A possibility of light pulse transformation by transient reflecting medium is investigated theoretically. After solving 1D problem of such a reflection one estimates such a transformation in a plane optical waveguide with time-dependent conductivity of one of the reflected media. Three types of the conductivity time-dependences are considered: harmonic, Bessel-like and splash-like ones. Obtained results show a possibility of pulse splitting under an influence of time-harmonic conductivity and pulse collapse vy the other considered nonstationarities

Marciniak, Marian

Vorgul, Irena Yu

Biblioteka Cyfrowa Instytutu Łączności / National Institute of Telecomunications: Digital Library

Paper Optical pulse splittingunder temporal variationsof reflecting mediumIrena Yu. Vorgul and Marian MarciniakAbstract — A possibility of light pulse transformation bytransient reflecting medium is investigated theoretically. Af-ter solving 1D problem of such a reflection one estimatessuch a transformation in a plane optical waveguide with time-dependent conductivity of one of the reflected media. Threetypes of the conductivity time-dependences are considered:harmonic, Bessel-like and splash-like ones. Obtained resultsshow a possibility of pulse splitting under an influence of time-harmonic conductivity and pulse collapse vy the other consid-ered nonstationarities.Keywords — optical pulse splitting, time dependent conductivity,reflecting boundary.Motivation and common formulationof the problemIn optics a transformation of electromagnetic field of lighthas two reasons to be one of the basic problem. The firstone is that light sources could not radiate all types of re-quired fields, laser sources radiate fields of a finite num-ber of frequencies. The second one is, as in radio rangeelectromagnetics, a necessity to modulate field for infor-mation transmission. Optical waverange field modulationis usually made based on electro-optical Pokker’s effect,magneto-optical Faradey’s or Kerr’s effects and on acousto-optical effect. In all these methods the field is modulatedduring the transmission through the correspondent mod-ulating media, which is connected with additional lossesand additional element of transmission tract. For informa-tion transmission it is not convenient to use frequency orphase light modulation, because in the existing light sourcesfor optical communication there is not enough coherence.Besides, such a modulation techniques are well-developedonly for harmonic initial fields.The present work has a goal to find a possibility to trans-form light pulses by transient reflecting medium and toestimate such a transformation in a plane optical waveg-uide where one of the reflecting media has time-dependentconductivity.One has chosen the conductivity as a transient parameterbecause it can be more easily changed than permittivitywhose change besides can leads to destructuring of the fullreflection in optical waveguide.To estimate pulse transformation in a plane optical waveg-uide with time-dependent conductivity of one of the reflect-ing media (Fig. 1) one firstly solves rigorously one-spaceFig. 1. Plane waveguide with nonstationary reflecting mediumdimensional problem of pulses reflection from a transientconductive half-space for an initial pulse propagating nor-mally to the reflection boarder (Fig. 2).Fig. 2. Formulation of 1D problemFor the conductivity time-dependences considered in 1Dcase two-space-dimensional problem is investigated. It issolved approximately for the initial pulse falling angle pro-viding a full reflection, in assumption of constant polarisa-tion of the initial and scattered fields (Fig. 9). Then oneestimates multiply re-reflection of the transformed pulsefrom the transient and stationary media forming a planelight waveguide.One-space-dimensional problemProblem formulationThe formulation of one-space-dimensional problem is indetermination of the reflected field by the reflected fieldand the initial one and by time-dependence of the relectedmedium conductivity.It is considered that the conductivity change starts after themoment t = 0 (Fig. 2). Before and after this moment the66JOURNAL OF TELECOMMUNICATIONSAND INFORMATION TECHNOLOGY1-2/2000Optical pulse splitting under temporal variations of reflecting mediumfields are called the initial and scattered fields, correspond-ingly. The fields are assumed to have only those compo-nents, which are normal to the x-axis and independent onthe y- and z-coordinates.Mathematically, the problem is formulated in terms of theVolterra integral equation for the electrical component ofelectromagnetic field [6], which has the following form forthe internal field (inside the transient region) for t > 0,x > 0:Ein(t,x) = A(t,x)+−2piεvΘ(vt− x)∫ tt−x/vdt ′σ(t ′)Ein(t ′,x− v(t− t ′))+−2piεvΘ(x− vt)∫ t0dt ′σ(t ′)Ein(t ′,x− v(t− t ′))+−2piεv∫ t0dt ′σ(t ′)Ein(t ′,x+ v(t− t ′))(1)where A(t,x) is known because it is determined by the ini-tial field and prehistory of its interaction with the medium:A(t,x) = E0(t,x)+−Θ(x− vt) 2piεv∫ 0t−x/vdt ′σ(t ′)E1(t ′,x− v(t− t ′))+−2piεv∫ 0−∞dt ′σ(t ′)E1(t ′,x+ v(t− t ′))and for the external field for t > 0, x < 0:Eex(t,x) = B(t,x)+−2piεvΘ(vt+ x)∫ t+x/v0dt ′σ(t ′)Ein(t ′,v(t− t ′)+ x) (2)whereB(t,x) = E0(t,x)+−2piεvΘ(vt+ x)∫ 0−∞dt ′σ(t ′)E1(t ′,v(t− t ′)+ x)+−2piεvΘ(x+ vt)∫ t+x/v−∞dt ′σ(t ′)E1(t ′,v(t− t ′)+ x)where ε is the dielectric permittivity, v= c/√ε is the lightvelocity in considered medium and the conductivity time-dependence (or time-spatial dependence) Eex(t,x) is a func-tion to be found.Analytical solution of the problemTo obtain an equation for the external field, we firstly solvethe equations (1) and (2) jointly to express the initial fieldafter the external one.As one can see from (2), the external field is determinedby the sum of the known function B and a function of onevariable t − x/v. This fact, which is due to the assumedhomogeneity and losslessness of the external half-space,makes impossible to express the internal field through theexternal one directly from this expression. However, it al-lows to obtain a non-integral formula for the external fielddetermined by the internal field on the boundary x = 0.For this purpose we introduce a new function F of onevariable as:F(t) =− 2piεv∫ t0d t ′σ(t ′,v(t− t ′))Ein(t ′,v(t− t ′)), (3)which determines the external field in the external region−vt < x < 0 by the expressionEex(t,x)−B(t,x) = F(t+ x/v).From (1) we can obtain that the internal field on the bound-ary is determined by the same function F :Ein(t,0)−A(t,0) = F(t).Considering this expression for shifted time moment t +y/v, −vt < y < 0, we haveEin(t+ y/v,0)−A(t+ y/v,0) = F(t+ y/v).After comparing this formula with (2) we obtain the fol-lowing expression for the external field:Eex(t,x) = B(t,x)+Ein(t+ x/v,0)−A(t+ x/v,0),−vt < x < 0. (4)Introduce another new functionΦ(t,x)≡ Ein(t− x/v,x)−A(t− x/v,x) (5)for 0 < x < vt, satisfying the following equation obtainedfrom (1):Φ′x(t,x)+2vΦ′t(t,x) =−4piεv∂∂ t∫ t−x/v0d t ′σ(t ′,v(t−t ′))Ein(t ′,v(t−t ′)) (6)with the boundary and initial conditions:Φ(t,0) = F(t), and Φ(x/v,x) = Ein(0,x)−A(0,x) = 0 .Knowing the external field at any point, the field in thewhole external region can be determined, including the re-gion close to the boundary:Eex(t,x)−B(t,x) = Eex(t− x1/v,x+ x1)+−B(t− x1/v,x+ x1) = F(t+ x/v)(7)So it would be enough to obtain the solution for externalfield close to the boundary at the points where |x|  vt.Under this approximation, we can solve Eq. (6), hence ex-pressing the internal field through the external one, becausethe integral at the right-hand part of (6) will be equal toF(t):Φ′x(t,x)+2vΦ′t(t,x)≈ 2F ′(t) . (8)JOURNAL OF TELECOMMUNICATIONSAND INFORMATION TECHNOLOGY 1-2/2000 67Irena Yu. Vorgul and Marian MarciniakAfter substitution of this equation solution into (3) we ob-tain the conductivity time-dependence in the half-space de-termined by the scattered field:σ(t)(Eex(t− x/v,x)−E0(t− x/v,x)+E0(t,0))== σ0A(0,0) ,(9)where σ0 = σ(0) is the known value of initial conductivity,and x means an arbitrary point coordinate (not only |x| vt) inside the external region −vt < x < 0.Pulse splitting with amplification by time-harmonicconductivity of the reflecting half-spaceFor the rectangular pulse E0 = Θ(t−x/v)−Θ(t−x/v−t0) scattering on the homogeneous half-space with time-harmonic conductivity, computer analysis revealed the scat-tered pulse features dependence on the conductivity fre-quency. When it is comparable with reverse incident pulseduration then the pulse of scattered field just changes a littlein its shape under the same duration.When the conductivity frequency is more then four times asmuch as reverse pulse duration, the scattered pulse has deepvalleys (Fig. 2). Their number grows with the frequencyincreasing. Thus, it becomes a consequence of pulses withjoint duration less then that of the initial pulse. Thesepulses amplitude can be more then ten times as much as thatof the initial pulse. The more is the conductivity frequency(Fig. 3(a)), the more is the number of reflected field pulses(Fig. 3(b)).Fig. 3. Time-harmonic conductivity of the reflecting half-spaceleading to (b) pulse splitting with amplification byPulse transformation by reflecting half-space withconductivity changing with time as Bessel functionBessel-type conductivity time-dependence provides the ini-tial rectangular pulse transformation into a pulse with dura-tion 10−3 as much as the initial one, following by reducingoscillating tail (Fig. 4).Such a pulse collapse can be also followed by the reflectedpulse amplification, as Fig. 5 shows.Consider the same pulse scattering on the half-space withtime-splashing conductivity (Fig. 6(a)) with its time depen-dence described by a difference of reducing exponents.For the time less then a pulse duration after it began itsinteraction with the half-space. Figure 6(b) shows the re-flected pulse front which then will save its shape and size,Fig. 4. Reflected pulse amplitude time-dependence for differentamplitudes of Bessel-type conductivity change(initial pulse length=1 cm)Fig. 5. Reflected pulse amplitude time-dependence for Bessel-type conductivity change (initial pulse length = 0.5 cm)moving with the correspondent to the medium light veloc-ity (circle incision in Fig. 7). With time this pulse of asmall amplitude moving away from the boundary leaves afield trace of a high amplitude (Fig.7). After the end of theconductivity splash the trace amplitude decreases formingso a splash-like pulse. The front of this pulse moves with avelocity lower then that of light for the considered medium.This trace evolution with time leads to its transformationFig. 6. Conductivity splash (a) and initial stage of the pulsereflection (b)Fig. 7. Reflected field front and trace68JOURNAL OF TELECOMMUNICATIONSAND INFORMATION TECHNOLOGY1-2/2000Optical pulse splitting under temporal variations of reflecting mediumFig. 8. Short pulse formed from the reflected field traceinto a shot pulse of a very high amplitude which is like toDirac delta function as Fig. 8 shows.Obtained solutions enable one to analyse a large number ofscattering on the transient conductive half-space problems.Results for the special cases can be useful, for example, forcreation short pulses of high amplitude.The obtained results demonstrate different possibilities ofinitial pulse splitting or time-compression by time variationof the reflecting medium.They also show possibilities of the reflected pulse amplifi-cation.For the case of time-harmonic conductivity in the reflectinghalf-space one can obtain an easy read information whenthe modulated signal amplitude and frequency are trans-lated into amplitudes and a number of pulses which are theresult of the initial pulse subdivision.2D ProblemIn 3D nonstationary media an integral equation for electro-magnetic field is [5]:E (t,r) =E 1(t,r)−2piσεv(graddiv− ∂2∂ t2)××∫ t0dt ′∫∞−∞dr ′σ(t ′,r ′)Θ(t− t ′−|r−r ′|/v)|r−r ′| E (t′,r ′),(10)where r ≡ (x,y,z) and|r−r ′|=√(x− x′)2+(y− y′)2+(z− z′)2.For the considered 2D problem we assume σ(t,r) ≡σ(t)Θ(x) and E = Ezk. In this case we have the followingscalar integral equation inside the transient region:x≥ 0Ein(t,x,y) = E1(t,x,y)+−Θ(vt− x)∫ tt−x/vdt ′σ(t ′)∫ x+v(t−t ′)x−v(t−t ′)dx′G+−Θ(x− vt)∫∞0dt ′σ(t ′)∫ x+v(t−t ′)x−v(t−t ′)dx′G+−Θ(vt− x)∫ t−x/v0dt ′σ(t ′)∫ x+v(t−t ′)0dx′G (11)whereG≡ 2piεv∫ y+√v2(t−t ′)−(x−x′)2y−√v2(t−t ′)−(x−x′)2dy′ln v(t− t′)+√v2(t− t ′)2− (x− x′)2− (y− y′)2v(t− t ′)−√v2(t− t ′)2− (x− x′)2− (y− y′)2 Ein(t′,x′,y′)and the following formula which expresses the external fieldafter the internal one:x < 0Ein(t,x,y) = E1(t,x,y)+−Θ(vt+ x)∫ t+x/v0dt ′σ(t ′)∫ v(t−t ′)−x0dx′G .(12)We are interested in the external field for which we alsocan obtain a usual wave equation with constant coefficientsdirectly from Maxwell’s equations or from (12):(∆− 1v20∂ 2∂ t2)Eex(t,x,y) = 0 . (13)However, boundary conditions here will contain the un-known internal field.Carrying out the procedures analogous with those in 1Dproblem solution, the following approximate integral equa-tion for the function F determining the field on the bound-ary asE(t,0,y) = E1(t,0,y)+F(t,y)can be obtained:F(t,y) =2piεv∫ t0dt ′σ(t ′)∫ y+v(t−t ′)y−v(t−t ′)dy′ F(t′,y′)(y− y′)v2(t− t ′)2− (y− y′)2 .(14)This equation can be solved numerically for concrete typesof the conductivity time-dependence using standard proce-dures. After obtaining the field on the boundary, we canestimate the reflected field by solving the wave equation(13).Estimating solution of this 2D problem has shown that therewill be the same effects of pulse splitting, compression andamplification as in 1D problem.These effects are not influenced later by multiply re-reflection, because the obtained pulses of short durationalready does not feel the conductivity nonstationarity ofcorrespondingly low frequency. It means that for pulsetransformation by nonstationarity of the reflecting mediumin optical waveguide a homogeneous simultaneous changeof the conductivity is not necessary, but it is enough tocreate local nonstationarity.References[1] F. A. Harfoush and A. Taflov, „Scattering of electromagnetic wavesby a material half-space with time-varying conductivity”, IEEE Trans.Anten. Propagat., vol. 39, no. 7, pp. 898–906, 1991.JOURNAL OF TELECOMMUNICATIONSAND INFORMATION TECHNOLOGY 1-2/2000 69Irena Yu. Vorgul and Marian Marciniak[2] A. G. Nerukh and I. Yu. Shavorykina, „Electromagnetic impulse re-turn from a conductive medium which has come into being”, Proc.Int. Symp. Anten. Propagat., vol. 2, pp. 585–588, 1992.[3] M. Marciniak, Łączność światłowodowa. Warszawa: WKŁ, 1998.[4] A. G. Nerukh and I. Yu. Shavorykina, „Transformation of radiationpulse in nonstationary conducting medium”, Radiophys. Quant. Elec-tron. (Consultants Bureau, New York), vol. 35, no. 3–4, pp. 203–209,1992.[5] A. G. Nerukh and N. A. Khizhnjak, „Modern Problems of TransientMacroscopic Electrodynamics”. Kharkov, Ukraine: Test-Radio, 1991.[6] S. Gonda and D. Seko. Optoelectronics in questions and answers,Leningrad: „Energoizdat” Publ., 1989 (Russian transl. from Japa-nees).[7] I. Vorgul and M. Marciniak, „Electromagnetic pulse scattering onhalf-space with continuously time-varying conductivity”, Prace IŁ,no. 110, pp. 29–42, 1998.Irena Yu. VorgulKharkov State University, 4 Svoboda Sq., Kharkov, 310077,UkraineMarian MarciniakNational Institute of Telecommunications, 1 Szachowa Str.,04-894 Warsaw, Poland, e-mail: M.Marciniak@itl.waw.pl70JOURNAL OF TELECOMMUNICATIONSAND INFORMATION TECHNOLOGY1-2/2000

Optical pulse splitting under temporal variations of reflecting medium, Journal of Telecommunications and Information Technology, 2000, nr 1,2

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Optical pulse splitting under temporal variations of reflecting medium, Journal of Telecommunications and Information Technology, 2000, nr 1,2

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