Bistable transmission of plane waves across two nonlinear delta functionsP. S. Moya,a� Max Ramírez,b� and M. I. Molinac�

Departamento de Física, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile

�Received 29 November 2006; accepted 16 August 2007�

We examine the transmission coefficient of plane waves across two nonlinear delta-function barrierswith positive opacity. It is shown that this simple system is capable of bistable behavior atsufficiently large input intensities. The bistable behavior is centered around the first transmissionresonances and is a simple example of a continuum nonlinear system that can display bistabilitywithout the presence of a feedback loop. © 2007 American Association of Physics Teachers.

�DOI: 10.1119/1.2785191�

I. INTRODUCTION

A common problem in optics and quantum mechanicscourses, which vividly illustrates the phenomenon of inter-ference and resonances, is plane wave transmission throughone or more delta-function-like potentials. A generalizationleads to a system of a periodic array of delta-function scat-terers, the “Dirac comb,” which is useful for introducing stu-dents to electron propagation in crystalline structures. Delta-like potentials are used to represent a localized potentialwhose energy scale is greater than any other and whose spa-tial extension is smaller than all other length scales. Someinteresting applications of the delta-function potential arefound in Ref. 1.

Delta functions and other simple potentials are now invogue due to recent advances in device nanofabrication. Thelatter allows a system of barriers at a mesoscopic or evensmaller scale to be designed, with the ultimate goal of pro-ducing materials with interesting properties. Applications inthis area go beyond electronic conduction because, in thescalar approximation, optical wave propagation in a mediumresembles the dynamics of an electron in a crystal lattice.

A common phenomenon observed in the transmission ofextended waves across a region with nonlinear response isbistability2: The existence of two stable transmission outputsfor the same input intensity. Sometimes, more than twotransmission states are possible. In this case we speak ofmultistability. Depending on the details of the system, a feed-back mechanism, typically found in continuum opticalsystems,3 is sometimes needed in addition to nonlinearity. Insome discrete systems, bistability4 and multistability5 havebeen observed only with nonlinearity. As the parameters ofthe system are varied, different transmission states are ac-cessed.

Figure 1 shows a typical case, the transmission of a planewave of a given wave vector �or wavelength� through a seg-ment of an optically active nonlinear medium. For a wavevector at or close to a certain value, a window of input in-tensities is observed where there seem to be three possibletransmission states for a given input intensity. It can beshown on general grounds2 that the portion with negativeslope �dashed� is unstable. Thus, starting from a small inten-sity value, the transmission grows with increasing intensity,until a threshold I= I1 is reached, where the transmissionjumps to a higher value. Afterward, further growth in theintensity increases the transmission monotonically. If we re-verse direction and start decreasing the intensity, we observea steady decrease in the transmission until a second threshold

I= I2 is reached, where the transmission jumps to a smaller

1158 Am. J. Phys. 75 �12�, December 2007 http://aapt.org/ajp

value. A further decrease in the intensity reduces the trans-mission monotonically. Thus, there is an intensity interval�I2 , I1� where there are two stable transmission values. Like-wise, we can observe a window of bistable behavior in atransmission versus wave vector plot for input intensitiesabove a certain level. In either case, the transmission cycleforms a hysteresis loop. This switching behavior withmemory constitutes the basis of various possible devices,which are driven by a change of intensity only.6

In this article we introduce a simple continuous nonlinearsystem that can display bistability without the need of a feed-back loop. It consists of an electron or electromagnetic wavepropagating through a medium that contains two nonlineardelta-function potentials. In an optical context, this systemmodels the case where small, nonlinear dielectric �Kerr� re-gions are embedded in a continuous, linear medium. Thesenonlinear regions are assumed to be much smaller than theirtypical mutual distance and smaller than the typical wave-length of the wave. In an electronic context, the nonlinearityis due to the strong coupling between the electron and thevibrational degrees of freedom of the embedded regions.Hereafter we will refer to the electronic context, but a simpletransformation will bring the system to its optical analoge.

II. THE MODEL

We consider a plane wave incident on two delta-function-like nonlinear barriers located at x=0 and x=L, as shown inFig. 2. The stationary state Schrödinger equation reads

d2��x�dx2 + k2��x� = �1��x����x���1��x� + �2��x − L�

����x���2��x� , �1�

where k��2mE /�2 is the electron wave vector, E is theelectron energy, �1 and �2 are the opacities of the deltabarriers, and �1 and �2 are the nonlinearity exponents asso-ciated with barriers one and two, respectively. The problemlooks similar to the usual two delta barriers problem, exceptfor the nonlinear terms ����1 and ����2, which modulate thestrength of the barrier opacities, depending on the value ofthe probability at the barriers. Typically, both �1 and �2 areequal to two,7 but we shall leave them unspecified and studythe dependence of the transmission coefficient on �1, �2, �1,and �2. For simplicity, we choose �1�0 and �2�0 toavoid complications such as modulational instability.8

Because we are interested in plane wave transmission, we

set

1158© 2007 American Association of Physics Teachers

��x� = �R0 exp�ikx� + R exp�− ikx� x � 0

A exp�ikx� + B exp�− ikx� 0 � x � L

T exp�ikx� x � L .

�2�

From the continuity of ��x� and discontinuity of ���x� at x=0 and x=L, we obtain

R0 + R = A + B �3�

ik�A − B� − ik�R − R� = � �R + R��1�R + R� �4�

Fig. 1. Example of transmitted intensity versus input intensity curve of abistable device.

0 1 0 0

a� � 1 + ��2/2k��R0� t sin�2kL� , �10�

defined resonances where t=1.

1159 Am. J. Phys., Vol. 75, No. 12, December 2007

A exp�ikL� + B exp�− ikL� = T exp�ikL� �5�

− ikT exp�ikL� + ik�A exp�ikL� − B exp�− ikL��

= − �2�T��2 exp�ikL� . �6�

From Eqs. �5� and �6�, we express A and B in terms of T:A=T�1+ i��2 /2k� �T��2� and B=−�i�2 /2k� �T��2 exp�2ikL�.Next, we insert Eq. �3� on the right-hand side of Eq. �4�. Onthe left-hand side, we express R0−R as −�R+R0−2R0�=2R0− �A+B�. We are finally left with a transcendental equa-tion for T only. We define the transmission coefficient as t��T�2 / �R0�2 and obtain after some algebra the followingequation for t:

Fig. 2. Two nonlinear delta-function barriers.

t =4

a + a� − b��1

k�R0��1t�1/2�a�2 + b�2��1/22

+ b + b� + a��1

k�R0��1t�1/2�a�2 + b�2��1/22 , �7�

where

a � 1 − ��2/2k��R0��2t�2/2 sin�2kL� , �8�

b � ��2/2k��R0��2t�2/2�1 + cos�2kL�� , �9�

�2 �2/2

b� � ��2/2k��R0��2t�2/2�1 − cos�2kL�� . �11�

Hereafter, we take L=1 as the length scale.

A. Linear regime

In the absence of nonlinearity ��1=0=�2�, Eq. �7� gives t

as a function of �1, �2, k, and L:

t =4

�2 − ��1/k���2/2k��1 − cos�2kL���2 + ���2/k� + ��1/k��1 + ��2/2k�sin�2kL���2 . �12�

In this case t does not depend on the intensity of the incidentwave R0 but depends on the ratios �1 /k and �2 /k. Thisdependence means that as k is increased, the effect of theopacity is reduced, which implies more “transparent” barri-ers, that is, higher transmission. The dashed curve in Fig. 3shows the transmission coefficient t versus wave vector k for�1=�2=2. Note the presence of an infinite number of well-

B. Nonlinear regime

In the presence of nonlinearity ��1 ,�2�0�, the transmis-sion coefficient depends on the input intensity: The opacityof each barrier is modulated by a factor �R0��1t�1/2 �barrierone� and �R0��2t�2/2 �barrier two�. Because the effective opac-ity now depends on the intensity of the input wave, so willthe transmission t. Thus, the equation for t is now nonlinear,

and must be solved by one of several numerical methods.

1159Moya, Ramírez, and Molina

Before we discuss the numerical results, there is an inter-esting observation concerning Eq. �7�: If we set t=1, that is,a resonance, then the resulting equation for k for the posi-tion�s� of the resonance�s� is �are� identical to the equationwe would obtain in the absence of nonlinearity if we replacethe opacity coefficients �1, �2 by effective opacities �1

e

��1 �R0��1 and �2e ��2 �R0��2. The position of the reso-

nances �when they exist� will be shifted with respect to thelinear case: For �R0 � �1 ��1�, the resonance will occur atsmaller �larger� k values. For R0=1, all resonances will occurat the same k value as in the linear case. Even though theposition of the resonances coincides with the ones obtainedfor the linear barriers problem, the system is far from beinglinear, as is evident from the shape of the nonlinear transmis-sion curve in Fig. 3. Far from resonance, t�1 and the effec-tive opacity coefficient becomes smaller, thus decreasing theoverall opacity of the barriers. As a result, the transmissionfor any wavevector increases. This effect is stronger thelarger the nonlinearity exponents.

In what follows we take �1=�2=2. After multiplying Eq.�7� by the denominator of the right-hand side, a polynomialequation for t is obtained. All the roots were numericallyobtained using MATHEMATICA, and the real ones between 0and 1 were selected. The smooth transmission curves werethen constructed by interpolation.

The solid curve in Fig. 3 shows the transmission coeffi-cient t versus the wave vector k for �1

�e�=�2�e�=2. With this

choice of parameters, the positions of the resonances coin-

Fig. 3. Transmission coefficient versus wave vector. Dashed curve: linearregime ��1=�2=0, �1=�2=2�. Solid curve: weakly nonlinear regime ��1

=�2=2, �1�e�=�2

�e�=2�.

Fig. 5. Magnification of the selected regions ar

1160 Am. J. Phys., Vol. 75, No. 12, December 2007

cide with the linear case �dashed line�. As predicted, thetransmission is higher, with broader resonances. In this case,there is no bistable behavior.

In Fig. 4 we show the same plot as in Fig. 3 but for �1�e�

=�2�e�=20. Notice the presence of narrow bistability win-

dows around the first two resonances. They are clearly seenin Fig. 5, which shows the regions near the first threemaxima. The bistable behavior extends only to the first twomaxima. We have verified that for larger effective opacities,more and more of the resonances are affected by bistability.Also, because an increase in opacity also decreases t, exceptat the very resonance position, the end result at high opaci-ties is a set of narrow bistable peaks.

We see from the right-hand side of Eq. �7� that the inputintensity is always multiplied by the transmission coefficient,implying that nonlinearity effects will be most noticeablenear resonance t�1, if ��1 /k� �R0��1 and ��2 /k� �R0��1 arelarge enough. For a given R0, �1, and �2, this dependenceon 1/k favors the first resonances �smaller k values�. Thus,we expect bistability to be present, if it exists, near the reso-nances with small k and to disappear altogether at highenough k.

Where does the bistability comes from? A simple explana-tion closely follows the treatment of Janssen for the anhar-monic oscillator.9 For simplicity we consider the case �1=�2=�. The main effect of nonlinearity is to redefine theeffective opacity of the barriers. Thus it makes sense to re-place � ��2 by �1

�e� when the system is far from any reso-

Fig. 4. Transmission coefficient versus wave vector for a strongly nonlinearregime ��1=�2=2, �1

�e�=�2�e�=20�. We observe bistability around the first

two resonances.

ound the first three resonances in Fig. 4.

1160Moya, Ramírez, and Molina

nance ���2�1�, and by �2�e� near a resonance ���2�1�. Of

course, we must have �1�e���2

�e�. Let us call t* the value ofthe transmission that separates what we call the nonresonantregion from the resonant one. Figure 6�a� shows the trans-mission near and around a given resonance for �1

�e�=6 and�2

�e�=7. Figure 6�b� shows the actual �stable� transmissioncurves around a resonance: Starting far from the resonanceregion, the system follows the transmission curve character-ized by �1

�e�. As the transmission coefficient becomes largerthan t*, the system will tend to follow the curve correspond-ing to �2

�e�. A further increase in wave vector �or frequency�will eventually lead the system back to t*, where the systemwill revert to the original curve. As a result, the transmissionwill suddenly jump down at a certain k=k2. For k�k2 thesystem stays on the �1

�e� curve �until it gets close to anotherresonance�. The path followed is ABCDE. If we now revertthe path and decrease k from k�k2, the system will stay onthe �1

�e� curve until reaching a certain value k=k1. A slightdecrease in k will bring the transmission above t*, and thesystem will jump to curve �2

�e�. As k continues decreasing,the system will follow the path EFCBA. As a result, thesystem acquires a wave vector �or frequency� window wherethere are two stable transmission values. The specific valuewill depend on whether k is increasing or decreasing �hyster-esis�.

This simplified treatment explains the main transmissionfeatures observed in our system of delta-function barriers. Inparticular, the stable transmission states correspond to the

Fig. 6. �a� Transmission curves around a single resonance for a linear sys-tem with effective opacities �1

�e� and �2�e�. �b� Stable transmission paths

�arrows and solid black curves� followed by the system around a singleresonance.

1161 Am. J. Phys., Vol. 75, No. 12, December 2007

highest and lowest portions of the transmission curves insidethe bistability windows shown in Figs. 5�a� and 5�b�.

III. CONCLUSIONS

We have examined the passage of a plane wave across twononlinear �cubic� delta-function barriers of positive opacityand have computed its transmission coefficient. This simplesystem, which lacks any external feedback mechanism, iscapable of displaying bistability in the transmission coeffi-cient for large input intensities. The main effect of nonlinear-ity occurs around the transmission resonances �t�1� whereit redefines the effective opacities, making them proportionalto the input intensities. This proportionality leads to the cre-ation of small bistability windows around the first reso-nances.

ACKNOWLEDGMENTS

This work was supported by a Comisión Nacional deCiencia y Tecnología �CONICyT, Chile� doctoral fellowship�M. Ramirez and P. S. Moya� and was supported in part byFONDECYT Grant 1050193 �M. I. Molina�.

a�Electronic mail: pmoya@zeth.ciencias.uchile.clb�Electronic mail: mramirez@zeth.ciencias.uchile.clc�Electronic mail: mmolina@uchile.cl1See, for instance, D. A. Atkinson, “An exact treatment of the Dirac deltafunction potential in the Schrödinger equation,” Am. J. Phys. 43, 301–304 �1975�; M. Lieber, “Quantum mechanics in momentum space: Anillustration,” Am. J. Phys. 43, 486–491 �1975�; A. K. Jain, S. K. Deb,and C. S. Shastry, “The problem of several delta-shell potentials in theLipmann-Schwinger formulation,” Am. J. Phys. 46, 147–151 �1978�; D.Lessie and J. Spadaro, “One dimensional multiple scattering in quantummechanics,” Am. J. Phys. 54, 909–913 �1986�; D. W. L. Sprung, HuaWu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J.Phys. 61, 1118–1124 �1993�; I. Mitra, A. DasGupta, and B. Dutta-Roy,“Regularization and renormalization in scattering from Dirac delta poten-tials,” Am. J. Phys. 66, 1101–1109 �1998�; M. I. Molina and C. A.Bustamante, “The attractive nonlinear delta-function potential,” Am. J.Phys. 70, 67–70 �2002�.

2B. E. A. Saleh and M. C. Teich, Fundamental of Photonics �John Wiley &Sons, New York, 1991�.

3L. A. Lugiato, “Theory of optical bistability,” in Progress in Optics,edited by E. Wolf �North-Holland, Amsterdam, 1984�, Vol. 21.

4M. I. Molina and G. P. Tsironis, “Nonlinear impurities in a linear chain,”Phys. Rev. B 47, 15330–15333 �1993�.

5F. Delyon, I.-E. Levy, and B. Souillard, “Nonperturbative bistability inperiodic nonlinear media,” Phys. Rev. Lett. 57, 2010–2013 �1986�.

6H. M. Gibbs, Optical Bistability: Controlling Light with Light �AcademicPress, Orlando, 1985�; A. C. Walker, “Application of bistable opticallogic gate arrays to all-optical digital parallel processing,” Appl. Opt. 25,1578–1585 �1986�; Claus Klingshirn, Semiconductor Optics �Springer,Berlin, 2005�, pp. 645–673.

7The value of � is related to the nature of the vibrational degree of free-dom that interacts with the electron inside the embedded regions. If wemodel those vibrations as Einstein oscillators, then for the case of har-monic oscillators that adapt instantly to the presence of the electron, itcan be shown that �=2� M. I. Molina, “New DNLS equations for anhar-monic vibrational impurities,” Mod. Phys. Lett. B 17, 111–119 �2003��.Other � values correspond to “soft” or “hard” anharmonic oscillators.

8B. A. Malomed and M. Ya. Azbel, “Modulational instability of a wavescattered by a nonlinear center,” Phys. Rev. B 47, 10402–10406 �1993�.

9H. J. Janssen, R. Serneels, L. Beerden, and E. L. M. Flerackers, “Experi-mental demonstration of the resonance effect of an anharmonic oscilla-tor,” Am. J. Phys. 51, 655–658 �1983�.

1161Moya, Ramírez, and Molina