Fast control of the reflection of a ferroelectric by an extremely short pulse
aa r X i v : . [ phy s i c s . op ti c s ] O c t Fast control of the reflection of a ferroelectric by an extremely short pulse
J.-G. Caputo , A.I. Maimistov , and E.V. Kazantseva , : Laboratoire de Math´ematiques, INSA de Rouen,Avenue de l’Universite,Saint-Etienne du Rouvray, 76801 France : Department of Solid State Physics and Nanostructures,Moscow Engineering Physics Institute,Kashirskoe sh. 31, Moscow, 115409 Russia : Department of General Physics,Moscow Institute for Physics and Technology,Institutskii lane 9, Dolgoprudny,Moscow region, 141700 Russia : Department of Condensed Matter Physics,Moscow Institute of Radiotechnics,Electronics and Automation,Vernadskogo pr. 78, Moscow, 119454 RussiaE-mails: [email protected],[email protected], [email protected] (Dated: May 21, 2018)We propose a new type of optical switch based on a ferroelectric. It is based on the gap whichexists for waves propagating from a dielectric to a ferroelectric material. This gap depends on thepolarization of the ferroelectric. We show that it can be shifted by a control electromagnetic pulseso that the material becomes transparent. This device would shift much faster than the relaxationtime of the ferroelectric (1 nano s). Estimates are given for a real material. PACS numbers: Ferroelectric materials, 77.84.-s ,Switching in ferroelectrics, 77.80.Fm Optical switches,42.79.Ta
I. INTRODUCTION
The rapid control of light is a problem that has beenstudied for a long time, in particular because of the ap-plications. Ferroelectric materials are good candidatesfor this control because they can be activated using anelectric field. The principle is the following. The reflec-tion and transmission properties of a ferroelectric mate-rial depend on its state of polarization, in particular it’sspontaneous polarization which exists in the absence ofelectric field. The controlling electric field will shift thispolarization so that then we can control the reflectionand transmission of any electromagnetic wave.A first idea is to use a constant electric field such as in[1]. However then we need to take into account the relax-ation of the ferroelectric which is about one nanosecondand this limits considerably the possibilities for appli-cations. Another idea is to use a fast electromagneticpulse so that the relaxation of the ferroelectric can beneglected. Here there are two frequency regions, one isthe high frequency signal we want to control, the other,of lower frequency is the control signal. To fix ideas wechoose the high frequency signal to be about a femtosec-ond in period and the control signal ten times slower.The time scales are such that we can neglect the relax-ation of the polarization. This way we achieve light con-trol with light, using an ultra-short pulse. In this article,we show that the reflection coefficient is equal to one in a frequency region, a ”gap” whose position depends onthe state of the polarization and the control. Shiftingthe control, we move the gap and render the ferroelectricopaque or transparent to femtosecond waves.After deriving the model equations for the field and po-larization, we linearize them around a functioning pointand derive the reflection and transmission coefficients us-ing a scattering formalism. These are then discussed asa function of the control to show how light can be drivenby the femtosecond pulse.
II. BASIC EQUATIONS
The Maxwell equations in a dielectric medium aretaken in the following form ∇ × E = − B ,t , ∇ · E = − ǫ − ∇ · P ,c ∇ × B = E ,t + ǫ − P ,t , ∇ · B = 0 , where the subscripts indicate the time derivative. Fromthese equation the wave equation results in c ∇ E − E ,tt = ǫ − [ P ,tt − ∇ ( ∇ · P )] . The geometry is shown in Fig. 1, it is a dielectriclayer for z < z ≥
0. Weassume the electric field to be polarized along x . Thespontaneous polarization of the ferroelectric is supposed y ωω δ E~ x Dielectric δ E E Ferroelectric z PE ~ c E FIG. 1: Top panel: schematic drawing of the dielectric-ferroelectric interface. The electric field E is incident nor-mally to the interface. It is polarized along x and the polar-ization is also along x . We show the control pulse E andthe controlled small amplitude wave δE of frequency ω c , thecarrier frequency (see text for details). The bottom panelindicates the Fourier spectra of these waves. to be also along x and to depend only on z . Because ofthis the wave equation above reduces to c E ,zz − E ,tt = P ,tt ǫ . (1)In particular the term ∇ ( ∇ · P ) is zero because the po-larization does not vary along z , the direction of propa-gation of the pulse. The linear dielectric material ( z < P follows P tt + ω P = ǫ ω p E, (2)where ω is the polarization frequency and ω p the plasmafrequency. The polarization in the ferroelectric is givenby the Landau-Kalatnikov equation [3] τ P ,tt − AP + BP = ǫ E, (3)where A = α ( T c − T ) > B > τ is the characteristictime of the polarization. The three equations (1,2,3) de-scribe completely the field and polarization of the media.For zero electric field, the spontaneous polarization ofthe ferroelectric is given by − AP + BP = 0 , → P = ± r AB . (4)Here the + (resp. − ) sign is for a polarization in thedirection of + x (resp. − x ). Now assume a field E con-stant during a time interval t p . The polarization will then shift and satisfy − AP + BP = ǫ E . (5)For small E we can estimate P using E as a perturba-tion. We get P = ± r AB + ǫ E A + O ( E ) . (6)Note that we did not consider the polarization close to0 because it is unstable. During the time interval t p wecan send a small electromagnetic wave δE . This will shiftthe polarization by δP . Assuming E = E + δE, P = P + δP where | δE | << E | δP | << P in equations(1,2,3) yields the linear system in the ferroelectric c δE ,zz − δE ,tt = ǫ − δP ,tt , (7) τ δP ,tt − AδP + 3 BP δP = ǫ δE. (8)In the dielectric layer P = 0 so that the second equationshould be replaced by δP ,tt + ω δP = ǫ ω p δE. (9)The three linear equations (7,9) represent the small os-cillations ( δE, δP ) around the functioning point ( E , P )which exists during the time t p . III. SOLUTIONS OF THE WAVE EQUATIONSIN FREQUENCY DOMAIN
The system of linear equations above can now be solvedcompletely using Fourier transforms in z and matching δE and its derivative at the interface z = 0. Taking theFourier transform in z we get for z < δE ,zz + k ˜ δE = − k ε − ˜ δP , (10)˜ δP = ε ˜ E ω p ω − ω . (11)Plugging the second equation into the first one, we obtain˜ δE ,zz + k ω p ω − ω ! ˜ δE = 0 . (12)In the ferroelectric medium for z >
0, starting from equa-tions (7) and following a similar procedure as for z < δE ,zz + k (cid:18) − τ ω − A + 3 BP (cid:19) ˜ δE = 0 . (13)Substituting the expression (6) of the spontaneous polar-ization P we finally get˜ δE ,zz + k A ± ǫ E p B/A − τ ω ! ˜ δE = 0 . (14)Thus we have a piecewise wave equation for z < z > ε diel ( ω ) = 1 + ω p ω − ω in the dielectric layer and ε ferr ( ω ) = 1 + 12 A ± ǫ E p B/A − τ ω in the ferroelectric layer. The solution in the two differentregions is E ( z, t ) = e iωt − ik z z < , E ( z, t ) = e iωt − ik z z > k and k are k = k ω p ω − ω ! / , (15) k = k A ± ǫ E p B/A − τ ω ! / . (16)These formulas show us that the external electromag-netic pulse can control the dielectric properties of theferroelectric material. IV. SCATTERING OF LINEAR WAVES OFFTHE INTERFACE z = 0 . The reflection and transmission coefficients of har-monic waves can be computed as a function of the controlfield E . Note that the two orientations of polarizationwill give the same reflection coefficient for E = 0. Onlyadding the control E is one able to distinguish the twostates of polarization. We set up the scattering formal-ism assuming an incident wave from the left, a reflectedwave and a transmitted wave,˜ δE ( z, ω ) = E in ( ω ) e ik z + E r ( ω ) e − ik z , in the dielectric and˜ E ( z, ω ) = E t ( ω ) e ik z , in the ferroelectric medium. In the absence of the sur-face charges and currents, the jump conditions on theinterface read [4]˜ E (0 − , ω ) = ˜ E (0+ , ω ) , ˜ E ,z (0 − , ω ) = ˜ E ,z (0+ , ω ) . Using these conditions and the solution of the wave equa-tion one can find the Fresnel relations connecting the | R | τ ω FIG. 2: Square of the modulus of the reflection coefficient | R | as a function of the reduced frequency τ ω for threedifferent values of the normalized control 3 ǫ E /P s = 0 (con-tinuous line, red online), 0 . − . A = 1 , B = 1. | R | ε E /P s FIG. 3: Square of the modulus of the reflection coefficient | R | for a fixed reduced frequency τ ω = 0 . ǫ E /P s . amplitudes of the incident wave E in ( ω ), reflected wave E r ( ω ) and transmitted wave E t ( ω ). E r ( ω ) = k − k k + k E in ( ω ) , (17) E t ( ω ) = 2 k k + k E in ( ω ) . (18)These relations are correct for any low amplitude waves,both solitary waves and for harmonic waves. The reflec-tion and transmission coefficients are then respectively R = E r E in = k − k k + k , T = E t E in = 2 k k + k , (19) V. DISCUSSION
Fig. 2 shows the modulus of the reflection coefficient | R | as a function of the reduced frequency τ ω for threedifferent values of the control E = 0 (continuous line, redonline), E = 0 . E = − . E > E < k (15). To illustrate how the field E can be used to block a wave we have plotted in Fig.3 | R | as a function of E for τ ω = 0 .
8. For E > . E is decreased | R | increases sharply and reaches 1 for E = 0 .
2. Belowthat value the ferroelectric is opaque to this particularfrequency.To show how this scheme can work in reality, we ex-amine parameters for a real material. Consider the studyby Noguchi et al [8] on the Bi T i O SrBi T i O in-tergrowth ceramics. This material was shown to havea large spontaneous polarization. In addition its Curietemperature is high so that it is stable. To estimate A and B from the measurements of [8] we recall that AB = P c , A r A B = ǫ E c , where P c and E c are respectively the coercitive polariza-tion and coercitive field. Using these values from [8] weget A = 210 − , B = 10 − m C − . This value of A defines a resonant frequency ω r such that τ ω r = 2 A. We get τ ω r = 4 . − The value of τ given by estimates of the inertia of molec-ular assemblies is about τ = 10 − [9]. This gives ω r = 4 . Hz.
An important point is that at resonance the two terms2 A and τ ω are almost equal so their difference is verysmall. Then a small shift due to the term 3 ǫ E p B/A will displace the resonance. Let us estimate the field E needed to shift the resonance from ω r to ω r /
2. We have3 ǫ E r BA = τ ω r . This gives E ≈ V m − . This value of the electric field can be achieved using alaser.
VI. CONCLUSION
The reflection of the electromagnetic wave on the in-terface between linear dielectric medium and ferroelec-tric was considered. We assume that the electromagneticwave is a superposition of a high frequency wave and aspike-like electromagnetic signal. The spectrum of thespike is located near the zero frequency and can be con-sidered as low frequency. We showed that the spike-likesignal induced an extra contribution to the total ferro-electric polarization. This causes a fast change of thereflection coefficient in the high frequency domain. Inthe time duration of the spike-like signal the relaxationprocesses can be neglected. Thus one can achieve a rapidcontrol of light with light, using an extremely short, spikepulse. [1] E.D. Mishina, N.E. Sherstyuk, V.I. Stadnichuk, A. S.Sigov, V. M. Mukhorotov, Yu. I. Golovko, A. van Ette-ger and Th. Rasing, Appl. Phys. Lett. , 12 2402-2404(2003)[2] L. Rosenfeld, Theory of Electrons, New York: Dover Pub-lications, 1965, pp. 68[3] L.D. Landau, L.P. Pitaevskii, E.M. Lifshitz, Electro-dynamics of Continuous Media , Elsevier Butterworth-Heinemann, Oxford, (2000).[4] M.Born, E. Wolf, Principles of Optics: ElectromagneticTheory of Propagation, Interference and Diffraction of Light, seventh (expanded) ed., Cambridge Univ. Press,Cambridge, UK, 2003.[5] V.L. Ginzburg, JETP , 739 (1945) (in Russian)[6] A.F. Devonshire Philos. Mag. , 3639-3641, (2000).[9] J.-G. Caputo, A.I. Maimistov, E.D. Mishina, E.V. Kazant-seva, V.M. Mukhortov,Phys. Rev. B82