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PT -symmetric transport in non-PT -symmetric bi-layer optical arrays J. Ramirez-Hernandez,1 F. M. Izrailev,1,∗ N. M. Makarov,2,† and D. N. Christodoulides3 1Instituto de F´ısica, Benemérita Universidad Autońoma de Puebla, Apdo. Post. J-48, Puebla, Pue., 72570, México 2Instituto de Ciencias, Benemérita Universidad Autónoma de Puebla, Priv. 17 Norte No. 3417, Col. San Miguel Hueyotlipan, Puebla, Pue., 72050, México 3College of Optics and Photonics-CREOL, University of Central Florida, 32816, USA (Dated: January 12, 2016) We study transport properties of an array created by alternating (a,b) layers with balanced loss/gain characterized by the key parameter γ. It is shown that for non-equal widths of (a,b) layers, i.e., when the corresponding Hamiltonian is non-PT-symmetric, the system exhibits the 6 scatteringpropertiessimilartothoseoftrulyPT-symmetricmodelsprovidedthatwithoutloss/gain 1 the structure presents the matched quarter stack. The inclusion of the loss/gain terms leads to an 0 emergenceofafinitenumberofspectralbandscharacterizedbyrealvaluesoftheBlochindex. Each 2 spectral band consists of a central region where the transmission coefficient TN (cid:62) 1, and two side regions with T (cid:54) 1. At the borders between these regions the unidirectional reflectivity occurs. n N Also,thesetofFabry-PerrotresonanceswithT =1arefoundinspiteofthepresenceofloss/gain. a N J PACSnumbers: 11.30.Er,42.25.Bs,42.82.Et 0 1 Introduction - The possibility of deliberately inter- additionthissamesymmetrywasrecentlyutilizedtoen- ] s mixing loss and gain in optical structures as a means force single-mode behavior in laser micro-cavities as well c to attain physical properties that are otherwise out of as to judiciously control wave dynamics around excep- i t reach in standard arrangements has recently been ex- tional points and photonic crystals [15]. Finally, similar p o ploredinanumberofstudies[1–8]. Ingeneral,suchnon- concepts are nowadays used in other fields, like for ex- . conservativeconfigurationscandisplaysurprisingbehav- ample opto-mechanics, acoustics, nonlinear optics, plas- s c ior resulting from the possible presence of exceptional monics, metamaterials, and imaging to mention a few i points [9, 10] and mode bi-orthogonality. A particu- [16]. In view of these developments, the following ques- s y lar class of non-Hermitian systems that has recently at- tion naturally arises. To which extent could a system h tracted considerable attention is that respecting parity- depart from an exact parity-time-symmetry and still ex- p time (PT) symmetry [1–5]. In this regard, if a Hamilto- hibit PT-symmetric characteristics? Recently, a possi- [ nian is PT-symmetric, then its ensuing spectrum hap- bilityofsynthesizingcomplexpotentialsthatsupporten- 1 pens to be entirely real in spite of the fact that the tirely real spectra despite the fact that they violate the v potential involved is complex [9, 10]. Once however a necessaryconditionforPT-symmetryhasbeendiscussed 9 loss/gainparameterexceedsacriticalvalue,thissymme- inRef.[17]withinthecontextofopticalsupersymmetry. 5 2 trycanspontaneouslybreakandconsequentlytheeigen- In this paper we show that it is indeed possible to 2 value spectrum ceases to be real and hence enters the observe PT-scattering behavior even in systems that 0 complex domain [9]. . strictly speaking lack this symmetry. This is explic- 1 The availability of optical attenuation and amplifica- itly demonstrated in perfectly matched multilayer non- 0 tion makes optics an ideal ground where PT-symmetric Hermitianarrangementshavingunequallayerwidths. In 6 1 effects can be experimentally observed and investigated thiscase,theincorporationofloss/gaindomainsleadsto : [3, 4, 11]. In the paraxial regime, a necessary (but not a finite number of bands-all associated with a real Bloch v sufficient) condition to impose this symmetry in photon- indexthatisreminiscentofactualPT-symmetriclattices i X icsisthatthecomplexrefractiveindexdistributionobeys [12]. Special attention has been paid to an emergence of r n(x) = n∗(−x) [1, 2]. This latter condition directly im- unidirectional reflectivity. Our results indicate that a a pliesrealpartoftherefractiveindexpotentialmustbean PT-symmetric response can be quite robust and there- even function of position while its imaginary component fore can be observed in more involved structures even in must be antisymmetric. the absence of a strict PT-symmetry. Thus far, the ramifications of parity-time symmetry The model - We consider the propagation of an elec- on optical transport have been systematically consid- tromagneticwaveoffrequencyω througharegulararray ered in several theoretical and experimental investiga- of N identical unit (a,b) cells embedded in a homoge- tions. Processes emerging from PT-symmetry, like dou- neous c-medium, see Fig. 1. Each cell consists of two ble refraction, band merging, unidirectional invisibility, dielectric a and b layers with the thicknesses d and d , a b abrupt phase transitions and nonreciprocal wave propa- respectively, sothatd=d +d istheunit-cellsize. The a b gationhavebeenpredictedandobserved[1,3,11–14]. In a and b layers are made of the materials absorbing and 2 amplifying the electromagnetic waves, respectively. The the transfer matrix Mˆ(γ) has specific symmetry, E+➝L c a b a b a b a b c E➝+R M22(γ)=M1∗1(−γ), M21(γ)=M1∗2(−γ), (2) ... - L 1 N-1 N R - where“∗”standsforthecomplexconjugation. Thissym- E E ➝ L ➝ R metry can be compared with that emerging for various ➝ x PT-symmetric Hamiltonians [7, 19–21]. Clearly, it dif- fers from the standard one, M = M∗ , M = M∗ , 22 11 21 12 FIG. 1. (Color online) A sketch of the model. that originates from time-reversal symmetry. Frequencybandstructure-Thedispersionrelation fortheBlochindexµ determinedthroughtheeigenval- loss and gain in the layers are incorporated via complex B uesexp(±iµ )ofthetransferMˆ-matrix,isdefinedbythe permittivities,whilethemagneticpermeabilitiesµ are B a,b matrix trace, 2cosµ =M +M . Thus, one obtains assumedtoberealandpositiveconstants. Therefractive B 11 22 indices n , impedances Z and wave phase shifts ϕ a,b a,b a,b cosϕ+γ2cosh(γϕ) of a and b layers are presented as follows, cosµ = . (3) B 1+γ2 n =n(0)(1+iγ), Z =Z(1+iγ)−1, ϕ =ϕ(1+iγ)/2, a a a a This relation determines the dependence of the Bloch n =n(0)(1−iγ), Z =Z(1−iγ)−1, ϕ =ϕ(1−iγ)/2, b b b b wave number κ = µ /d on the frequency ω (or, on the B where phase shift ϕ∝ω) for the wave inside the scattering region. Oneoftheimportantconclusionsthatfollowsfrom Z =µ /n(0) =µ /n(0), ϕ=2ωn(0)d /c=2ωn(0)d /c. Eq. (3), is a finite number N of spectral bands de- a a b b a a b b band fined by the condition |cosµ |<1. This number N B band Here the parameter γ > 0 measures the strength of bal- depends on the parameter γ only, and for γ (cid:28) 1 can be anced loss and gain inside a and b layers, respectively, estimated as and ϕ = ϕ + ϕ is the purely real phase shift of the a b wave passing each of the unit cells. In the case of no N ≈(1/πγ)ln(1/γ). (4) band loss/gain (γ = 0) the structure is known as the matched quarter stack for which the a and b layers are perfectly Within these bands the Bloch index µ is real in spite B matched and have equal optic paths, n(0)d = n(0)d . of the presence of loss and gain. This property has a a b b As one can see, for γ (cid:54)= 0 and d (cid:54)= d the correspond- been widely discussed in view of the nature of the PT- a b ing Hamiltonian has no the PT-symmetry. A particular symmetric systems. As is known, the real-valued solu- PT-symmetric realization is achieved only when d =d tionsµ tothedispersionequation2cosµ =M +M a b B B 11 22 witWhiεth(a0i)n=anεy(b0)la,yaenrdaµnae=lecµtrbo.magnetic wave propagates cMân-meaxtisrtixoinslyrewalh-veanluthede,tir.aec.eifM11+M22 of the transfer alongthex-direction(Fig.1)perpendiculartothestrat- ImM =−ImM . (5) ification according to the 1D Helmholtz equation. Its 22 11 generalsolutioninsideeveryunitcellcanbepresentedas Inourmodelthisdeterminativeconditionisprovidednot a superposition of traveling forward and backward plane due to the PT-symmetry but merely due to the balance waves. By combining these solutions with the boundary between loss and gain. conditions for the wave at the interfaces between a and b layers, one can obtain the unit-cell transfer matrix Mˆ Outside the bands, where |cosµB| > 1, the index µB ispurelyimaginary,thuscreatingspectral gaps. Herethe relating the wave amplitudes of two adjacent unit cells waves are known as the evanescent Bloch states, attenu- (see, e.g., Ref. [18]). For our model the elements of the ated on the scale of the order of |µ |−1. Therefore, for a M-matrix read B sufficiently long structure, N|µ | (cid:29) 1, the transmission B exp(iϕ)+γ2exp(−γϕ) is exponentially small. It should be noted that in the M = , (1a) 11 1+γ2 absence of loss/gain (γ = 0), there are no spectral gaps iγ since all spectral bands are touching. M = [exp(−iϕ)−exp(γϕ)], (1b) 12 1+γ2 When γ exceeds the critical value γ = 1, there are cr iγ no spectral bands since the Bloch index µ (ω) becomes B M = [exp(iϕ)−exp(−γϕ)], (1c) 21 1+γ2 imaginary for any frequency ω. Two examples of the exp(−iϕ)+γ2exp(γϕ) dependenceµB(ω)areshowninFig.2fordifferentvalues M22 = 1+γ2 . (1d) ofγ. Itisquiteinstructivethatwithinanyspectralband themaximalvalueoftheBlochindexµ(max) <π forany B NotethatdetMˆ =1inaccordancewiththegeneralcon- nonzero γ. The smaller the parameter γ, the closer to π ditioninherentfortransfermatrices. Ontheotherhand, can be the Bloch index µ . On the other hand, for any B 3 24 6 1.0 5 18 4 0.8 π12 π3 ϕ/ ϕ/ 2 0.6 6 γ = 0.1 γ = 0.25 1 γ 01 0.80.60.40.2 00 0.20.40.60.8 1 01 0.80.60.40.2 00 0.20.40.60.8 1 0.4 Im µ /π Re µ /π Im µ /π Re µ /π Β Β Β Β 0.2 FIG. 2. The dependence (3) of the Bloch index µ on the B phase shift ϕ ∝ ω for two values of γ. Spectral bands and 0.0 0 1 2 3 4 5 6 7 8 9 10 gaps are shown by shaded areas. ϕ/π FIG. 4. Spectral bands and gaps in dependence on the given γ from the interval 0<γ <1 the higher the band loss/gain parameter γ. Full curve shows the border µB = 0 the smaller the value of µ(max). above which µB is imaginary (gaps). Below this curve µB is B real. Between full and dashed curve µ is real and T (cid:54) 1. Transmittance - With the standard transfer matrix B N Insidedashedregionsµ isrealandT (cid:62)1. Thedependence B N approach we derived analytical expression for the trans- ϕ=ϕ (γ) is depicted by the dashed curve on which T =1. s N mittanceTN ofourmodelconsistingofN unit(a,b)cells The horizontal dash-dotted line is shown for γ =0.25. and connected to perfect c-leads by the impedance Z, (cid:20) γ2sin2(Nµ ) (cid:21)−1 T = 1− B F(γ,ϕ)F(−γ,ϕ) , (6) N 4(1+γ2)2sin2µ B F(γ,ϕ)=γ2sinh(γϕ)−γsinϕ+2(cosϕ−e−γϕ). (7) PT-symmetric models (see, e.g., Ref. [22] and references Here F(−γ,ϕ) < 0 for any value of ϕ, unlike F(γ,ϕ) therein). However,inourcasetheBlochindexµ (γ,ϕ ) B s thatcanbeeitherpositiveornegativeasafunctionofϕ. does not vanish, in contrast to truly exceptional points Two examples of dependence TN(ϕ) are given in Fig. 3. for which both conditions, TN = 1 and µB = 0, are ful- filled. These conditions are known as that providing in- 4 visibility of a model in scattering process. The only sit- γ = 0.25, Ν = 5 2 uation for the invisibility to occur in our setup is when N T 0 two conditions, F(γ,ϕ)=0 and µB(γ,ϕ)=0, meet. As n one can see in Fig. 4, the sequence of exceptional points l -2 ϕ = ϕ ≡ ϕ (γ ) emerges for the corresponding se- sp s sp -4 quence of γ = γ for which two curves ϕ = ϕ (γ) and 0 1 2 3 4 5 sp s ϕ/π µB(γ,ϕ)=0 touch each other. Note that at every fixed 4 value ofγ =γ thecorrespondingexceptionalpoint ϕ sp sp 2 γ = 0.25, Ν = 10 is located on the top of the highest band. N T 0 ln -2 Apartfromthespecificpointsϕs(γ),theperfecttrans- mission, T = 1, occurs due to Fabry-Perot resonances N -4 0 1 2 3 4 5 located within spectral bands. They are defined by the ϕ/π condition sin(Nµ )/sinµ = 0 specifying the resonant B B valuesµres =mπ/N (m=1,2,3,...,N−1)oftheBloch B FIG.3. ThelogarithmofTN definedbyEq.(6)asafunction index µB. Since the spectrum µB(ω) is restricted by ofϕforγ =0.25anddifferentvaluesN ofunitcells. Shaded µ(max), the total number of such resonances in a given regionscorrespondtospectralbands,andcirclesstandforthe B spectralbandcanvaryfromzeroto2(N−1), depending borders between the regions with T (cid:54)1 and T (cid:62)1. N N on the band number and loss/gain parameter γ. Specif- ically, for a fixed γ the higher spectral band contains The analysis shows that a distinctive property of the the smaller number of the Fabry-Perot resonances. This transmittance is that every spectral band consists of the result is principally different from what happens in an central region with T (cid:62) 1 and those with T (cid:54) 1, N N array of bi-layers with real and positive optical parame- see Figs. 3 and 4. At the borders between these regions ters, where N−1 number of the Fabry-Perot resonances (circles in Fig. 3) the transmission is perfect, T = 1, N emerge in any spectral band. and this happens due to vanishing the function F(γ,ϕ). The corresponding values ϕ (γ) for which F(γ,ϕ ) = 0 s s canbecomparedwithexceptional pointsemerginginthe Unidirectionalreflectivity-Theexpressionsforthe 4 2000 Within the spectral gaps where the Bloch index is (L) N γ = 0.25 purely imaginary (µB = i|µB|), for a sufficiently long R structure, N|µ |(cid:29)1, the transmittance (6) is exponen- B / 1000 tially small. In such a situation the left reflectance (8) R) is larger than T , however smaller than 1. The right ( N N R reflectance (9) can be shown to exceed unity. Summa- 00 1 2 3 4 5 6 rizing, one can write, TN < RN(L) < 1 < RN(R). Both ϕ/π reflectances increase and eventually saturate inside the gapsforN →∞. However,bothR(L) (cid:28)1andR(R) (cid:29)1 N N FIG. 5. Ratio right-to-left reflectances (full curve) versus ϕ remain to be valid. Thus, the ratio R(L)/R(R) can reach N N for γ =0.25. Dashed regions are those where µB is real and extremely small values of the order of 10−7. T >1, see Fig. 4. N Conclusions-Wehaveanalyticallystudiedthemodel that reveals the PT-symmetric transport although its left and right reflectances, R(L) and R(R), read Hamiltonian, in general, is not the PT-symmetric one. N N Ouranalysisexhibitsthatsuchasituationemergesinthe R(L) γ2sin2(Nµ ) bi-layer optical setup which without loss/gain is known N = B F2(γ,ϕ); (8) TN 4(1+γ2)2sin2µB as the quarter stack, provided all layers are perfectly matched (no reflections due to boundary conditions). R(R) γ2sin2(Nµ ) N = B F2(−γ,ϕ). (9) Intheabsenceofloss/gainterms,allspectralbandsare TN 4(1+γ2)2sin2µB touchedthuscreatingaperfecttransmissionforanywave frequency. When the balanced loss and gain are alter- Note that T (−γ) = T (γ) and R(L)(−γ) = R(R)(γ). N N N N natingly included in all layers, the spectral gaps emerge It is remarkable that TN(γ) and RN(L)(γ),RN(R)(γ) satisfy between the bands, and the total number of bands is de- the famous relation, fined by the loss/gain parameter γ only. (cid:113) Inside the spectral bands the Bloch index µB appears |1−TN|= RN(L)RN(R), (10) to be real as it happens in the known models with the PT-symmetric Hamiltonians. The analysis shows that that is known to occur in PT-symmetric models (see, eachofthespectralbandsconsistsofcentralregionwhere e.g., Ref. [23]). the transmission coefficient is greater than one, T (cid:62)1, N Since at the specific points ϕ = ϕs(γ) the function and two side regions with T (cid:54) 1. At the borders be- N F(γ,ϕ) vanishes, the left reflectance (8) also vanishes, tween these regions the transmission is perfect, T = 1, N however, the right one (9) remains finite. For γ (cid:28)1 and however, µ does not typically vanish as happens in the B γexp(γϕ)(cid:29)1 it can be estimated as follows, known PT-symmetric models. These borders are of specific interest since one of the reflectances vanishes, thus R(R) ≈16γ−4sin2(Nµ ). (11) N B leading to the so-called ”unidirectional reflectivity”, the effect which is important both from theoretical and ex- Thiseffectisknownastheunidirectionalreflectivity [12]. perimental points of view. The ratio between left and It is one of the most important properties of scattering right reflectances is specified by the loss/gain parameter in the PT-symmetric systems. The ratio between right γ and the wave frequency ω, being independent of the and left reflectances, number N of cell units. It is important that a strong R(R)/R(L) =F2(−γ,ϕ)/F2(γ,ϕ), (12) difference between left and right reflectances occurs in a N N relativelylargewave-propagationregionwheretheBloch is shown in Fig. 5 in dependence on ϕ for γ =0.25. Re- index µ is real. B markably, this ratio is N-independent, whereas at the Theanalyticalexpressionsdisplaythatinsidethespec- Fabry-Perot resonances where sin(NµB)/sinµB = 0, tral bands the Fabry-Perot resonances emerge in spite both reflectances expectedly vanish, R(L) = R(R) = 0. of the presence of loss and gain. These resonances ex- N N ist both in the regions with T > 1 and T < 1. 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