Optical range electromagnetic field interaction with inhomogeneous finite dielectric media is investigated. New method for analysis of this interaction in 1D case based on integral equations for fields in transient media is proposed. Some special cases of quasi-periodical structures with doubled quasi-periodicity are investigated numerically. A possibility to obtain good filtering properties in such structures for small number of quasi-periods is shown

Marciniak, Marian

Vorgul, Irena Yu

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

Paper Dielectric photonic quasi-crystalswith doubled quasi-periodicityIrena Yu. Vorgul and Marian MarciniakAbstract — Optical range electromagnetic field interactionwith inhomogeneous finite dielectric media is investigated.New method for analysis of this interaction in 1D case basedon integral equations for fields in transient media is proposed.Some special cases of quasi-periodical structures with doubledquasi-periodicity are investigated numerically. A possibility toobtain good filtering properties in such structures for smallnumber of quasi-periods is shown.Keywords — photonic crystals, optical pulse transformation.IntroductionIn the last years research activities on photonic band gap(PBG) structures as artificial periodic and quasi-periodicstructures whose transmission properties exhibit frequencybands where the propagation of electromagnetic waves isforbidden [1], becomes one of the most actual direction inphotonics and optical communication. It is known also thatsuch structures can display properties of bandpass filters bydisruption of the periodicity [2, 3]. Classical periodic lay-ered structures was properly investigated theoretically andexperimentally for different wavebands [4, 5]. The condi-tions of wave propagation in them were defined as well astheir reflection characteristics. Detailed description of wavebehaviour in them is a fundamental base for all consequentinvestigations.However, these structures are not ideal for practical appli-cations and their development by increasing complexity ofthe structures (as, for example, dual periodicity) is towardthe improved model. The criteria for such an optimizationare the gap width and shape (preferably a rectangular-likeone) as well as the structure dispersion and simplicity oftheir fabrication.One of the promising way of the improvement is to use notperiodical but quasi-periodical structures. Such structuresin a case of small shift of thickness of the layers compos-ing the correspondent periodical structure were consideredin [6, 7].In spite of impressive progress in the new and emergingarea of PBG engineering in recent years, their developmentby increasing complexity of the structures (as, for example,dual periodicity) toward the improved model is still actual.The criteria for such an optimization are the gap widthand shape (preferably a rectangular-like one) as well asthe structure dispersion and simplicity of their fabrication.There are a lot of works on periodical and quasi-periodicalstructures with a great number of periods providing forsome bands of frequency a full wave reflection. We tried toobtain a high reflection with sharp frequency spectrum fromquasi-periodical structures considering a small number oflayers, which is much more difficult to obtain but whichsimplifies sufficiently the structure fabrication.Previously, we considered mainly semi-infinite struc-tures [8, 9]. In the present paper the investigation of 1Dquasi-periodic dielectric structures is carrying out towardfinding optimal structures for sharp frequency filter andphase transformer. We consider a planar structure as a di-electric layer with double-quasi-periodical permittivity. Anadditional complexity of the structure allows to obtain highreflection for a wide band or sharp resonances on definedfrequencies as well as a sufficient.Problem formulation and solutionThe considered dielectric layer 0≤ x≤ a has a permittivitymathematically determined byε(x) ==N∑n=0(α−(1β)n)[θ(x−an)−θ(x−a(n+1))]++M∑m=0(γ−(1η)m)[θ(x−bm)−θ(x−b(m+1))] ,(1)Fig. 1. Algorithm of analytical solving procedure for the problemJOURNAL OF TELECOMMUNICATIONSAND INFORMATION TECHNOLOGY 1-2/2000 77Irena Yu. Vorgul and Marian MarciniakFig. 2. Permittivity profile in the layer (a); reflection coefficient amplitude in dependence on the incident wave frequency (b), (c);reflection coefficient phase, equal to pi at its maximum (d)being essentially a superposition of two quasi-periodicalstructures with the composing layers widths equal to a andb, correspondingly, and slight deviation of the layers per-mittivity from the periodical ones.The problem is solved based on integral equations for elec-tromagnetic fields in nonstationary media [10]. There aresome advantages in using this approach even for stationarystructures because the method is based on Volterra integralequation which can be solved by iterations with improvedconvergence unlike Friedholm and singular equations. Sowe can easily control the obtained results accuracy. Initialpoint of the problems solutions are Volterra integral equa-tions [10] which can be obtained from Maxwell equationfor the electrical component of electromagnetic field ob-tained by Green function of corresponding wave equationwith all nonstationarities picked up at its right hand part.In the considered 1D case it has the following form: forthe internal field (x > 0, t > 0):Ein(t,x) = E0(t,x)+−θ(vt− x)∫ tt−x/vdt ′ j(t ′,x− v(t− t ′))+−θ(x− vt)∫ t0dt ′ j(t ′,x− v(t− t ′))−∫ t0dt ′ j(t ′,x+ v(t− t ′))and for the external field (x < 0):Eex(t,x) == B(t,x)−θ(vt + x)∫ t+x/v0dt ′ j(t ′,v(t− t ′)+ x) , (2)where j(t,x) = ∂∂x[ε2(t,x)−ε1ε1Ein(t,x)]for dielectricmedium, θ is Heaviside step function and v = c/√ε .Analytical iteration formula for the reflected field deter-mined by the layer permittivity distribution was obtainedaccording to the scheme in Fig. 1. The iteration algo-rithm was realized as a computer program enabling one toinvestigate band gap and filtering properties of the consid-ered structures reflection for transient and stationary (as along-time approximation for the transient one) cases. Thisapproach is worth to be used even in a case of a stationaryinhomogeneous layer, because for nonstationary problemwe have a Volterra-type integral equation, for which a con-vergence of the iterative procedure is proved, so we can besure in the results.Calculation resultsCalculation for different layer parameters showed that forbig deviation of the structure permittivity from one ofthe periodic structure the reflection coefficient oscillatesslightly about the value, which is not appropriate neitherfor anti-reflection coating nor for a good reflector.We considered big deviations of the structure permittivityfrom one of periodic structure as well as small ones. Inthe first case, we found the oscillations of the reflectioncoefficient seeming not applicable to optical devices.For small deviations of the structure from the periodical oneit is possible to choose such structure parameters whichprovide good filtering and band gap characteristics of it(Fig. 2).78JOURNAL OF TELECOMMUNICATIONSAND INFORMATION TECHNOLOGY1-2/2000Dielectric photonic quasi-crystals with doubled quasi-periodicityThe reflection coefficient amplitude for the consideredstructures with doubled quasi-periodicity exceeds its max-imal value for simple quasi-periodical structures, as calcu-lated by the same program supposing the existence of onlyone sum in (1) (Fig. 3).Fig. 3. Permittivity profile in the layer (a); reflection coefficientamplitude in dependence on the incident wave frequency (b)ConclusionAn additional complexity of the structure allows to obtainhigh reflection for a wide band or sharp resonances ondefined frequencies as well as a sufficient reflected fieldphase shift for some parameters combinations.References[1] E. Yablonovich, J. Opt. Soc. Amer. B, no. 10, pp. 283–295, 1993.[2] H. J. De Los Santos, „On the design of photonic band gap crystalfilters”, in Proc. 1998 URSI Symp., Atlanta, USA, 1998, p. 121.[3] N. A. Khizhnyak and K. Yu. Kramarenko, „Propagation of Electro-magnetic Waves in Space-Varying Structures with Dual Periodicity”,Ukrain. Phys. J., vol. 10, p. 1256–1259, 1997.[4] L. Brillouin and M. Parodi, Propagation des Ondes dans les MilieuxPeriodiques. Paris: Masson, 1956.[5] L. M. Brekhovskih, Waves in Layered Media. Moscow: USSRAcademy of Sciences Publ., 1957.[6] F. Zolla, D. Felbacq, and B. Guizal, „A remarkable diffractive prop-erty of photonic quasi-crystals”, Opt. Commun., vol. 148, pp. 6–10,1998.[7] D. Felbacq, F. Zolla, and B. Guizal, „Wave propagation inone-dimensional photonic crystal”, Opt. Commun., vol. 152,pp. 119–126, 1998.[8] I. Yu. Vorgul and A. G. Nerukh, „Inverse problems for media withtransient conductivity”, Microw. Opt. Technol. Lett., vol. 19, no. 3,pp. 148–150, 1998.[9] I. Yu. Vorgul and M. Marciniak, „Electromagnetic pulse scatteringon half-space with continuously time-varying conductivity”, PraceIL, no. 110, pp. 29–42, 1998.[10] A. G. Nerukh and N. A. Khizhnyak, Modern problems of transientmacroscopic electrodynamics. Kharkov: Test-Radio Publ., 1991.[11] P. J. Roberts et al., „2D photonic band-gap structures as quasi-metals”, Opt. Lett., vol. 21, pp. 507–509, 1996.Irena Yu. Vorgul,Kharkov State University,4 Svoboda Sq., Kharkov 310077, Ukrainee-mail: yuts@ira.kharkov.uaMarian Marciniak,National Institute of Telecommunications,1 Szachowa Str., 04-894 Warsaw, Poland,e-mail: M.Marciniak@itl.waw.plJOURNAL OF TELECOMMUNICATIONSAND INFORMATION TECHNOLOGY 1-2/2000 79

Dielectric photonic quasi-crystals with doubled quasi-periodicity, Journal of Telecommunications and Information Technology, 2000, nr 1,2

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Dielectric photonic quasi-crystals with doubled quasi-periodicity, Journal of Telecommunications and Information Technology, 2000, nr 1,2

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