Abstract
The transmission properties through a saturable cubicquintic nonlinear defect attached to lateral linear chains is investigated. Particular attention is directed to the possible nonreciprocal diodelike transmission when the paritysymmetry of the defect is broken. Distinct cases of parity breaking are considered including asymmetric linear and nonlinear responses. The spectrum of the transmission coefficient is analytically computed and the influence of the degree of saturation analyzed in detail. The transmission of Gaussian wavepackets is also numerically investigated. Our results unveil that spectral regions with high transmission and enhanced diodelike operation can be achieved.
Similar content being viewed by others
Introduction
The investigation of controlled transport of energy/mass has remained a hot area of research since a long time and it has recently gained considerable revival due to its direct implications in manufacturing technological devices for controlled transport. The diode is one of the basic devices which can directionally control transport allowing it to occur mainly in a preferential direction. Besides the most traditional diodes that operate by rectifying electric current, there have been several demonstrations for the capability of achieving such nonreciprocal transport of heat flow^{1,2,3}, acoustic^{4,5,6} and electromagnetic^{7,8,9,10} waves.
In linear systems with timereversal symmetry, no symmetry breaking of transmission can be generated according to the reciprocity theorem^{11,12,13}. Timereversal symmetry can be broken in magnetooptical devices by an applied magnetic field to generate nonreciprocal transport of optical waves^{10}. An analog mechanism has been developed to produce nonreciprocal isolators of acoustic waves^{14}. On the other hand, nonlinear processes can lead to asymmetric wave propagation without the need of an external magnetic field. Recent experimental advances have demonstrated that nonlinear optical lattices with alternating gain and loss atomic configurations are ideal realizations of nonHermitian paritytime symmetric systems that can be explored to better understand the peculiar physical properties of these nonHermitian systems in atomic settings^{15,16}.
By exploring to own nonlinear properties of the underlying medium, the nonreciprocal propagation of elastic and optical waves have been demonstrated. In particular, a pronounced rectifying factor for the scattering of harmonic waves was reported in a multilayered system represented by a set of discrete nonlinear Schrödinger equations with a cubic nonlinearity restricted to take place just in two nonlinear asymmetric layers^{17}. Recently, the effect of a higher order quintic nonlinearity on the nonreciprocal transport has been investigated, unveiling that the combined effect of cubic and quintic terms is nonadditive^{18}. The nonlinear dynamics of soliton formation in laserinduced optical gratings has been experimentally probed under the competing action of cubic and quintic nonlinear contributions^{19,20}, evidencing mutual transformations among dropletlike fundamental, dipole, and azimuthally modulated vortex solitons.
However, when very intense waves propagate in matter, it recovers its linear behavior but with a distinct group velocity as compared to that of a lowintensity wave^{21}. This phenomenon is known as the saturation of the nonlinearity and is associated with the emergence of several nontrivial optical phenomena^{21,22,23,24,25,26,27,28,29,30,31,32}. Within the context of nonreciprocal transmission through nonlinear asymmetric layers, it has been shown that bistability can be enhanced by saturation of a cubic nonlinearity, thus favoring the nonreciprocal diodelike transmission. Also, saturation was shown to have opposite impacts of the rectifying action over short and long wavelength harmonic signals^{33}. Further, nonlocal nonlinear responses were shown to generally act by reducing the diode action^{34}.
Motivated by the nonadditivity of cubic and quintic nonlinear effects on the nonreciprocal transmission of harmonic waves and by the beneficial trend produced by saturation of the nonlinear responses, we will address to the question of how these aspects can be put together to enhance the diodelike scattering of waves by a nonlinear asymmetric dimer. Within a tightbinding approach, we will analytically compute the spectrum of transmission through a saturable nonlinear asymmetric dimer coupled to linear sidechains. We will explicitly consider the saturation of both cubic and quintic nonlinear contributions and explore distinct scenarios for breaking the parity symmetry. We also numerically investigate the transmission of Gaussian incoming waves. In particular, we discuss the possibility of achieving large rectification action with high transmission in appropriated regions of the model’s parameters space.
The paper is organized as follows: In section II, we introduce the model and develop the main formalism used to obtain the spectral transmission properties. In section III, we explore the saturation effect on the multistability and transmission for various(distinct) symmetry breaking scenarios. Section IV devotes to the analysis of the saturation effect on the rectification factor. In section V, we provide some numerical results concerning the nonreciprocal transmission on incoming Gaussian wavepackets. Finally, in section VI, we summarize and draw our main conclusions.
Saturable CubicQuintic DNLS Chain
In the present work, we study the stationary scattering solutions of plane waves in a discrete linear chain with a pair of saturable nonlinear defects. The defects will be described as a saturable cubicquintic discrete nonlinear Schrödinger dimer. The saturable cubicquintic discrete nonlinear Schrödinger (sCQDNLS) equation with a saturated nonlinearity is given by
where μ_{3} and μ_{5} are the control parameters for the saturation of both the onsite cubic and quintic nonlinear responses, respectively. In what follows we will assume μ_{3} = μ_{5} = μ. V_{n} is the potential at site n. The offdiagonal coefficient is chosen to be unity without loss of generality. The parameters γ_{n} and ν_{n} are the local cubic and quintic nonlinear responses of the system, respectively, that are the main nonlinear contributions at low wave amplitudes. The stationary solutions of Eq. (1) are given by
Using these in Eq. (1), we arrive at a stationary cqDNLS equation
which can be rearranged in a backward iterative scheme^{17,35} as follows:
where ω is the spatial frequency and ϕ_{n} is the complex mode amplitude at site n. Eq. (4) is a map for the amplitudes at site n − 1 of the timeindependent discrete nonlinear Schrodinger equation. This map is helpful for obtaining the transmission formulae, since Eq. (4) provides the amplitude at site n − 1 given the amplitudes at site n and n + 1. Note that we are interested in examining the scattering by a nonlinear dimer, i.e., two sites carrying the nonlinear effects at the center of an infinite one dimensional chain. This implies that the full cqDNLS in Eq. (1) applies to just these two sites. Therefore, the last two terms in Eq. (1) will not contribute when an input/output signal is away from the dimer. In that case, the transmission will be governed by just the linear part of Eq. (1). Hence the input/output signal will not feel any nonlinear response from the lattice in that region and can propagate freely.
We will focus on the scattering properties of plane wave solutions of the following form
where, R_{0}, R and T are the amplitudes of incoming, reflected and transmitted waves, respectively. Note that the nonlinear region in the lattice corresponds to the sites n = 1, 2. Thus the sitedependent coefficients carrying nonlinear effects ν_{n} and γ_{n} are absent in the linear part of the lattice. Therefore, the model corresponds to a nonlinear dimer connected to otherwise linear side chains. Further, we will consider V_{n} = 0 on both side chains.
The purpose of this work is to evaluate the capability of the saturated cqDNLS dimer to produce efficient asymmetric scattering and hence to operate as a wave diode. The desired effect (asymmetric transmission) arises when one breaks the translational symmetry of the lattice^{17,18,33,34}. This symmetry breaking combined with the nonlinearity leads to a nonreciprocal transmission of the input signal. In a onedimensional setting, this can be done in different ways which we discuss in details in the following sections.
Using the backward transfer map, the transmission coefficient for a rightpropagating wave (k > 0, i.e., the wave is incident from the left of the dimer) is given by
where,
and
When a harmonic wave is incident from the right of the nonlinear dimer (\(k < 0\)), i.e., a leftpropagating wave, the transmission coefficients can be computed in the same way simply by exchanging the subscripts 1 and 2, which amounts to flipping the lattice^{17,18,33,34}.
Effects of Saturation on Multistability and Transmission
We are interested in exploring a nonreciprocal transmission of input signals which will lead us to the desired diodelike action. In particular, we focus on the influence of the simultaneous saturation of both cubic and quintic nonlinear responses. We begin by showing the typical relation between incoming and transmitted intensities for varying strengths of the saturation parameter μ.
Figure 1 is a plot for transmitted intensity T^{2} as a function of incoming intensity R_{0}^{2} for the nonlinear dimer with V_{0} = −2.5 and with γ = 1, ν = 0.5 representing the local (onsite) cubic and quintic nonlinear response of the lattice respectively, μ determines the extent to which we saturate these nonlinear responses. Note that we have defined the asymmetry in onsite potentials ε_{v} whose strength determines the extent to which the onsite potentials differ from each other V_{1} = V_{0}(1 + ε_{v}) and V_{2} = V_{0}(1 − ε_{v}). In the case at hand we deal with a dimer and the difference between (two) onsite potentials is taken to be ε_{v} = 0.05. Similarly, we will later utilize the asymmetry in other site dependent parameters with a relevant corresponding denotation. For no saturation (μ = 0, top left in Fig. 1), there are two bistability regions, i.e., regions where a single input intensity gets transmitted with different intensities representing distinct defect modes. The first window occurs at lower incident intensities and it is significantly broad compared to the second window occurring at higher incoming intensities (see^{18} for details). When the saturation parameter μ increases, the multistability regions are suppressed, going from a single bistability window at intermediate values of μ) to the usual single mode behavior at strongly saturable nonlinearities.
Figure 2 shows the transmission coefficient t(k, T^{2}) as a function of transmitted intensity T^{2} for a sCQDNLS dimer with the same parameter values as in Fig. 1. The shift between the left and rightincident transmission peaks due to the asymmetry in onsite potentials ε_{v} is apparent. It is also important to mention that the transmission coefficient is comparatively larger in the ranges of intensity which lead to a significantly asymmetric transmission. The asymmetry between the left and right incidence transmission is suppressed when the nonlinear dimer is strongly saturable.
Figure 3 is the transmission coefficient density plot for the sCQDNLS dimer with asymmetric onsite potentials and with the parameter values for the onsite nonlinear responses being the same as in the previous figures. The transmission increases with increasing saturation from μ = 0 to μ = 0.5 (see the plots in the top panel in Fig. 3), then the transmission gradually decreases when we further increase the saturation from μ = 0.5 to μ = 1.5 (see the plots in the bottom panel in Fig. 3).
When the lattice’s symmetry is broken threefold with all three site dependent parameters chosen to be different simultaneously at each dimer site (different onsite potential, cubic and quintic response), it is possible to maintain higher transmission at increased levels of asymmetry and saturation. Figure 4 is produced to depict this case with the parameter values chosen as described in the caption. The threefold symmetry breaking parameters ε_{v}, ε_{c} and ε_{q} are nonzero simultaneously, where ε_{v} describes the asymmetry in onsite potentials, ε_{c} and ε_{q} describe the asymmetry in cubic and quintic response, respectively. In the regime of high saturation and 3fold asymmetry, high transmission occurs at intermediate wavenumbers k ≈ 0.8~1.8. Note however that in this regime, there is no transmission for very small wavenumbers k ≈ 0~0.2 or very large wavenumbers k ≈ 2.7~π. It is important to stress that higher transmission at higher asymmetry is a welcome feature with regard to a diodelike transmission. In previous works, higher asymmetry renders a better diode effect but usually with an overall lower transmission^{17,18,34}. The present result shows that the simultaneous asymmetry on the potential and cubic and quintic nonlinear contributions can overcome the problem of lower transmission at better diodelike modes.
Distinct Asymmetry Scenarios
As mentioned before, there are three distinct ways to break the translational symmetry of the underlying onedimensional sCQDNLS lattice system in order to support the nonreciprocal transmission. It is of interest to study possible implications of saturating the nonlinear responses of the lattice on transmission of input signals under various distinct possibilities to induce asymmetry. The sCQDNLS dimer model has two local nonlinear responses. In this section, we will report on the effects of saturation on the nonreciprocal transmission under three different possible parity breaking schemes. The three different cases correspond to the three distinct choices of onsite parameters asymmetries. For instance, one could choose either of the following

(i)
different onsite potentials for the two dimer sites.

(ii)
different onsite cubic nonlinear response.

(iii)
different onsite quintic nonlinear response.
Additionally, one can also have all three factors acting simultaneously, which would amount to a higher degree of broken symmetry, i.e., a threefoldsymmetry breaking or threefoldasymmetry case, which we will also discuss in detail in the upcoming subsections.
Different onsite potentials
This is the first of three possibilities to break the translational symmetry of the one dimensional chain under consideration. Figure 5 presents the transmission coefficient as a function of saturation μ and the transmitted intensity T^{2}.
As shown in Fig. 5, bulk of the transmission occurs at saturation μ = 2~3. For smaller saturation, \(\mu < 0.5\), the transmission is very small. To reinforce this conclusion, we plot the transmission coefficient as function of saturation parameter μ and asymmetry in onsite potentials ε_{v} in Fig. 6 which confirms that the transmission decreases at higher asymmetry, i.e., ε_{v} > 0.2 and is entirely suppressed at ε_{v} ≥ 0.4.
Different onsite cubic nonlinearities
As discussed above, it is also possible to break the lattice’s symmetry by choosing different cubic nonlinearities for the two sites under consideration i.e., γ_{1} ≠ γ_{2}, the resulting effects on transmission are depicted in Fig. 7.
Comparing the plots in Figs 5 and 7, it is evident that the first (left) plot from both figures are quite similar. Hence, the transmission remains almost the same in both cases for small asymmetry. However, the plot on the right is very different in the two figures. It shows that the trend of diminishing transmission with increasing asymmetry in onsite potentials (ε_{v} ≠ 0) does not hold in the case when asymmetry is between the cubic nonlinear responses (ε_{c} ≠ 0). It is clear from Fig. 7 (right) that even at a sufficiently larger asymmetry (ε_{c}), the system still allows significant transmission roughly for the saturation values of μ = 2.5~3.8.
The corresponding plot of transmission coefficient as function of μ and ε_{c} is plotted in Fig. 8, which confirms our analysis that a significant transmission indeed survives for large enough ε_{c} (Notice the scale on vertical axes in Figs 6 and 8).
Different onsite quintic nonlinearities
Now we consider the effect of saturation on transmission when the lattice symmetry is broken by different onsite quintic nonlinearity i.e., ν_{1} ≠ ν_{2}. Figure 9 shows the transmission in the μ versus T^{2} plane for varying asymmetry ε_{q} in the quintic response. The corresponding plot for μ versus ε_{q} is given in Fig. 10.
Figure 10 confirms that when the lattice symmetry is broken by choosing different quintic responses (ε_{q} ≠ 0), transmission of the input signal remains pronounced for even higher asymmetry compared with the cases discussed before.
Threefold symmetry breaking
From Fig. 6 we learn that for those parameter regimes, bulk of the transmission occurs for asymmetry ε_{v} ≤ 0.2, and for saturation μ ≈ 2.5~3.5. The cases of symmetry breaking by means of nonlinearity Figs 8 and 10 suggest that it is possible to achieve higher transmission regimes for increased asymmetry levels which could be interesting to explore in the context of a diodelike transmission because, as discussed earlier in this paper and other studies^{17,18,33,34}, higher asymmetry renders a better diodelike action but with reduced overall transmission.
Below we show the density plot of transmission coefficient as a function of T^{2} and ε_{v} (left) and a corresponding plot of transmission coefficient as a function of ε_{v} and μ (right), for the case of a threefold broken symmetry of the onedimensional sCQDNLS lattice.
It is clear from Fig. 11 that the threefold broken symmetry can help lift the transmission regimes up towards higher asymmetry. However, bulk of the transmission still occurs for saturation values around μ = 2.5~3.5.
In summary, we can conclude from the results presented in Figs 6, 8, 10 and 11 that the transmission is more susceptible to asymmetry when the broken translational symmetry (or asymmetry leading to nonreciprocal transmission) corresponds to different onsite potentials (i.e., ε_{v} in our notation), as compared to either of the other two types of asymmetries (i.e., ε_{c} or ε_{q} in our notation).
Effects of Saturation on Rectification
To extend our analysis on the diodelike transmission through the nonlinear dimer, we will present some results for the socalled rectifying factor in this section, which will allow us to identify regions with maximal asymmetry in the transmission. The rectifying factor is defined as^{17}
where t_{L} and t_{R} are the transmission coefficients of the waves coming from the left and right of the nonlinear dimer, respectively. The rectifying factor \( {\mathcal R} \) has a maximum value of ±1, which corresponds to perfect diodelike action. In what follows we will present the rectifying action for the cases when (i) asymmetry in the lattice is due to different onsite potentials, i.e., V_{1} ≠ V_{2} (ii) asymmetry due to different onsite cubic nonlinear response, i.e., γ_{1} ≠ γ_{2}, (iii) asymmetry due to different onsite quintic nonlinear response, i.e., ν_{1} ≠ ν_{2}. We will highlight the effect of saturation on rectification for these three cases and also for the threefoldasymmetry case.
Asymmetry due to different onsite potentials
Rectifying action with different onsite potentials under the effect of varying saturation is plotted in Fig. 12. Note that the asymmetry between onsite potentials remains fixed at ε_{v} = 0.2, cubic response at γ_{1} = γ_{2} = 1 and quintic response at ν_{1} = ν_{2} = 0.5. The transmission becomes more symmetric as the saturation coefficient increases, specially for μ > 0.4.
For the cases of μ ≤ 0.4, a diodelike action is apparent, with transmission of both rightpropagating (brighter) and leftpropagating (darker) branches. This action seems to be more pronounced for input waves with larger wavenumbers for saturation up to μ = 0.2. For saturation μ > 0.2 it mostly favors the asymmetric transmission of input waves with small wavenumbers, with the exception of a couple of faint branches exhibiting some diodelike action at large wavenumbers k~π at μ = 0.4. Let us denote this kind of action as “positive diodeaction”, for reasons which will become clear in the upcoming sections.
Furthermore, it is also apparent that the diodeaction is more pronounced for larger wavenumbers, particularly for smaller saturation values as evidenced in the top panel of Fig. 12. Note, however, that even though a diodelike effect increases with increasing asymmetry, the overall transmission reduces significantly, as noted before in^{17,18,34}.
Asymmetry due to different onsite cubic nonlinear responses
Rectifying action with different onsite cubic nonlinearity and under the effect of varying saturation is plotted in Fig. 13. The nonlinearity is taken to be different on the two dimer sites to induce the required asymmetry in the system. The asymmetry in cubic response is fixed at ε_{c} = 0.2. Other sitedependent parameters as ν_{1} = 0.5 = ν_{2} and V_{1} = −2.5 = V_{2}. The wave fills the nonlinear responses as γ_{1, 2} = γ(1 ± ε_{c}) at the first and second dimer site respectively. Comparing results from Figs 12 and 13, it becomes clear that the diodeaction is reversed for this case.
Asymmetry due to different onsite quintic nonlinear responses
It is important to mention that for the case of asymmetric onsite quintic nonlinearity with ν = 0.5, γ = 1, ε_{q} = 0.2, ν_{1, 2} = ν_{0}(1 ± ε_{q}), we get the same rectifying action as that in the case of asymmetric cubic nonlinearity shown in Fig. 13. Therefore, we can conclude that, for a specific value of asymmetry between cubic response and quintic response, the rectifying action is similar. So it is possible to achieve a similar diodelike action for either of the cases discussed above with same sitedependent parameter strengths. This trend can carry on for various strengths of saturation up to μ~1. Therefore, due to the above reasoning and the fact that it is possible to maintain pronounced transmission at higher asymmetry, we present the rectifying plots for this case with a higher value of asymmetry between onsite quintic nonlinearity ε_{q} = 0.8, as shown in Fig. 14.
Note that Fig. 14 is produced for a higher asymmetry value i.e., ε_{q} = 0.8 and the corresponding diodelike action is visible, which remains pronounced for values of saturation up to \(\mu \, \sim \,0.4\). In the limit of higher saturating, the pattern seems to align with the asymmetric cubic case discussed in Fig. 13.
It is also important to note that the diodelike action is reversed in comparison to the case of asymmetric onsite potentials, (see Fig. 12). The leftpropagating (dark region) waves with large wavenumbers are transmitted while rightpropagating (clear region) waves with smaller wavenumbers get through. This pattern carries on for all saturation strengths. Let’s denote this kind of action as negative diodeaction, which will be a handy denotation in the upcoming section.
Threefold asymmetry
The threefold asymmetry case provides us with a variety of control parameters to be able to manipulate for the desired kind of diodeaction. The asymmetry parameters related to nonlinearity (cubic and quintic) i.e., ε_{q}, ε_{c} compete with the asymmetry parameter ε_{v} (asymmetry in onsite potential), for the kind of diodelike action, discussed in previous subsections. Both ε_{q} and ε_{c} favor a negative diodeaction, while strengthening ε_{v} helps produce a positive diodeaction. Moreover, we also report that the positive action is more susceptible to ε_{v} as compared to ε_{q} and ε_{c}, i.e., comparatively larger values of ε_{q} and ε_{c} are required to suppress the positive action produced by a smaller ε_{v} value. This conclusion also confirms our results from sectionIV.
Plots in Fig. 15 are produced for a fixed value of saturation μ = 0.2. The branches exhibiting the positive and negative diodeaction are apparent which clearly demonstrate the reverse action.
Propagation of a Gaussian WavePacket
It is instructive to consider the implications of the above considerations on the transmission of a Gaussian wavepacket. In this section, we consider the timedependent dynamics of a Gaussian wavepacket propagating in the sCQDNLS lattice. A Gaussian wavepacket is taken as the initial condition^{17,18}
where B is the amplitude and σ the width of the initial wavepacket chosen to be σ = 56 in the numerical results reported in this section. With the sCQDNLS dimer situated at the center, the scattering of an initial input signal constituting of a Gaussian wavepacket is given in Fig. 16 for asymmetric onsite potential. The case for both left and right incidences are shown.
It is evident that, due to the broken parity symmetry in this system, right incidence gets a higher transmission (see Fig. 16(a)) as compared to the left incidence in Fig. 16(b). The corresponding transmission coefficients are t_{k > 0} = 0.576647 and \({t}_{k < 0}=0.157794\). Note that the wavepacket scattering plot in Fig. 16 is produced for asymmetric onsite potentials with the asymmetry ε_{v} = 0.05, saturation μ = 0.1, cubic and quintic nonlinearities at γ = 1 and ν = 0.5, respectively.
One can conclude from Fig. 17 that the wavepacket seems to have a much improved transmission when we saturate both the nonlinearities to a higher level μ = 0.5, although with a degraded rectification action. The incident wavepacket also seems to have maintained its shape after transmission through the two nonlinear layers i.e., the dimer. The corresponding transmission coefficients are t_{k > 0} = 0.861496 and \({t}_{k < 0}=0.762887\).
Threefold symmetry breaking
Here, we present the results of the wavepacket dynamics when the underlying lattice system exhibits a threefold symmetry breaking. i.e., all three sitedependent parameters V, γ and ν are different for the two dimer sites, and all other parameters are the same as in the previous cases of Figs 16 and 17.
The wavepacket transmission coefficients for the case discussed in Fig. 18 are t_{k > 0} = 0.133193 and \({t}_{k < 0}=0.669277\) which, together with Fig. 17, suggests that the left propagating wavepackets have a significantly higher transmission rate as compared to the right propagating ones. Hence the trend is reversed under the threefold symmetry breaking which again confirms our analysis on the reverse diodeaction discussed in detail in the previous section. It occurs for the case with higher saturation with roughly the same difference between the left and right transmission coefficients as in case of Fig. 17.
Summary and Conclusion
In summary, we studied the transmission properties of an infinitely long one dimensional lattice carrying a dimer in the center modeled by a saturable cubicquintic discrete nonlinear Schrödinger equation. The saturated cubicquintic DNLS dimer was tested for asymmetric transmission of the input signal. With three possible ways to break the parity symmetry, we showed that transmission is more susceptible when broken symmetry corresponds to different onsite potentials. We also showed that if the lattice symmetry is broken by means of different cubic/quintic nonlinear parameters, the system supports better transmission at higher asymmetry. The rectifying action was computed to characterize the diodelike transmission and we reported that the rectifying action is reversed with regard to the transmission of right and left moving signal for the two types of symmetry breaking mechanisms (onsite asymmetric potential and/or onsite asymmetric nonlinearity). Further, we unveiled that, under the threefold broken symmetry, the diodelike action can be tuned to be positive or negative. Finally, the dynamics of a Gaussian wavepacket was considered numerically for the cases of asymmetry due to onsite potentials and the threefold asymmetry. The wavepacket scattering analysis confirms that the diodeaction is reversed under the threefold asymmetry case. The above results demonstrate that the simultaneous control of the asymmetries on the linear and nonlinear parameters of a dimer defect can allow for an efficient diodelike action is specific spectral regions featuring both high transmission and large rectifying factor. We believe that the phenomenology predicted in our work can be experimentally probed, considering that several aspects of nonHermitian and highorder nonlinear contributions have been unveiled in recent experiments regarding optical pulse propagation on laserinduced atomic gratings^{15,16,19,20}. Efforts along this direction would bring valuable new insights to this exciting subject area.
References
Chang, C. W., Okawa, D., Majumdar, A. & Zettl, A. SolidState Thermal Rectifier. Science 314, 1121–1124 (2006).
Sun, T., Wang, J. X. & Kang, W. Ubiquitous thermal rectification induced by nondiffusive weak scattering at low temperature in onedimensional materials: Revealed with a nonreflective heat reservoir. Europhys. Lett. 105, 16004 (2014).
Wang, Y. et al. Phonon Lateral Confinement Enables Thermal Rectification in Asymmetric SingleMaterial Nanostructures. Nano Lett. 14, 592–596 (2014).
Li, X. F. et al. Tunable Unidirectional Sound Propagation through a SonicCrystalBased Acoustic Diode. Phys. Rev. Lett. 106, 084301 (2011).
Boechler, N., Theocharis, G. & Daraio, C. Bifurcationbased acoustic switching and rectification. Nat. Mater. 10, 665–668 (2011).
Yuan, B., Liang, B., Tao, J. C., Zou, X. Y. & Cheng, J. C. Broadband directional acoustic waveguide with high efficiency. Appl. Phys. Lett. 101, 043503 (2012).
Gallo, K., Assanto, G., Parameswaran, K. R. & Fejer, M. M. Alloptical diode in a periodically poled lithium niobate waveguide. Appl. Phys. Lett. 79, 314 (2001).
Fan, L. et al. An AllSilicon Passive Optical Diode. Science 335, 447–450 (2012).
Roy, D. Fewphoton optical diode. Phys. Rev. B 81, 155117 (2010).
Lira, H., Yu, Z. F., Fan, S. H. & Lipson, M. Electrically Driven Nonreciprocity Induced by Interband Photonic Transition on a Silicon Chip. Phys. Rev. Lett. 109, 033901 (2012).
Rayleigh, J. The Theory of Sound. Dover Publications, New York (1945).
Figotin, A. & Vitebsky, I. Nonreciprocal magnetic photonic crystals. Phys. Rev. E 63, 066609 (2001).
Khanikaev, A. B. & Steel, M. J. Lowsymmetry magnetic photonic crystals for nonreciprocal and unidirectional devices. Opt. Express 17, 5265–5272 (2009).
Fleury, R. et al. Sound Isolation and Giant Linear Nonreciprocity in a Compact Acoustic Circulator. Science 343, 516–519 (2014).
Zhang, Z. et al. Observation of ParityTime Symmetry in Optically Induced Atomic Lattices. Phys. Rev. Lett. 117, 123601 (2016).
Zhang, Z. et al. ParityTimeSymmetric Optical Lattice with Alternating Gain and Loss Atomic Configurations. Laser Photonics Rev. 12, 1800155 (2018).
Lepri, S. & Casati, G. Asymmetric Wave Propagation in Nonlinear Systems. Phys. Rev Lett. 106, 164101 (2011).
Wasay, M. A. Asymmetric wave transmission through one dimensional lattices with cubicquintic nonlinearity. Sci. Rep. 8, 5987 (2018).
Zhang, Y. et al. FourWave Mixing Dipole Soliton in LaserInduced Atomic Gratings. Phys. Rev. Lett. 106, 093904 (2011).
Wu, Z. et al. Cubicquintic condensate solitons in fourwave mixing. Phys. Rev. A 88, 063828 (2013).
Agrawal, G. P. Nonlinear Fiber Optics. Academic Press, San Diego (1995).
Gatz, S. & Herrmann, J. Soliton propagation in materials with saturable nonlinearity. J. Opt. Soc. Am. B 8, 2296–2302 (1991).
Lyra, M. L. & GouveiaNeto, A. S. Saturation effects on modulational instability in nonKerrlike monomode optical fibers. Opt. Commun. 108, 117–120 (1994).
Zhong, X. & Xiang, A. Crossphase modulation induced modulation instability in singlemode optical fibers with saturable nonlinearity. Opt. Fiber Technol. 13, 271–279 (2007).
Nithyanandan, K., Raja, R. V. J., Porsezian, K. & Uthayakumar, T. A colloquium on the influence of versatile class of saturable nonlinear responses in the instability induced supercontinuum generation. Opt. Fiber Technol. 19, 348–358 (2013).
da Silva, G. L., Gleria, I., Lyra, M. L. & Sombra, A. S. B. Modulational instability in lossless fibers with saturable delayed nonlinear response. J. Opt. Soc. Am. B 26, 183–188 (2009).
Lyra, M. L. & GouveiaNeto, A. S. Evolution of Coherent States in a Dispersionless Fibre with Saturable Nonlinearity and the Generation of Macroscopic Quantumsuperposition States. J. Mod. Opt. 41, 1361 (1994).
Hu, S. & Hu, W. Defect solitons in saturable nonlinearity media with paritytime symmetric optical lattices. Physica B 429, 28–32 (2013).
Cao, P., Zhu, X., He, Y. J. & Li, H. G. Gap solitons supported by paritytimesymmetric optical lattices with defocusing saturable nonlinearity. Opt. Commun. 316, 190–197 (2014).
GuzmánSilva, D. et al. Multistable regime and intermediate solutions in a nonlinear saturable coupler. Phys. Rev. A 87, 043837 (2013).
Shi, W. et al. Intrinsic localized modes in a nonlinear electrical lattice with saturable nonlinearity. Europhys. Lett. 103, 30006 (2013).
Samuelsen, M. R., Khare, A., Saxena, A. & Rasmussen, K. Ø. Statistical mechanics of a discrete Schrödinger equation with saturable nonlinearity. Phys. Rev. E 87, 044901 (2013).
Assunção, T. F., Nascimento, E. M. & Lyra, M. L. Nonreciprocal transmission through a saturable nonlinear asymmetric dimer. Phys. Rev. E 90, 022901 (2014).
Wasay, M. A. Nonreciprocal wave transmission through an extended discrete nonlinear Schrödinger dimer. Phys. Rev. E 96, 052218 (2017).
Tsironis, G. & Hennig, D. Wave transmission in nonlinear lattices. Phys. Rep. 307, 333–342 (1999).
Acknowledgements
This work was supported by the ICT R & D program of MSIT/IITP (1711073835: Reliable cryptosystem standards and core technology development for secure quantum key distribution network) and GRI grant funded by GIST in 2018. MLL acknowledges the financial support from CNPq, CAPES, and FINEP (Federal Brazilian Agencies), and FAPEAL(Alagoas State Agency).
Author information
Authors and Affiliations
Contributions
M.A.W. conceived the presented idea. M.A.W. developed the theory, performed the analytical calculations and then the computations to produce the graphical results. M.L.L. verified the analytical results and analysed the graphical results. B.S.H. reviewed the manuscript critically for important intellectual content, encouraged M.A.W. and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wasay, M.A., Lyra, M.L. & Ham, B.S. Enhanced nonreciprocal transmission through a saturable cubicquintic nonlinear dimer defect. Sci Rep 9, 1871 (2019). https://doi.org/10.1038/s41598019388725
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598019388725
This article is cited by
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.