Abstract

The lateral shifts of the wave reflected and transmitted from PT-symmetric one-dimensional multilayer-structures are investigated near the coherent-perfect-absorption (CPA)-laser point and the exceptional points. We predict that at the CPA-Laser point, the reflections from both sides and transmission as well as the related shifts are all very large, reaching their negative (or positive) maxima. Moreover, we show that although the reflections are direction-dependent in the PT-symmetric multilayer-structure, the related lateral shifts have same behaviors from both sides. Additionally, one may realize the reversal of the lateral shift through the suitable adjustment of the incident angle and the layer numbers. Numerical simulations for Gaussian incident beams are performed, and reasonable agreement between the theoretical results and numerical simulations is found.

© 2017 Optical Society of America

1. Introduction

It is well known that Hermitian exhibits real energy eigenvalues, which is just one of the fundamental principles in the field of quantum mechanics. Moreover, theoretical investigations [1–3] indicated that non-Hermitian Hamiltonians can also exhibit real eigenvalues as long as they meet the weaker condition of PT symmetry. PT symmetry is the invariance of a system under the parity operations r→-r (i.e. space inversion), and time-reversal operations t→-t (or complex conjugation in the time-harmonic regime). In the past few years, several novel phenomena were predicted and observed in PT symmetrical structures. Although the idea of PT symmetry is derived from quantum mechanics [4–7], it was immediately developed in the field of optics—called the optical PT symmetry, which was reported in the systems with balanced gain and loss (i.e. n(r)=n(r) [8–20]. Along this line, a series of intriguing optical effects were explored, such as negative refraction [10], double refraction [11, 12], unidirectional invisibility [13–19], and coherent perfect laser absorbers [20, 21].

On the other hand, since the lateral shifts of the reflected wave (i.e. Goos-Hänchen shift) in the interface of two different dielectric media were discovered by Goos and Hänchen in 1947 [22], much work concentrated on the GH shifts both theoretically [23–26] and experimentally [27–29]. In addition, GH shifts from some complex configurations including absorption [30, 31], nonlinear responses [32], temperature effect [33], epsilon-near-zero material [34, 35] and photonic crystals [36–40] were widely reported. In general, in conventional photonic crystals, it was found that there is an obvious GH shift near the edge of the photonic band gap structures [37], and giant lateral shift at the defect mode [38]. More recently, in comparison with the phenomena found in conventional photonic crystals, the GH shift for wave scattering from complex crystals was predicted to be extremely large inside the reflection band as the PT symmetry-breaking threshold was approached [40]. Incidentally, the influence of PT symmetry on the GH shift of atomic medium in a cavity was investigated, and a giant enhancement for the GH shift was revealed when the PT-symmetry condition was satisfied [41].

In this paper, we would like to investigate the lateral shifts of the wave reflected and transmitted through a periodic multilayer-structure with PT symmetry. Especially, we aim at studying the lateral shifts of reflected and transmitted beams near the CPA laser point and the exceptional points, exhibited by the PT-symmetry structure. To one’s interest, we shall show that although the propagation properties of PT-symmetry structure are highly direction-dependent (i.e., the reflections from two sides are quite different), their lateral shifts from two sides always coincide with each other, manifesting highly direction-independent. In addition, we shall show that at the CPA laser point, where both reflections and transmission achieve large magnitudes, the corresponding lateral shifts are very large, and even reach their negative (or positive) maxima. Particularly, they are very sensitive to the incident angle and the number of periods. We believe that these phenomena may lead to some potential applications for designing and producing optical devices in the future.

2. Model and theory

Let us consider a TM-polarized wave propagating in a periodic multilayer-structure, consisting of 2N slabs with the same thickness L. Adjacent slabs have the complex-conjugate relative permittivities such as ε1=ε'iε'' andε2=ε'+iε'', which satisfy the PT-symmetry requirement in optics. For simplicity, the structure is surrounded by vacuum, and the incident beam is illuminated from the left (or right) side with angle θ0, as shown in Fig. 1(a).

 

Fig. 1 (a) Geometry of the light beam propagating through a periodic multilayer-structure with complex-conjugate permittivities in linear homogeneous medium, and (b) schematic diagram of the model for S-matrix.

Download Full Size | PPT Slide | PDF

For a TM wave, the magnetic field obeys the Helmholtz equation,

2H(x,z)εjμj2z+(k02kx2εjμj)H(x,z)=0,
and the above solution is given by H(x,z)=(Aeikzz+Beikzz)eiωt+ikxx, where εj and μj are the relative permittivity and permeability, A and B are amplitudes of the magnetic field. Hence, the magnetic field distribution in the surrounding vacuum can be expressed as,
H0(x,z)=eiωt+ik0xx{Hfeik0zz+Hbeik0zz,z<NLHf+eik0zz+Hb+eik0zz,z>NL
wherek0z=k0cosθ0 represents the z-component of the wave vector k0(=2π/λ), Hf,Hf+,Hband Hb+ are amplitudes of the forward and backward propagating waves outside the multilayer-structure.

When a wave propagates along the z-axis illuminates the structure, the scattering properties can be calculated through transfer matrix method (TMM) [42, 43]. These amplitudes can be found by interrelating the fields at the structure outer interfaces,

(Hf+Hb+)=Q(HfHb)=(c2kzωε2μ2c2kzωε2μ211)1M(c2kzωε1μ1c2kzωε1μ111)(HfHb),
where M=(m1m2)Nis the total transfer matrix for a wave propagating through the whole structure with N unit cells, and mj is the transfer matrix in the jth layer, which can be expressed as:
mj=(cos(kjzL)iωεjμjc2kjzsin(kjzL)ic2kjzωεjμjsin(kjzL)cos(kjzL)),(j=1,2)
where kjz=k0nj2n02sin(θ0)2 represents the z component of the wave vector k in jth (j = 1,2) slab, c and n0 are the light speed and relative refractive index in vacuum respectively.

In the case of injection only from the left side, the reflection coefficients coefficients with transmission are written as rL=Hb/Hf,tL=Hf+/Hf; and in the case of injection only from the right side, the reflection coefficients with transmission coefficients are rR=Hf+/Hb+, tR=Hb/Hb+, where the subscript L(R) indicates the wave incident from the left (right) side. As a consequence, one yields the expressions of the complex reflection/transmission coefficients in terms of the elements of transfer matrix Q, rL=Q21/Q22, rR=Q12/Q22, tL=tR=1/Q22. The reflectivity and transmissivity are RL,R=|rL,R|2, TL,R=|tL,R|2, respectively. Especially, rLandrR are different while tLand tR are equal in a PT symmetric structure, and these observables are related by a generalized unitarity relation ||t|21|=|rLrR| [18].

Based on the stationary phase method, the lateral shifts of the reflected and transmitted beams can be derived from [23],

Δr,t=dφr,tdk0x,
whereφr,t are the phases of the reflection/transmission coefficients [i.e. r=|r|exp(iφr) and t=|t|exp(iφt)], and k0x=k0sin(θ0)is the x-component of the wave vector k0 in vacuum. After some direct simplifications, one yields,

Δr,t=λ2πcos(θ0)dφr,tdθ0.

In the PT symmetric system, as the simultaneous gain and loss are balanced, there is an unique feature that the growth and decay modes are tightly linked with each other due to the existence of the breaking phase. When the system permits self-lasing at some frequencies, one may observe the coherent perfect absorption [44]. For a laser oscillator without injected signal, the boundary conditions Hf=Hb+=0 are adopted, which yieldsQ22=0. In contrast, for a perfect absorber, the boundary conditions Hb=Hf+=0 result inQ11=0. In this connection, we take Fig. 1(b) for a better understanding. In general, there is always a real frequencyω0at which Q11(ω)=0, and it follows that at the same frequency one has Q22(ω)=0. In other words, the lasing medium also behaves as a CPA under the condition when Q11(ω)=Q22(ω)=0.

On the other hand, it is also necessary to illustrate the S-matrix, which is defined as,

(HbHf+)=S(Hb+Hf)=(tRrLrRtL)(Hb+Hf).
Actually, S-matrix is quite helpful for us to determine the exceptional points, at which the reflectivity of the PT symmetric periodic structure is zero from one end while it is nonzero from the other end, and the transmissivity is unitary. This is just the so-called unidirectional invisibility. For a PT symmetric structure, the eigenvalues of the S-matrix take form,
λ1,λ2=t±rLrR2.
The eigenvalues of the S-matrix are either reciprocal moduli pair (|λ1|=1/|λ2|1) or unimodular pair (|λ1|=1/|λ2|=1). For the exceptional points, they are exactly at the transition points between reciprocal moduli pair and unimodular pair.

3. Numerical results

We are now in a position to present numerical results on the lateral shifts of the reflected/transmitted waves from the PT symmetric multilayer-structure. At first, we aim at the lateral shift near the CPA-Laser point. For simplicity, the relevant parameters are chosen as L=125um,ε'=0.1,ε''=0.1, and θ0=5deg, to realize the CPA-Laser condition.

In Fig. 2, we show the reflectivity/transmissivity and their lateral shifts for a PT bilayer-structure (N = 1) as a function of the frequency ranging from ω=1.25×1013Hz to ω=1.75×1013Hz. We find that the lateral shifts of the reflection from two directions and the transmission are indeed the same [see Fig. 2(b)], although the reflections from both sides and the transmission are quite different [see Fig. 2(a)]. Actually, the same shifts can be understood from the same variation tendencies of the φr,L,φr,R and φt with the incident angle, as shown in the insert in Fig. 2(b). In addition, large negative shifts arise at the CPA-Laser point ω=1.49×1013Hz. These results are quite meaningful since one may obtain large lateral shifts, accompanied with large reflections and transmission, which makes the shifts observable and detectable practically near the CPA-Laser point.

 

Fig. 2 (a) Reflectivity/transmissivity and (b) the related shifts for the PT bilayer-structure (N = 1) as a function of the frequency, for the thickness of each slab L=125um, the incident angle θ0=5deg, the permittivities ε1=0.10.1iand ε2=0.1+0.1i. The insert in Fig. 2(b) shows the phases of reflectivity/transmissivity as a function of incident angle θ0at the CPA-Laser point.

Download Full Size | PPT Slide | PDF

For a given PT bilayer-structure, the frequency for the CPA-laser point varies slightly with different incident angle, as shown in Fig. 3(a). Correspondingly, the peak of the lateral shifts moves at the same time, as shown in Fig. 3(b). Note that for relatively large incident angle such as θ0=5deg, one achieves negatively large lateral shifts. On the contrary, large lateral shifts with positive value is found for relatively small incident angle. Therefore, through the fine adjustment of incident angle, one may expect novel phenomena for the lateral shifts at the CPA-laser points. For instance, there is a crossover incident angle, above and below which, the direction of the lateral shift changes at the CPA-laser point. Furthermore, near the crossover incident angle such as θ0=2.8deg/2.9deg, the lateral shift as a function of the frequency exhibits a Fano-like shape, whose physical mechanism deserves further careful investigations.

 

Fig. 3 (a) Reflectivity from left side and (b) the lateral shifts as a function of the frequency for different incident angles. Other parameters are L=125um,ε1=0.10.1i, ε2=0.1+0.1i.

Download Full Size | PPT Slide | PDF

To discuss the effect of the stack periodicity on the lateral shifts near the CPA-laser points, we further increase the parameter N to 5 and 10, as shown in Fig. 4. It can be observed that, with increasing the number of unit cells, the CPA-laser points appear more than one, so as the poles of reflections and related shifts. Furthermore, some of the related shifts become much larger in comparison with the case for N = 1. Note that there are some minima of reflections due to the Bragg resonances, at which small maximal and positive shifts are found [see in the insert in Figs. 4(b) and 4(d)].

 

Fig. 4 Reflectivity from left side and the related lateral shifts for (a, b) N = 5 and (c, d) N = 10 as a function of the frequency. Other parameters are the same as those in Fig. 2.

Download Full Size | PPT Slide | PDF

Next, we shall discuss the lateral shifts of the reflected wave from the PT symmetric structure near the exceptional points. In this case, we choose the permittivities of the structure to be ε'=0.0001,ε''=0.001, which is in the epsilon-near-zero limit. General speaking, the exceptional points can be observed for the incident angle slightly larger than the critical incident angle θc=arcsin(ε'2+ε''2)/2ε'=4.08deg, at which the zero-reflection is fulfilled [19]. Hence, the incident angle is set as θ0=5deg.

In Fig. 5(a), we show the eigenvalues of the S-matrix as a function of the incident frequency for PT-symmetric bilayer-structures, i.e. N = 1. It is evident that ω=2.95×1013s1 and ω=3.05×1013s1correspond to the exceptional points, at which unidirectional invisibility appears. This can be clearly observed from the reflectivity/transmissivity spectra, as shown in Fig. 5(b). In addition, we show the phase of rLandrR in Fig. 5(c). Note that the phases of rLandrR at the exact exceptional points are non-continuous. This is understandable because the related reflectivity rLor rRis equal to zero, and the related lateral shift is meaningless at these points accordingly.

 

Fig. 5 (a) Eigenvalues of the S matrix, (b) Reflectivity/transmissivity and (c) phases of reflection coefficients as a function of the frequency, for L=125um, θ0=5deg, ε1=0.00010.001iand ε2=0.0001+0.001i.

Download Full Size | PPT Slide | PDF

Figure 6 shows the reflectivity and the related lateral shifts as a function of the frequency for different stack periodicity. Again, for PT symmetric-structure, although the behavior of reflections from two sides is quite different in PT symmetric structure, the lateral shifts of reflection from the opposite sides are absolutely equal. In addition, the lateral shifts are non-monotonic with increasing the incident frequency, and achieve their positive/negative maxima near the exceptional points (marked with A, B and C). The reason is that near the exceptional points, the incident wave penetrates more light into the medium, resulting in large lateral shift.

 

Fig. 6 Reflectivity and the related shifts from left (blue solid line) and right (red dashed line) sides for (a, d)N = 1 (b, e)N = 45 and (c, f) N = 90 as a function of the frequency. Other parameters are the same as those in Fig. 5.

Download Full Size | PPT Slide | PDF

From Fig. 6, we further find that the peak positions around the exceptional points do not move with the increase of the periodicity. This is due to the fact that the exceptional points are really independent of the parameter N. However, the maximal value of lateral shifts near the exceptional points is positive for N = 1, becomes negative for N = 45, and is positive again for N = 90. In other words, the directions of the lateral shifts depend on the parameter N. Besides the maximal shifts near the exceptional points, we get more maxima (both positive and negative) of the lateral shifts thanks to the Bragg resonances. As one knows, at these Bragg-resonance points, reflections are weak for both sides of incidence, when the thickness for the whole stack equals an integer number of Block half-waves, i.e., 2NLk¯=πq(q=0,±1,±2,), where k¯ is the Bloch wavenumber. Actually, this phenomenon is quite common in the case of normal 1D-photonic crystal, and the shifts reach their maxima near those resonances when the related reflections almost vanish. In general, the number of the Bragg resonant peaks depends not only on the number of periods but also on the permittivity of the periodic structure. Since the permittivity here is near-zero, one observes only a few reflectionless transmission/Bragg resonant peaks even for large number periods. On the contrary, a large number of Bragg resonances take place for the ten-layered PT structures with ε1=0.10.1i and ε2=0.1+0.1i [see Fig. 4(c)]. In addition, the maximal values of the lateral shifts in Bragg resonances are positive below the exceptional points, and negative above the exceptional points. We believe that the existence of the exceptional points changes the original tendency of both reflection phases [see Fig. 5(c)]. Such a tendency implies a sharp slope of phase from positive value to negative one, and thereby changing the usual direction of the lateral shifts. Hence, we conclude that one may change the direction of the lateral shift by adjusting the incident frequency and the periodic number.

To illustrate the effect of the stack periodicity, we pay attention to the lateral shifts near the exceptional points with frequency ω=3×1013s1. From Fig. 7, we find that, with increasing the number of unit cells, the lateral shift increases at first, reaching the positive maximum, and then, the lateral shift becomes negative at layer number Nc=14, reaching the negative maximum. After that, the periodic variation takes place. Accordingly, the strongest shifts are periodically observed for Nc=14,28,42, and so on. For instance, we observe that the maximal value of the lateral shift can be as large as 10000 times of the incident wavelength for Nc=14. As the period of the structure increases, the Bragg resonances appear more and more, which leads to the transforms of phases of the reflections and transmission at the exceptional points, and hence the lateral shifts at the exceptional points become positive and negative.

 

Fig. 7 Dependence of the lateral shift of reflection from left on the stack periodicity N for ω=3×1013s1. Other parameters are the same as those in Fig. 6.

Download Full Size | PPT Slide | PDF

In the end, the reflectivity/transmissivity and the lateral shifts in both CPA-laser point and exceptional points are shown in Fig. 8. Note that, CPA-laser phenomenon occurs at θ0=5deg, and unidirectional invisibility phenomena occurs around θ0=20deg [see Figs. 8(a) and 8(b)]. The related shifts (marked with A and B) appear to be very large, as expected.

 

Fig. 8 (a,b) Reflectivity/transmissivity and (c) the related shifts for the PT bilayer-structure (N = 1) as a function of the incident angle, for L=125um, ω=1.49×1013s1, ε1=0.10.1i and ε2=0.1+0.1i.

Download Full Size | PPT Slide | PDF

4. Simulations

To verify the validity of our theory, we take one step forward to carry out numerical simulations by using COMSOL Multiphysics (RF module), based on finite element method. For simplicity, we consider a Gaussian beam incident from air to the PT symmetric bilayer.

Figure 9 shows the magnetic field amplitude distribution (a,c,e) and (b,d,f) at the CPA-Laser point. From Figs. 9(a), 9(c), and 9(e), one can observed that the Gaussian profile of the reflected beam and transmitted beam are both very strong, and hence the magnetic field of incident beam is much weaker to be observed. Since the incident beam is much weak in comparison with the reflected and transmitted ones, for the sake of clearness in Figs. 9(b), 9(d), and 9(f), we have rescaled the incident field by a amplification factor 10. In Figs. 9(a) and 9(b), we chose θ0 = 2deg, and ω=1.452×1013s1. It is evident that both reflected and transmitted waves exhibit large positive shifts. For instance, the simulated results show the lateral shifts of reflection and transmission are Δr=22.18λ and Δt=22.21λ. On the other hand, the theoretically results are Δr,t=25.02λ. Taking into account the incident beam being a Gaussian beam, such a deviation is quite acceptable. For the other case with θ0 = 5deg, and ω=1.493×1013s1, large negative shifts are observed [see Figs. 9(c) and 9(d)]. The simulated results show the lateral shifts are Δr=14.03λ and Δt=14.12λ, in comparison with the theoretical results Δr,t=15.50λ. In addition, we have exchanged the permittivities of bilayer slabs with ε1=0.1+0.1iand ε2=0.10.1i, which corresponds to the right incidence on the bilayer slabs with ε1=0.10.1i and ε2=0.1+0.1i. The results are shown in Figs. 9(e) and 9(f), and the relevant lateral shifts are almost the same with those in Figs. 9(c) and 9(d). In detail, the simulated results show the lateral shifts areΔr=14.15λ and Δt=14.20λ. Therefore, the prediction that lateral shifts are direction-independent is numerically verified.

 

Fig. 9 Magnetic field amplitude distribution of all the calculated apace and in the surface of reflection and transmission for (a, b) θ0 = 2deg, ω=1.452×1013s1, (c, d) θ0 = 5deg, ω=1.493×1013s1, withε1=0.10.1iand ε2=0.1+0.1i, and (e, f) θ0 = 5deg, ω=1.493×1013s1, withε1=0.1+0.1iand ε2=0.10.1i.

Download Full Size | PPT Slide | PDF

5. Conclusions

To summarize, we have outlined the stationary-phase method to the analysis of GH shift of the reflected/transmitted beams from the PT symmetric structures near the CPA-laser point and the exceptional points. It is found that the shifts of reflections from both sides have the same behaviors although the reflections behave diversely. Large shifts are evidently observed near the CPA-laser point and the exceptional points. To one’s interest, at the CPA-Laser points, the lateral shifts are quite large and even reach their negative or positive maxima, accompanied with large reflection and transmission. Moreover, they are very sensitive to the incident frequency and the number of periods. As a consequence, one may realize the switch between large positive and large negative shift by adjusting the incident frequency and the number of period slightly. Our investigation may be helpful in many potential applications for designing and producing optical devices, for instance, increasing the sensitivity of optical devices.

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 11374223), the Natural Science Foundation of Jiangsu Province (Grant No. BK20161210), the Qing Lan project, “333” project (Grant No. BRA2015353), and PAPD of Jiangsu Higher Education Institutions.

References and links

1. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [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, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002). [CrossRef]   [PubMed]  

4. C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252(6), 272–276 (1999). [CrossRef]  

5. Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282(6), 343–348 (2001). [CrossRef]  

6. A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43(1), 205–214 (2002). [CrossRef]  

7. Y. C. Lee, M. H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local PT Symmetry Violates The No-Signaling Principle,” Phys. Rev. Lett. 112(13), 130404 (2014). [CrossRef]   [PubMed]  

8. 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]  

9. S. Savoia, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89(8), 085105 (2014). [CrossRef]  

10. R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014). [CrossRef]   [PubMed]  

11. 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]  

12. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010). [CrossRef]  

13. 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]  

14. 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]  

15. X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38(15), 2821–2824 (2013). [CrossRef]   [PubMed]  

16. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef]   [PubMed]  

17. 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(2), 1760–1767 (2014). [CrossRef]   [PubMed]  

18. X. F. Zhu, Y. G. Peng, and D. G. Zhao, “Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry,” Opt. Express 22(15), 18401–18411 (2014). [CrossRef]   [PubMed]  

19. O. Shramkova and G. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18(10), 105101 (2016). [CrossRef]  

20. Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011). [CrossRef]   [PubMed]  

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

22. F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. 1(7-8), 333–346 (1947). [CrossRef]  

23. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1‐2), 87–102 (1948). [CrossRef]  

24. R. H. Renard, “Total reflection: a new evaluation of the Goos–Hänchen shift,” J. Opt. Soc. Am. A 54(10), 1190–1197 (1964). [CrossRef]  

25. J. Cowan and B. Aničin, “Longitudinal and transverse displacements of a bounded microwave beam at total internal reflection,” J. Opt. Soc. Am. A 67(10), 1307–1314 (1977). [CrossRef]  

26. R. Riesz and R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A 2(11), 1809–1817 (1985). [CrossRef]  

27. E. Pfleghaar, A. Marseille, and A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos-Hänchen shift,” Phys. Rev. Lett. 70(15), 2281–2284 (1993). [CrossRef]   [PubMed]  

28. P. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78(5), 851–854 (1997). [CrossRef]  

29. B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998). [CrossRef]  

30. J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50(21), 1664–1667 (1983). [CrossRef]  

31. H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002). [CrossRef]   [PubMed]  

32. D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97(4), 041903 (2010). [CrossRef]  

33. B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009). [CrossRef]   [PubMed]  

34. Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015). [CrossRef]   [PubMed]  

35. Y. L. Ziauddin, Y. L. Chuang, S. Qamar, and R. K. Lee, “Goos-Hänchen shift of partially coherent light fields in epsilon-near-zero metamaterials,” Sci. Rep. 6(1), 26504 (2016). [CrossRef]   [PubMed]  

36. D. Felbacq and R. Smaâli, “BLOCH modes dressed by evanescent waves and the generalized Goos-Hänchen effect in photonic crystals,” Phys. Rev. Lett. 92(19), 193902 (2004). [CrossRef]   [PubMed]  

37. D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28(18), 1633–1635 (2003). [CrossRef]   [PubMed]  

38. L. G. Wang and S. Y. Zhu, “Giant lateral shift of a light beam at the defect mode in one-dimensional photonic crystals,” Opt. Lett. 31(1), 101–103 (2006). [CrossRef]   [PubMed]  

39. J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14(7), 3024–3029 (2006). [CrossRef]   [PubMed]  

40. S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011). [CrossRef]  

41. Y. C. Ziauddin and R. K. Lee, “Giant Goos-Hänchen shift using PT symmetry,” Phys. Rev. A 92(1), 013815 (2015). [CrossRef]  

42. M. Born and E. Wolf, Principles of Optics (New York, 1999).

43. W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1), 016601 (2004). [CrossRef]   [PubMed]  

44. B. Zhu, R. Lü, and S. Chen, “PT-symmetry breaking for the scattering problem in a one-dimensional non-Hermitian lattice model,” Phys. Rev. A 93(3), 032129 (2016). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
    [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, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
    [Crossref] [PubMed]
  4. C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252(6), 272–276 (1999).
    [Crossref]
  5. Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282(6), 343–348 (2001).
    [Crossref]
  6. A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43(1), 205–214 (2002).
    [Crossref]
  7. Y. C. Lee, M. H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local PT Symmetry Violates The No-Signaling Principle,” Phys. Rev. Lett. 112(13), 130404 (2014).
    [Crossref] [PubMed]
  8. 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]
  9. S. Savoia, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89(8), 085105 (2014).
    [Crossref]
  10. R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
    [Crossref] [PubMed]
  11. 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]
  12. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
    [Crossref]
  13. 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]
  14. 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]
  15. X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38(15), 2821–2824 (2013).
    [Crossref] [PubMed]
  16. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
    [Crossref] [PubMed]
  17. 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(2), 1760–1767 (2014).
    [Crossref] [PubMed]
  18. X. F. Zhu, Y. G. Peng, and D. G. Zhao, “Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry,” Opt. Express 22(15), 18401–18411 (2014).
    [Crossref] [PubMed]
  19. O. Shramkova and G. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18(10), 105101 (2016).
    [Crossref]
  20. Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
    [Crossref] [PubMed]
  21. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
    [Crossref]
  22. F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. 1(7-8), 333–346 (1947).
    [Crossref]
  23. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1‐2), 87–102 (1948).
    [Crossref]
  24. R. H. Renard, “Total reflection: a new evaluation of the Goos–Hänchen shift,” J. Opt. Soc. Am. A 54(10), 1190–1197 (1964).
    [Crossref]
  25. J. Cowan and B. Aničin, “Longitudinal and transverse displacements of a bounded microwave beam at total internal reflection,” J. Opt. Soc. Am. A 67(10), 1307–1314 (1977).
    [Crossref]
  26. R. Riesz and R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A 2(11), 1809–1817 (1985).
    [Crossref]
  27. E. Pfleghaar, A. Marseille, and A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos-Hänchen shift,” Phys. Rev. Lett. 70(15), 2281–2284 (1993).
    [Crossref] [PubMed]
  28. P. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78(5), 851–854 (1997).
    [Crossref]
  29. B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
    [Crossref]
  30. J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50(21), 1664–1667 (1983).
    [Crossref]
  31. H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002).
    [Crossref] [PubMed]
  32. D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97(4), 041903 (2010).
    [Crossref]
  33. B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009).
    [Crossref] [PubMed]
  34. Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
    [Crossref] [PubMed]
  35. Y. L. Ziauddin, Y. L. Chuang, S. Qamar, and R. K. Lee, “Goos-Hänchen shift of partially coherent light fields in epsilon-near-zero metamaterials,” Sci. Rep. 6(1), 26504 (2016).
    [Crossref] [PubMed]
  36. D. Felbacq and R. Smaâli, “BLOCH modes dressed by evanescent waves and the generalized Goos-Hänchen effect in photonic crystals,” Phys. Rev. Lett. 92(19), 193902 (2004).
    [Crossref] [PubMed]
  37. D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28(18), 1633–1635 (2003).
    [Crossref] [PubMed]
  38. L. G. Wang and S. Y. Zhu, “Giant lateral shift of a light beam at the defect mode in one-dimensional photonic crystals,” Opt. Lett. 31(1), 101–103 (2006).
    [Crossref] [PubMed]
  39. J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14(7), 3024–3029 (2006).
    [Crossref] [PubMed]
  40. S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
    [Crossref]
  41. Y. C. Ziauddin and R. K. Lee, “Giant Goos-Hänchen shift using PT symmetry,” Phys. Rev. A 92(1), 013815 (2015).
    [Crossref]
  42. M. Born and E. Wolf, Principles of Optics (New York, 1999).
  43. W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1), 016601 (2004).
    [Crossref] [PubMed]
  44. B. Zhu, R. Lü, and S. Chen, “PT-symmetry breaking for the scattering problem in a one-dimensional non-Hermitian lattice model,” Phys. Rev. A 93(3), 032129 (2016).
    [Crossref]

2016 (3)

O. Shramkova and G. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18(10), 105101 (2016).
[Crossref]

Y. L. Ziauddin, Y. L. Chuang, S. Qamar, and R. K. Lee, “Goos-Hänchen shift of partially coherent light fields in epsilon-near-zero metamaterials,” Sci. Rep. 6(1), 26504 (2016).
[Crossref] [PubMed]

B. Zhu, R. Lü, and S. Chen, “PT-symmetry breaking for the scattering problem in a one-dimensional non-Hermitian lattice model,” Phys. Rev. A 93(3), 032129 (2016).
[Crossref]

2015 (2)

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
[Crossref] [PubMed]

Y. C. Ziauddin and R. K. Lee, “Giant Goos-Hänchen shift using PT symmetry,” Phys. Rev. A 92(1), 013815 (2015).
[Crossref]

2014 (5)

Y. C. Lee, M. H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local PT Symmetry Violates The No-Signaling Principle,” Phys. Rev. Lett. 112(13), 130404 (2014).
[Crossref] [PubMed]

S. Savoia, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89(8), 085105 (2014).
[Crossref]

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[Crossref] [PubMed]

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(2), 1760–1767 (2014).
[Crossref] [PubMed]

X. F. Zhu, Y. G. Peng, and D. G. Zhao, “Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry,” Opt. Express 22(15), 18401–18411 (2014).
[Crossref] [PubMed]

2013 (2)

X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38(15), 2821–2824 (2013).
[Crossref] [PubMed]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

2012 (1)

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]

2011 (3)

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

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]

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

2010 (4)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

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

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97(4), 041903 (2010).
[Crossref]

2009 (1)

2008 (1)

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

2007 (1)

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

2006 (2)

2004 (2)

D. Felbacq and R. Smaâli, “BLOCH modes dressed by evanescent waves and the generalized Goos-Hänchen effect in photonic crystals,” Phys. Rev. Lett. 92(19), 193902 (2004).
[Crossref] [PubMed]

W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1), 016601 (2004).
[Crossref] [PubMed]

2003 (1)

2002 (3)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43(1), 205–214 (2002).
[Crossref]

H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002).
[Crossref] [PubMed]

2001 (1)

Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282(6), 343–348 (2001).
[Crossref]

1999 (1)

C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252(6), 272–276 (1999).
[Crossref]

1998 (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]

B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

1997 (1)

P. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78(5), 851–854 (1997).
[Crossref]

1993 (1)

E. Pfleghaar, A. Marseille, and A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos-Hänchen shift,” Phys. Rev. Lett. 70(15), 2281–2284 (1993).
[Crossref] [PubMed]

1985 (1)

1983 (1)

J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50(21), 1664–1667 (1983).
[Crossref]

1977 (1)

J. Cowan and B. Aničin, “Longitudinal and transverse displacements of a bounded microwave beam at total internal reflection,” J. Opt. Soc. Am. A 67(10), 1307–1314 (1977).
[Crossref]

1964 (1)

R. H. Renard, “Total reflection: a new evaluation of the Goos–Hänchen shift,” J. Opt. Soc. Am. A 54(10), 1190–1197 (1964).
[Crossref]

1948 (1)

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1‐2), 87–102 (1948).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. 1(7-8), 333–346 (1947).
[Crossref]

Ahmed, Z.

Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282(6), 343–348 (2001).
[Crossref]

Almeida, V. R.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Al-Rashed, A. A. R.

B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

Alu, A.

S. Savoia, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89(8), 085105 (2014).
[Crossref]

Alù, A.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[Crossref] [PubMed]

Anicin, B.

J. Cowan and B. Aničin, “Longitudinal and transverse displacements of a bounded microwave beam at total internal reflection,” J. Opt. Soc. Am. A 67(10), 1307–1314 (1977).
[Crossref]

Artmann, K.

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1‐2), 87–102 (1948).
[Crossref]

Balcou, P.

P. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78(5), 851–854 (1997).
[Crossref]

Bender, C. M.

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

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252(6), 272–276 (1999).
[Crossref]

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

Bersch, C.

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]

Birman, J. L.

J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50(21), 1664–1667 (1983).
[Crossref]

Boettcher, S.

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

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

Cao, H.

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]

Castaldi, G.

S. Savoia, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89(8), 085105 (2014).
[Crossref]

Chan, C. T.

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
[Crossref] [PubMed]

Chan, S. W.

Chen, H.

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
[Crossref] [PubMed]

Chen, S.

B. Zhu, R. Lü, and S. Chen, “PT-symmetry breaking for the scattering problem in a one-dimensional non-Hermitian lattice model,” Phys. Rev. A 93(3), 032129 (2016).
[Crossref]

Chen, Y. F.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Chong, Y. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

Christodoulides, D. N.

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]

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]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

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]

Chuang, Y. L.

Y. L. Ziauddin, Y. L. Chuang, S. Qamar, and R. K. Lee, “Goos-Hänchen shift of partially coherent light fields in epsilon-near-zero metamaterials,” Sci. Rep. 6(1), 26504 (2016).
[Crossref] [PubMed]

Cowan, J.

J. Cowan and B. Aničin, “Longitudinal and transverse displacements of a bounded microwave beam at total internal reflection,” J. Opt. Soc. Am. A 67(10), 1307–1314 (1977).
[Crossref]

Della Valle, G.

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

Dunne, G. V.

C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252(6), 272–276 (1999).
[Crossref]

Dutriaux, L.

P. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78(5), 851–854 (1997).
[Crossref]

Eichelkraut, T.

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]

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

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]

Engheta, N.

S. Savoia, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89(8), 085105 (2014).
[Crossref]

Fegadolli, W. S.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Felbacq, D.

D. Felbacq and R. Smaâli, “BLOCH modes dressed by evanescent waves and the generalized Goos-Hänchen effect in photonic crystals,” Phys. Rev. Lett. 92(19), 193902 (2004).
[Crossref] [PubMed]

D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28(18), 1633–1635 (2003).
[Crossref] [PubMed]

Feng, L.

Flammia, S. T.

Y. C. Lee, M. H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local PT Symmetry Violates The No-Signaling Principle,” Phys. Rev. Lett. 112(13), 130404 (2014).
[Crossref] [PubMed]

Fleury, R.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[Crossref] [PubMed]

Galdi, V.

S. Savoia, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89(8), 085105 (2014).
[Crossref]

Gao, D.

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97(4), 041903 (2010).
[Crossref]

Gao, L.

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97(4), 041903 (2010).
[Crossref]

B. Zhao and L. Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express 17(24), 21433–21441 (2009).
[Crossref] [PubMed]

Ge, L.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

Goos, F.

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. 1(7-8), 333–346 (1947).
[Crossref]

Hänchen, H.

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. 1(7-8), 333–346 (1947).
[Crossref]

He, J.

He, S.

Hsieh, M. H.

Y. C. Lee, M. H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local PT Symmetry Violates The No-Signaling Principle,” Phys. Rev. Lett. 112(13), 130404 (2014).
[Crossref] [PubMed]

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

Jost, B. M.

B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

Kip, D.

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]

Kottos, T.

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]

Lai, H. M.

Lee, R. K.

Y. L. Ziauddin, Y. L. Chuang, S. Qamar, and R. K. Lee, “Goos-Hänchen shift of partially coherent light fields in epsilon-near-zero metamaterials,” Sci. Rep. 6(1), 26504 (2016).
[Crossref] [PubMed]

Y. C. Ziauddin and R. K. Lee, “Giant Goos-Hänchen shift using PT symmetry,” Phys. Rev. A 92(1), 013815 (2015).
[Crossref]

Lee, R.-K.

Y. C. Lee, M. H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local PT Symmetry Violates The No-Signaling Principle,” Phys. Rev. Lett. 112(13), 130404 (2014).
[Crossref] [PubMed]

Lee, Y. C.

Y. C. Lee, M. H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local PT Symmetry Violates The No-Signaling Principle,” Phys. Rev. Lett. 112(13), 130404 (2014).
[Crossref] [PubMed]

Li-Gang, W.

W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1), 016601 (2004).
[Crossref] [PubMed]

Lin, Z.

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]

Longhi, S.

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

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

Lu, M. H.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Lü, R.

B. Zhu, R. Lü, and S. Chen, “PT-symmetry breaking for the scattering problem in a one-dimensional non-Hermitian lattice model,” Phys. Rev. A 93(3), 032129 (2016).
[Crossref]

Makris, K. G.

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]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

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]

Marseille, A.

E. Pfleghaar, A. Marseille, and A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos-Hänchen shift,” Phys. Rev. Lett. 70(15), 2281–2284 (1993).
[Crossref] [PubMed]

Meisinger, P. N.

C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252(6), 272–276 (1999).
[Crossref]

Miri, M. A.

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]

Moreau, A.

Mostafazadeh, A.

A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43(1), 205–214 (2002).
[Crossref]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

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]

Nian-Hua, L.

W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1), 016601 (2004).
[Crossref] [PubMed]

Oliveira, J. E.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Onishchukov, G.

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]

Pattanayak, D. N.

J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50(21), 1664–1667 (1983).
[Crossref]

Peng, Y. G.

Peschel, U.

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]

Pfleghaar, E.

E. Pfleghaar, A. Marseille, and A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos-Hänchen shift,” Phys. Rev. Lett. 70(15), 2281–2284 (1993).
[Crossref] [PubMed]

Puri, A.

J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50(21), 1664–1667 (1983).
[Crossref]

Qamar, S.

Y. L. Ziauddin, Y. L. Chuang, S. Qamar, and R. K. Lee, “Goos-Hänchen shift of partially coherent light fields in epsilon-near-zero metamaterials,” Sci. Rep. 6(1), 26504 (2016).
[Crossref] [PubMed]

Qiang, L.

W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1), 016601 (2004).
[Crossref] [PubMed]

Ramezani, H.

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]

Regensburger, A.

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]

Renard, R. H.

R. H. Renard, “Total reflection: a new evaluation of the Goos–Hänchen shift,” J. Opt. Soc. Am. A 54(10), 1190–1197 (1964).
[Crossref]

Riesz, R.

Rüter, C. E.

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]

Saleh, B. E.

B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

Savoia, S.

S. Savoia, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89(8), 085105 (2014).
[Crossref]

Scherer, A.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Segev, M.

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]

Shi-Yao, Z.

W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1), 016601 (2004).
[Crossref] [PubMed]

Shramkova, O.

O. Shramkova and G. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18(10), 105101 (2016).
[Crossref]

Simon, R.

Smaâli, R.

D. Felbacq and R. Smaâli, “BLOCH modes dressed by evanescent waves and the generalized Goos-Hänchen effect in photonic crystals,” Phys. Rev. Lett. 92(19), 193902 (2004).
[Crossref] [PubMed]

D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28(18), 1633–1635 (2003).
[Crossref] [PubMed]

Sounas, D. L.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[Crossref] [PubMed]

Staliunas, K.

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

Stone, A. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011).
[Crossref] [PubMed]

Tsironis, G.

O. Shramkova and G. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18(10), 105101 (2016).
[Crossref]

Wang, L. G.

Wang, Y.

Weis, A.

E. Pfleghaar, A. Marseille, and A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos-Hänchen shift,” Phys. Rev. Lett. 70(15), 2281–2284 (1993).
[Crossref] [PubMed]

Xu, Y.

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
[Crossref] [PubMed]

Xu, Y. L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Yang, S.

Yi, J.

Yin, X.

Zhang, P.

Zhang, X.

Zhao, B.

Zhao, D. G.

Zhu, B.

B. Zhu, R. Lü, and S. Chen, “PT-symmetry breaking for the scattering problem in a one-dimensional non-Hermitian lattice model,” Phys. Rev. A 93(3), 032129 (2016).
[Crossref]

Zhu, H.

Zhu, S. Y.

Zhu, X.

Zhu, X. F.

Ziauddin, Y. C.

Y. C. Ziauddin and R. K. Lee, “Giant Goos-Hänchen shift using PT symmetry,” Phys. Rev. A 92(1), 013815 (2015).
[Crossref]

Ziauddin, Y. L.

Y. L. Ziauddin, Y. L. Chuang, S. Qamar, and R. K. Lee, “Goos-Hänchen shift of partially coherent light fields in epsilon-near-zero metamaterials,” Sci. Rep. 6(1), 26504 (2016).
[Crossref] [PubMed]

Ann. Phys. (2)

F. Goos and H. Hänchen, “A new and fundamental experiment on total reflection,” Ann. Phys. 1(7-8), 333–346 (1947).
[Crossref]

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437(1‐2), 87–102 (1948).
[Crossref]

Appl. Phys. Lett. (1)

D. Gao and L. Gao, “Goos–Hänchen shift of the reflection from nonlinear nanocomposites with electric field tunability,” Appl. Phys. Lett. 97(4), 041903 (2010).
[Crossref]

J. Math. Phys. (1)

A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43(1), 205–214 (2002).
[Crossref]

J. Opt. (1)

O. Shramkova and G. Tsironis, “Scattering properties of PT-symmetric layered periodic structures,” J. Opt. 18(10), 105101 (2016).
[Crossref]

J. Opt. Soc. Am. A (3)

R. H. Renard, “Total reflection: a new evaluation of the Goos–Hänchen shift,” J. Opt. Soc. Am. A 54(10), 1190–1197 (1964).
[Crossref]

J. Cowan and B. Aničin, “Longitudinal and transverse displacements of a bounded microwave beam at total internal reflection,” J. Opt. Soc. Am. A 67(10), 1307–1314 (1977).
[Crossref]

R. Riesz and R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A 2(11), 1809–1817 (1985).
[Crossref]

Nat. Mater. (1)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Nat. 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,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Nature (1)

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]

Opt. Express (4)

Opt. Lett. (4)

Phys. Lett. A (2)

C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252(6), 272–276 (1999).
[Crossref]

Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282(6), 343–348 (2001).
[Crossref]

Phys. Rev. A (5)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

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

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84(4), 042119 (2011).
[Crossref]

Y. C. Ziauddin and R. K. Lee, “Giant Goos-Hänchen shift using PT symmetry,” Phys. Rev. A 92(1), 013815 (2015).
[Crossref]

B. Zhu, R. Lü, and S. Chen, “PT-symmetry breaking for the scattering problem in a one-dimensional non-Hermitian lattice model,” Phys. Rev. A 93(3), 032129 (2016).
[Crossref]

Phys. Rev. B (1)

S. Savoia, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers,” Phys. Rev. B 89(8), 085105 (2014).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

W. Li-Gang, L. Nian-Hua, L. Qiang, and Z. Shi-Yao, “Propagation of coherent and partially coherent pulses through one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1), 016601 (2004).
[Crossref] [PubMed]

Phys. Rev. Lett. (12)

D. Felbacq and R. Smaâli, “BLOCH modes dressed by evanescent waves and the generalized Goos-Hänchen effect in photonic crystals,” Phys. Rev. Lett. 92(19), 193902 (2004).
[Crossref] [PubMed]

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014).
[Crossref] [PubMed]

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]

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

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref] [PubMed]

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]

Y. C. Lee, M. H. Hsieh, S. T. Flammia, and R.-K. Lee, “Local PT Symmetry Violates The No-Signaling Principle,” Phys. Rev. Lett. 112(13), 130404 (2014).
[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(9), 093902 (2011).
[Crossref] [PubMed]

E. Pfleghaar, A. Marseille, and A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos-Hänchen shift,” Phys. Rev. Lett. 70(15), 2281–2284 (1993).
[Crossref] [PubMed]

P. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78(5), 851–854 (1997).
[Crossref]

B. M. Jost, A. A. R. Al-Rashed, and B. E. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

J. L. Birman, D. N. Pattanayak, and A. Puri, “Prediction of a resonance-enhanced laser-beam displacement at total internal reflection in semiconductors,” Phys. Rev. Lett. 50(21), 1664–1667 (1983).
[Crossref]

Rep. Prog. Phys. (1)

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

Sci. Rep. (2)

Y. Xu, C. T. Chan, and H. Chen, “Goos-Hänchen effect in epsilon-near-zero metamaterials,” Sci. Rep. 5, 8681 (2015).
[Crossref] [PubMed]

Y. L. Ziauddin, Y. L. Chuang, S. Qamar, and R. K. Lee, “Goos-Hänchen shift of partially coherent light fields in epsilon-near-zero metamaterials,” Sci. Rep. 6(1), 26504 (2016).
[Crossref] [PubMed]

Other (1)

M. Born and E. Wolf, Principles of Optics (New York, 1999).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 (a) Geometry of the light beam propagating through a periodic multilayer-structure with complex-conjugate permittivities in linear homogeneous medium, and (b) schematic diagram of the model for S-matrix.
Fig. 2
Fig. 2 (a) Reflectivity/transmissivity and (b) the related shifts for the PT bilayer-structure (N = 1) as a function of the frequency, for the thickness of each slab L = 125 u m , the incident angle θ 0 = 5 d e g , the permittivities ε 1 = 0.1 0.1 i and ε 2 = 0.1 + 0.1 i . The insert in Fig. 2(b) shows the phases of reflectivity/transmissivity as a function of incident angle θ 0 at the CPA-Laser point.
Fig. 3
Fig. 3 (a) Reflectivity from left side and (b) the lateral shifts as a function of the frequency for different incident angles. Other parameters are L = 125 u m , ε 1 = 0.1 0.1 i , ε 2 = 0.1 + 0.1 i .
Fig. 4
Fig. 4 Reflectivity from left side and the related lateral shifts for (a, b) N = 5 and (c, d) N = 10 as a function of the frequency. Other parameters are the same as those in Fig. 2.
Fig. 5
Fig. 5 (a) Eigenvalues of the S matrix, (b) Reflectivity/transmissivity and (c) phases of reflection coefficients as a function of the frequency, for L = 125 u m , θ 0 = 5 d e g , ε 1 = 0.0001 0.001 i and ε 2 = 0.0001 + 0.001 i .
Fig. 6
Fig. 6 Reflectivity and the related shifts from left (blue solid line) and right (red dashed line) sides for (a, d)N = 1 (b, e)N = 45 and (c, f) N = 90 as a function of the frequency. Other parameters are the same as those in Fig. 5.
Fig. 7
Fig. 7 Dependence of the lateral shift of reflection from left on the stack periodicity N for ω = 3 × 10 13 s 1 . Other parameters are the same as those in Fig. 6.
Fig. 8
Fig. 8 (a,b) Reflectivity/transmissivity and (c) the related shifts for the PT bilayer-structure (N = 1) as a function of the incident angle, for L = 125 u m , ω = 1.49 × 10 13 s 1 , ε 1 = 0.1 0.1 i and ε 2 = 0.1 + 0.1 i .
Fig. 9
Fig. 9 Magnetic field amplitude distribution of all the calculated apace and in the surface of reflection and transmission for (a, b) θ0 = 2deg, ω = 1.452 × 10 13 s 1 , (c, d) θ0 = 5deg, ω = 1.493 × 10 13 s 1 , with ε 1 = 0.1 0.1 i and ε 2 = 0.1 + 0.1 i , and (e, f) θ0 = 5deg, ω = 1.493 × 10 13 s 1 , with ε 1 = 0.1 + 0.1 i and ε 2 = 0.1 0.1 i .

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

2 H ( x , z ) ε j μ j 2 z + ( k 0 2 k x 2 ε j μ j ) H ( x , z ) = 0 ,
H 0 ( x , z ) = e i ω t + i k 0 x x { H f e i k 0 z z + H b e i k 0 z z , z < N L H f + e i k 0 z z + H b + e i k 0 z z , z > N L
( H f + H b + ) = Q ( H f H b ) = ( c 2 k z ω ε 2 μ 2 c 2 k z ω ε 2 μ 2 1 1 ) 1 M ( c 2 k z ω ε 1 μ 1 c 2 k z ω ε 1 μ 1 1 1 ) ( H f H b ) ,
m j = ( cos ( k j z L ) i ω ε j μ j c 2 k j z sin ( k j z L ) i c 2 k j z ω ε j μ j sin ( k j z L ) cos ( k j z L ) ) , ( j = 1 , 2 )
Δ r , t = d φ r , t d k 0 x ,
Δ r , t = λ 2 π cos ( θ 0 ) d φ r , t d θ 0 .
( H b H f + ) = S ( H b + H f ) = ( t R r L r R t L ) ( H b + H f ) .
λ 1 , λ 2 = t ± r L r R 2 .

Metrics