proper imaginary parts of refractive indices and thicknesses of the slabs are satisfied. Moreover, the unidirectional invisibility can be converted to unidirectional reflection when the imaginary parts of the refractive indices are tuned to their odd symmetric forms.

© 2014 Optical Society of America

1. Introduction

In the past few years, systems exhibiting parity-time (PT) symmetry have attracted considerable attention in various areas, ranging from quantum field theory [1

1. S. Longhi and G. Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A 85(1), 012112 (2012). [CrossRef]

], mathematical physics [2

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

,3

3. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [CrossRef]

], solid-state [4

4. N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77(3), 570–573 (1996). [CrossRef] [PubMed]

,5

5. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009). [CrossRef] [PubMed]

] and atomic physics [6

6. M. Hiller, T. Kottos, and A. Ossipov, “Bifurcations in resonance widths of an open Bose-Hubbard dimer,” Phys. Rev. A 73(6), 063625 (2006). [CrossRef]

], to linear and nonlinear optics [7

7. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007). [CrossRef] [PubMed]

9

9. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43(26), 265305 (2010). [CrossRef]

]. In PT symmetric systems, physical properties of quantum mechanics and quantum field theories can be created. By exploiting optical modulation of the complex refractive index, the constructed optical PT structures can lead to a series of intriguing optical phenomena, such as double refraction, power oscillations [10

10. 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]

,11

11. M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010). [CrossRef]

], absorption enhanced transmission [12

12. A. Guo, G. J. Salamo, 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(9), 093902 (2009). [CrossRef] [PubMed]

], non-reciprocity of light propagation [10

10. 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]

], and coherent perfect absorbing-lasing [13

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

]. Recently, unidirectional invisibility has also been demonstrated in PT symmetric periodic structures [14

14. 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]

], which shows that at the exceptional points, reflection of a PT symmetric periodic structure is diminished from one end while it is enhanced from the other end. This exotic property is of practical significance and it opens the door to a new generation of photonic devices, for example, chip-scale optical network analyzers based on unidirectional photonic implement [15

15. 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 (2012). [CrossRef] [PubMed]

]. Additionally, the general design principle in the optical domain can also be extended to other frequencies and classical wave systems, such as unidirectional microwave invisibility for military applications and ultrasonic equipment for marine exploration. Nevertheless, as the complex refractive index obeyingn(r)=n*(r), i.e., even/odd symmetry for real/imaginary part of the refractive index, is strictly demanded in a PT structure and it is not easy to control it precisely in practice. Fabrication of such complex periodic structure still remains challenging.

In this paper, a simple two-layer slab structure is proposed to realize unidirectional invisibility. Theoretical analysis and numerical calculations illustrate that unidirectional invisibility can be achieved in such conventional non-PT-symmetric structure, and no special relationship between the real parts of the refractive indices are demanded except they cannot be in same values. Particularly, unidirectional invisibility can be converted into unidirectional reflection when the imaginary parts of the refractive indices are converted into the odd symmetric forms.

2. Design and theory

The proposed one-dimensional two-layer slab structure is schematically shown in Fig. 1(b) inset, wheren1, n2 andl1,l2represent the refractive indices and thicknesses of the slabs respectively.
Fig. 1 (a) n2 derived based on Eq. (5) for unidirectional invisibility of one-dimensional two-layer slab in (b) inset as m = 10, 20, 30 and n1=1.444. Groups (i) and (ii) correspond to solutions of unidirectional invisibility for left and right incidence, respectively; at singular point A, reflectionlessness happens for both right and left incidences. Based onn2, the corresponding effective length (b)L2 can be obtained from Eq. (5). It is shown that no mattern2 belongs to group (i) or group (ii) in (a), the same n2' will correspond to a sameL2.
As light propagates along the z direction, the transmission and reflection properties of such structure can be calculated by transfer matrix method (TMM) [16

16. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef] [PubMed]

,17

17. Y. Shen and G. P. Wang, “Gain-assisted time delay of plasmons in coupled metal ring resonator waveguides,” Opt. Express 17(15), 12807–12812 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-15-12807. [CrossRef] [PubMed]

], and the transfer matrices are read as
Mi=[cosδijηisinδijsinδi/ηicosδi](i=1,2),
(1)
withδi=ω/cnili and ηi=(ε0/μ0)1/2ni. In those equations, ωis the frequency, c,ε0 andμ0 are velocity of light in vacuum, permittivity and permeability of free space, respectively. As light propagates along the positive z direction, i.e., incident from the left side in Fig. 1(b) inset, the corresponding transmission coefficient tL and reflection coefficient rL can be deduced fromM1M2. Conversely, if light propagates along the negative z direction, i.e., incident from the right side, then tR andrRare offered byM2M1. In this way, results of tR/tL=1and rR/rL being the function of (η0,η1,η2,δ1,δ2) are obtained when background refractive index n0 for left and right sides are considered, whereη0=(ε0/μ0)1/2n0. In other words, tLis always equal to tR no matter whatn1andn2 might be, but rLis generally different withrR and their ratio rR/rL is a complicated fraction formula. When numerator of this fraction equals zero and simultaneously the denominator does not, rR/rL (rL/rR) will come up to zero (infinite), which implies that only light incident from the left side of the two-layer structure has reflection. Similarly, as the denominator equals zero and numerator does not, with rL/rR(rR/rL) coming up to zero (infinite), only the right side has reflection. On every account, the phenomenon of unidirectional invisibility, which shows property of reflectlessness, can happen.

To simplify the discussion and clearly demonstrate the unidirectional invisibility phenomena, in our paper we setδ1 satisfying
δ1=2pπ+π/2,(p=0,1,2...),
(2)
i.e., lossless n1 and suitable l1 are chosen. We note that in practice the requirement for a lossless n1 can be met by selecting active material with fitting gain coefficient, or ordinary material working in transparent regions, for example, SiO2 at 1550nm [18

18. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

]. Under such δ1conditions, rR/rL can be written as:
rRrL=(n2/n1n1/n2)sinδ2+j(n1/n0n0/n1)cosδ2(n1/n2n2/n1)sinδ2+j(n1/n0n0/n1)cosδ2.
(3)
The solutions for numerator and denominator polynomial of Eq. (3) being zero can be expressed as:
ej2L2n2=±(n22n12)+n2(n121)±(n22n12)n2(n121).
(4)
In this equation, plus and minus signs correspond to the solution of numerator and denominator of Eq. (3) being zero respectively. In this calculation, L2=ωl2/c is defined as effective length and background refractive index n0=1is assumed. For solution expression Eq. (4), we can further write it to
L2=1j2n2[log±(n22n12)+n2(n121)±(n22n12)n2(n121)+j2mπ],(m=0,1,2...).
(5)
Consequently, when effective length L2, refractive indices n1 and n2 together satisfy the relationship of Eq. (5), rR/rL of Eq. (3) possibly comes up to zero or infinite, leading to unidirectional invisibility for light incident from right side and left side of two-layer structure shown in Fig. 1(b) inset, respectively. It is important to note that in practice the lengthl2, and so L2 must be real values. Due to this, relationships amongL2, n2 and n1 can be eventually determined.

3. Simulation and discussion

Specifically, in this paper we choosen11.444for example, which corresponds to the refractive index of SiO2 for 1550 nm [18

18. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

] and meets the assumed lossless condition for Eq. (2). Then n2that satisfies the demand of L2being real values can be derived according to Eq. (5), and the relations between its real and imaginary parts for m = 10, 20, 30 are illustrated in Fig. 1(a). In this figure, the group (i) is deduced from Eq. (5) with the plus signs taken, which leads to numerator of Eq. (3) being zero and implies the reflectionless property for light incident from right side of two-layer structure [Fig. 1(b) inset]. The group (ii) is deduced with minus signs taken and leads to the denominator being zero, implying the reflectionless property of light from the left side. Here, it is a remarkable fact that the group (i) and (ii) are symmetric by line of n2'' = 0, while n2=n2'+in2''. Namely, as a n2 [see, group (i)] makes rR = 0, rL0 [see, Eq. (3)] and offers unidirectional invisibility (reflection) for right (left) incidence, its conjugate value [see, group (ii)] can make rR0, rL = 0 [see Eq. (3)] and offers unidirectional invisibility (reflection) for left (right) incidence. This also implies that the unidirectional invisibility for one side can be converted into unidirectional reflection when we tune n2''to its opposite value. Next, due to n2, the corresponding effective length L2 can be obtained from Eq. (5), and the dependences of L2on n2' for m = 10, 20, 30 are illustrated in Fig. 1(b). It is shown that no mattern2 belongs to group (i) or group (ii) in Fig. 1(a), same n2' will correspond to a sameL2. Particularly, we note that in Fig. 1(a), singular points exist at n2 = n1for both group (i) and (ii). Here, it is at n2 = 1.444, which combines with its corresponding L2in Fig. 1(b) and leads to rR = 0, rL = 0 [see, Eq. (3)], and provides reflectionless property for both right and left incidences. Additionally, in our calculation to derive n2 based on Eq. (5), we assume that the demand of L2being real values is satisfied when imaginary part of L2 is less than105. In other words, the truncation error of L2we set for transcendental Eq. (5) isj105.

To verify the above results, the following takes the reflections of structure with n2and corresponding L2 severally located at m = 10 in Fig. 1(a) (i) and Fig. 1(b) as an instance. As we know, in this case the numerator of Eq. (3), except for n2 = 1.444, should be zeros and leads to rR = 0, rL0, resulting in the unidirectional invisibility for right incidence. For n2 = 1.444, the numerator and denominator of Eq. (3) should be both zeros, and rR = 0, rL = 0 are simultaneously provided. With TMM method, the calculated reflections being dependent on n2' are depicted in Fig. 2.
Fig. 2 Reflections depend on n2' of structure with n2and corresponding L2 located at m = 10 in Fig. 1(a) (i) and Fig. 1(b) respectively. Reflections for right and left incidences, i.e., |rR|2and |rL|2, are illustrated by the blue dotted and red circle curves respectively. The results on linear scale are plotted in inset.
In which, reflections for right and left incidence, i.e., |rR|2and |rL|2, are illustrated by the blue dotted and red circle curves respectively. The results on linear scale are plotted in Fig. 2 inset. From Fig. 2 we can see that the values of right reflection (blue dotted) in the considered n2'region are all less than 1010and approximated to zero, except that region of n2' near 1.444, |rR|2/|rL|2<108 are provided in the considered region. In other words, the reflections of the structure for right and left incidence have enormous difference, indicating unidirectional invisibility (right side) and reflection (left side) respectively. At n2' = 1.444, |rR|2=|rL|2in Fig. 2 both are approximately zeros. Those results well verify what expected from Fig. 1.

Furthermore, we can choose Si [18

18. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

] and active material Er-doped Zn2Si0.5Ge0.5O4 (ZSG:Er) [19

19. C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett. 84(9), 1462–1464 (2004). [CrossRef]

] as dielectrics n1 and n2 respectively, as an instance performed by finite-difference time-domain [17

17. Y. Shen and G. P. Wang, “Gain-assisted time delay of plasmons in coupled metal ring resonator waveguides,” Opt. Express 17(15), 12807–12812 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-15-12807. [CrossRef] [PubMed]

,20

20. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. AP-14, 302–307 (1966).

,21

21. Y. Shen and G. P. Wang, “Optical bistability in metal gap waveguide nanocavities,” Opt. Express 16(12), 8421–8426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-12-8421. [CrossRef] [PubMed]

] to demonstrate the realization of unidirectional invisibility in practice. For 1550 nm, refractive index n1 of Si is approximately 1.444, and n2of ZSG isn2' = 1.7514 with tunablen2'' able to be tuned by external field. In this instance from the curves of m = 10 in Fig. 1(a) (i) and Fig. 1(b) we can know, as n2'' = −0.03034, and L2 = 18.79797 (corresponding to l2 = 4.6373 μm), the unidirectional invisibility for right incidence will happen. We note that the length of dielectric n1 need to be selected to make Eq. (2) satisfied, and it is artificially set as l1 = 3.4886 μm here. Figure 3(a) shows the optical intensity distribution (normalized to the incident intensity) along z direction when light from left is incident on the structure with the above parameters.
Fig. 3 Normalized intensity distribution along z direction when 1550 nm light from (a) left and (b) right is incident on the structure composed of Si and active material ZSG. For 1550 nm, n1 of Si is approximately 1.444, and n2of ZSG isn2' = 1.7514. n2'',l2, and l1 are set as −0.03034, 4.6373 μm, and 3.4886 μm respectively, according to curves of m = 10 in Fig. 1(a) (i) and Fig. 1(b) and it demonstrates the realization of unidirectional invisibility for instance. In (a) and (b), pink dashed lines denote the left and right unidirectional incident planes respectively.
In this simulation, the left unidirectional incident plane is located at z = −10.083 μm, which has a 6.5943 μm distance from the left face of n1, and it is denoted with pink dashed line. Thus, we can detect the left reflected light intensity in the region of z<-10.083 μm and obtain|rL|2, which is |rL|20.70 here. In this way, when the right unidirectional incident plane, which locates at z = 11.232 μm and has a 6.5943 μm distance from the right face of n2, is incident on the structure, the resulted optical intensity distribution is as shown in Fig. 3(b). We detect the right reflection in the region of z>11.232 μm and obtain |rR|20.0125. Evidently, |rR|2<<|rL|2 and |rR|2/|rL|2 = 0.0179 are illustrated, which confirm the results in Fig. 2, demonstrating that the unidirectional invisibility for right incidence and unidirectional reflection for left incidence are offered. Incidentally, |tR|2 and |tL|2can be detected from z> 4.6373 μm and z<-3.4886 μm in Figs. 3(a) and (b) respectively, and tR/tL1 can be observed from Fig. 3, which well agrees with the results deduced from Eq. (1).

In the preceding lines, Fig. 2 verifies the solutions of n2 that locates at m = 10 in Fig. 1(a) (i) and offers unidirectional invisibility (reflection) for right (left) incidence. Similarly the conjugate solutions, which locate at m = 10 in Fig. 1(a) (ii) and should offer unidirectional invisibility (reflection) for left (right) incidence, can also be verified. The calculated reflections dependent on n2' are depicted in Fig. 4 by TMM method.
Fig. 4 Reflections depending on structural parameter n2' with n2and corresponding L2 located at m = 10 in Fig. 1(a) (ii) and Fig. 1(b) respectively. Reflections for right and left incidence, i.e., |rR|2and |rL|2, are illustrated by the blue dotted and red circle curves respectively. The results on linear scale are plotted in the inset.
In Fig. 4, reflections for right and left incidences, i.e., |rR|2and |rL|2, are illustrated by the blue dotted and red circle curves respectively. The results on linear scale are plotted in Fig. 4 inset. From Fig. 4 we can know, in whole considered n2'region, |rL|2<1010 and is approximated to zero. Except the region when n2' is near 1.444, all the other regions satisfy |rL|2/|rR|2<105, implying the unidirectional invisibility (left side) and reflection (right side). At n2' = 1.444, |rR|2=|rL|2both are approximately zeros. The results well verify what expected from Fig. 1. Comparing Fig. 4 with the results shown in Fig. 2, we can know that as n2 at m = 10 in Fig. 1(a) (i) converts into its conjugate form in Fig. 1(a) (ii), the right invisibility (Fig. 2, blue dotted) and left reflection (Fig. 2, red circle) will convert into right reflection (Fig. 4, blue dotted) and left invisibility (Fig. 4, red circle) respectively. To sum up, as the imaginary parts of n2convert into their odd symmetric forms, the unidirectional invisibility can be converted into unidirectional reflection.

Similarly, whenn1,l1, andn2'are the same as the structural parameters for Fig. 3, but n2''is tuned from −0.03034 [located at curve of m = 10 in Fig. 1(a) (i)] to 0.03034 [located at curve of m = 10 in Fig. 1(a) (ii)], the optical intensity distributions (normalized to the incident intensities) along z direction for left incidence and right incidence are illustrated in Figs. 5(a) and (b) respectively.
Fig. 5 Normalized intensity distribution along z direction when 1550 nm light from (a) left and (b) right is incident on the structure composed of Si and active material ZSG. For 1550 nm, n1 of Si is approximately 1.444, and n2of ZSG isn2' = 1.7514. n2'',l2, and l1 are set as 0.03034, 4.6373 μm, and 3.4886μm respectively according to curves of m = 10 in Fig. 1(a) (i) and Fig. 1(b) which demonstrate the realization of unidirectional invisibility for instance. In (a) and (b), pink dashed lines denote the left and right unidirectional incident planes respectively.
In the region of z<-10.083 μm in Fig. 5(a) and z>11.232 μm in Fig. 5(b), |rL|20.00148 and |rR|20.0631 can be detected respectively. Eventually, |rL|2<<|rR|2 and |rL|2/|rR|2 = 0.0235 are illustrated, which well agree with the results in Fig. 4, demonstrating the unidirectional invisibility for left incidence and unidirectional reflection for right incidence. Again, comparing Fig. 5 with Fig. 3, we can know that as the imaginary parts of n2convert into their odd symmetric forms (−0.03034 to 0.03034), the unidirectional invisibility [right side in Fig. 3(b)] and reflection [left side in Fig. 3(a)] can be converted into unidirectional reflection [right side in Fig. 5(b)] and invisibility [left side in Fig. 5(a)] respectively.

Once more, we note that in our calculation, n1=1.444 is chosen for instance. Accordingly, n2 that combines with n1 to offer unidirectional invisibility can be provided due to Eq. (5) and is shown in Fig. 1(a). It shows that in the considered region, n2' can be any value except the value ofn1. Practically, other values for lossless n1 can also be chosen, and similar results for n2' are obtainable. Given these points, no special relationship between the real parts of n1and n2is demanded as the necessary prerequisite, except they cannot be the same in value. In any case, the unidirectional invisibility can be achieved as parametersn0,n1,n2, l1andl2 together makerR/rL come up to zero or infinite. Here, rR/rLcan be directly deduced from Eq. (1).

4. Conclusion

In conclusion, a simple two-layer slab structure is proposed to realize unidirectional invisibility. Theoretical analysis and numerical calculations illustrate that the unidirectional invisibility can be practically achieved in such conventional non-PT-symmetric structure. Particularly, unidirectional invisibility can be converted into unidirectional reflection when the imaginary parts of the refractive indices are converted into the odd symmetric forms. This exotic property is of practical significance and it opens the door to a new generation of photonic devices, for example, chip-scale optical network analysers based on unidirectional photonic implement. Moreover, the general design principle in the optical domain can also be extended to other frequencies and classical wave systems, such as unidirectional microwave invisibility for military applications and ultrasonic equipment for marine exploration.

Acknowledgments

We thank Guo Ping Wang and Hui Fen Zhang for helpful discussion. This work is supported by the National Natural Science Foundation of China (Grant No. 61265002 and 11274247).

References and links

1.

S. Longhi and G. Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A 85(1), 012112 (2012). [CrossRef]

2.

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

3.

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [CrossRef]

4.

N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77(3), 570–573 (1996). [CrossRef] [PubMed]

5.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009). [CrossRef] [PubMed]

6.

M. Hiller, T. Kottos, and A. Ossipov, “Bifurcations in resonance widths of an open Bose-Hubbard dimer,” Phys. Rev. A 73(6), 063625 (2006). [CrossRef]

7.

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007). [CrossRef] [PubMed]

8.

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009). [CrossRef] [PubMed]

9.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A 43(26), 265305 (2010). [CrossRef]

10.

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]

11.

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010). [CrossRef]

12.

A. Guo, G. J. Salamo, 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(9), 093902 (2009). [CrossRef] [PubMed]

13.

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

14.

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]

15.

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 (2012). [CrossRef] [PubMed]

16.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef] [PubMed]

17.

Y. Shen and G. P. Wang, “Gain-assisted time delay of plasmons in coupled metal ring resonator waveguides,” Opt. Express 17(15), 12807–12812 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-15-12807. [CrossRef] [PubMed]

18.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

19.

C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett. 84(9), 1462–1464 (2004). [CrossRef]

20.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. AP-14, 302–307 (1966).

21.

Y. Shen and G. P. Wang, “Optical bistability in metal gap waveguide nanocavities,” Opt. Express 16(12), 8421–8426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-12-8421. [CrossRef] [PubMed]

OCIS Codes
(230.7400) Optical devices : Waveguides, slab
(310.6860) Thin films : Thin films, optical properties
(290.5839) Scattering : Scattering, invisibility

ToC Category:
Thin Films

History
Original Manuscript: April 29, 2014
Revised Manuscript: July 16, 2014
Manuscript Accepted: July 17, 2014
Published: August 4, 2014

Citation
Yun Shen, Xiao Hua Deng, and Lin Chen, "Unidirectional invisibility in a two-layer non-PT-symmetric slab," Opt. Express 22, 19440-19447 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-19440


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References

  1. S. Longhi and G. Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A85(1), 012112 (2012). [CrossRef]
  2. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett.80(24), 5243–5246 (1998). [CrossRef]
  3. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys.70(6), 947–1018 (2007). [CrossRef]
  4. N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett.77(3), 570–573 (1996). [CrossRef] [PubMed]
  5. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett.103(3), 030402 (2009). [CrossRef] [PubMed]
  6. M. Hiller, T. Kottos, and A. Ossipov, “Bifurcations in resonance widths of an open Bose-Hubbard dimer,” Phys. Rev. A73(6), 063625 (2006). [CrossRef]
  7. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett.32(17), 2632–2634 (2007). [CrossRef] [PubMed]
  8. S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett.103(12), 123601 (2009). [CrossRef] [PubMed]
  9. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Optical structures with local PT-symmetry,” J. Phys. A43(26), 265305 (2010). [CrossRef]
  10. 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]
  11. M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A82(1), 010103 (2010). [CrossRef]
  12. A. Guo, G. J. Salamo, 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(9), 093902 (2009). [CrossRef] [PubMed]
  13. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A82(3), 031801 (2010). [CrossRef]
  14. 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]
  15. 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 (2012). [CrossRef] [PubMed]
  16. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett.24(11), 711–713 (1999). [CrossRef] [PubMed]
  17. Y. Shen and G. P. Wang, “Gain-assisted time delay of plasmons in coupled metal ring resonator waveguides,” Opt. Express17(15), 12807–12812 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-15-12807 . [CrossRef] [PubMed]
  18. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
  19. C. C. Baker, J. Heikenfeld, Z. Yu, and A. J. Steckl, “Optical amplification and electroluminescence at 1.54 μm in Er-doped zinc silicate germanate on silicon,” Appl. Phys. Lett.84(9), 1462–1464 (2004). [CrossRef]
  20. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag.AP-14, 302–307 (1966).
  21. Y. Shen and G. P. Wang, “Optical bistability in metal gap waveguide nanocavities,” Opt. Express16(12), 8421–8426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-12-8421 . [CrossRef] [PubMed]

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