REFLECTION AND TRANSMISSION BAND STRUCTURES OF A ONE-DIMENSIONAL PERIODIC SYSTEM IN THE PRESENCE OF ABSORPTION

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1 Armenian Journal of Physics, 0, vol. 4, issue, REFLECTIO AD TRASMISSIO BAD STRUCTURES OF A OE-DIMESIOAL PERIODIC SYSTEM I THE PRESECE OF ABSORPTIO A. Zh. Khachatrian State Engineering University of Armenia, Yerevan, Armenia ashot.khachatrian@gmail.com Received 4 June, 0 Abstract On the base of the generalized transfer matrix method the roblem of electromagnetic wave roagation through an arbitrary eriodic one-dimensional absorbing medium is considered. Analytical exressions for transmission and reflection amlitudes are found. The obtained results are alied to a model system reresented by a onedimensional eriodic array of alternating layers of vacuum and a metal characterized by the comlex frequencydeendent dielectric function. It is shown that in an occuied whole sace eriodic system a harmonic time wave rocess cannot exist. Indeendently of the frequency value, this rocess always attenuates. Keywords: generalized transfer matrix, absorbing medium, transmission and reflection bands. Introduction Photonic crystals reresent a new kind of otical materials, which ossess many interesting roerties and render many novel alications ossible as well [ 6]. The existence of hotonic band gas in hotonic crystals, owing to multile Bragg scatterings, leads to many interesting henomena [ 9]. Most hotonic crystals fabricated so far are made from two dielectric materials. owadays, the dielectric hotonic crystals (DPC) have been widely studied and received a great deal of attention because of its erceived alications, roerties and new hysical henomena. Usually, hotonic band gas of dielectric hotonic crystals are not wide. Combinations of metallic and dielectric materials may lead to more interesting roerties comaring with dielectric hotonic crystals. The inclusion of metal sheets into dielectric hotonic crystal can increase hotonic band gas considerably [0 ]. For examle, the absortion of the bulk metal can be enhanced by inserting a dielectric layer eriodically to one-dimensional metal-dielectric hotonic crystals. By a roer choice of the structural and material arameters, one can obtain a large absortion enhancement in the visible and the infrared ranges [ 4]. In addition, the inclusion of eriodically saced thin metallic elements is becoming quite attractive at nanodimensions, where metamaterial have already shown quite interesting henomena. Metal-dielectric hotonic crystals, as basic stacks of a single dielectric with metallic insets at very low filling fractions (less than %), were first studied by Kuzmiak and Maradudin [5] using disersion relations as aroximate solutions. All those single dielectric structures have no aragon with dielectric hotonic crystals, where the aroach based on a disersion relation and the transfer matrix method are equivalent. Desite the fact that an investigation of one-dimensional eriodic structures consisting of absorbing layers was a subject of interest for many authors and has a great interest for many years,

2 Armenian Journal of Physics, 0, vol. 4, issue until now an analytical solution of this roblem is unknown. The roblem is that the oerator of a field wave is not Hermitian, so that the flux of electromagnetic wave energy is not conserved. In the standard method of transfer matrix [6 7], the comlex transmission and reflection amlitudes are derived from the elements of the transfer matrix, however the inverse relation between comlex scattering characteristics and transfer matrix elements is not established. We consider the mentioned matter as a main disadvantage of the standard transfer matrix method and it is a main reason why for a multilayer system all calculations are erformed by numerical methods only. In this work, we resent a general method of electromagnetic wave roagation in a onedimensional absorbing media of an arbitrary shae. We show that for a multilayered structure the roblem of determination of the scattering amlitudes is reduced to solution of a set of recurrent equations. In the case of absorbing media with a continually changing ermittivity the roblem is formulated as an initial condition roblem for the wave equation. The develoed method is alied for discussion of a hotonic crystal. amely, for transmission and reflection amlitudes of a hotonic crystal analytical exressions are found and absortion band structure is investigated.. Generalized transfer matrix method for an arbitrary one-dimensional absorbing medium We consider a harmonic in time electromagnetic waves (ex{ i t} ) in a one-dimensional absorbing medium. It is well known that the coordinate deendence of the field electric comonent is described by the following equation: where V( x) V ( x) iv ( x), k de k V( x) E 0 dx, () c, V k ( x), V k ( x) and ( x), ( x) are real and imagine arts of a ermittivity. Below we discuss a wave transmission roblem through irregularly absorbing slab when the dielectric constant has a form of, x x, ( x) i ( x), x x x,, x x. In the regions outside the slab the general asymtotic solution of Eq. () can be written as Ex ( ) A ex{ ikx} B ex{ ikx}, x x, A ex{ ikx} B ex{ ikx}, x x. If k 0 the quantities A, B and B, A are the amlitudes of convergent and divergent waves. It is clear that when k 0 then A, B become the amlitudes of divergent waves and B, A become the amlitudes of convergent waves. () (3) 9

3 Armenian Journal of Physics, 0, vol. 4, issue In accordance with the transfer matrix method (see [8]) between the amlitudes A, B and A, B a linear relation exists: A ˆ A M, Mˆ, B B Where,,, are the transfer matrix elements and (4). (5) Let us consider the left scattering roblem, when k 0 and the field asymtotic behavior has the form of ex{ ikx} r( k)ex{ ikx}, x x, Eleft (6) tk ( )ex{ ikx}, x x, where t ( k), r( k) are transmission and reflection amlitudes coefficients of the wave incident on the slab from the left. Below, mentioned transmission and reflection amlitudes will be defined in corresondence with the left scattering roblem (6). Comaring Eq. (3) and Eq. (6) it is easy to see that A, ( ) B r k and A t( k), B 0. (7) Substituting Eq. (7) into Eq. (4) and taking into account Eq. (5), one can get rk ( ) and. (8) tk ( ) tk ( ) The obtained result (8) is the basic formulas of the standard transfer matrix theory, which is usually alied for calculation of transmission, reflection and absortion coefficients of a dissiative medium. In the case of nonabsorbing media ( ( x) 0 ) between the transfer matrix elements the following nonalgebraic relations exist: *, *. Unfortunately Eq. (8) allows erforming numerical calculations only and these formulas are not effective to get analytical results. amely, until now analytical exressions for transmission and reflection amlitudes of an electromagnetic wave scattering on a eriodic absorbing medium are not obtained. In the recent aers [8, 9] a so-called generalized transfer matrix method is develoed, which as will be shown below, is very efficient both to erform numerical calculations and to obtain analytical results. In accordance with the main results of the above-mentioned aers, the transfer matrix for an arbitrary absorbing medium has the following form r( k) ˆ t( k) t( k) M. (9) rk ( ) tk ( ) tk ( ) 9

4 Armenian Journal of Physics, 0, vol. 4, issue As it is seen from Eq. (9) and Eq. (4) in general case the transformation between the transfer matrix elements is realized by mean of changing the sign of the quantity k. ote that for the case of nonabsorbing media this action ( k k ) is equivalent to a comlex conjugation action, i.e. * * t ( k) t( k), r ( k) r( k). For consideration of a wave roagation roblem for a multilayered system, in the aers [9, 0] the following matter is discussed: how the transfer matrix elements are changed when the slab is arallel transorted on a some distance L. As it was shown, and ex{ i kl}, ex{ i kl}, where,, and are the transfer matrix elements of the transorted slab. 3. Transmission through multilayered structure Let us consider the transfer matrix matrices of the system s searate layers: Mˆ of a -layered structures as a roduct of transfer ˆ n n n n M, (0) n n nn n In accordance with Eq. (9), the transfer matrix of the system can be written as where T and R ( k) ˆ T ( k) T ( k) M, () R ( k) T ( k) T ( k) R are transmission and reflection amlitudes of the system. A transfer matrix of a single layer written with hel of the scattering amlitudes can be resented as rn ( k) n n tn( k) tn( k), () n n rn ( k) tn( k) tn( k) where t n and r n are transmission and reflection amlitudes of a n-th single layer of the system. If one considers as a variable quantity, when on the base of Eq. (0) and Eq. () the roblem of determination of T and R can be reduced to the solution of a set of finite-differential equations with initial conditions. Indeed, from Eq. (0) and Eq. () one can write down Mˆ ˆ M, (3) where Mˆ is the transfer matrix of the system consisting only of the first layers (without the last layer) of the system. Using Eq. (3), Eq. () and Eq. (0) one has 93

5 Armenian Journal of Physics, 0, vol. 4, issue R ( k) r ( k) R ( k) T( k) T( k) t( k) tn( k) T ( k) T( k). R( k) r( k) R ( k) T( k) T( k) t( k) t( k) T ( k) T ( k) It is easy to see that the matrix equation (4) is equivalent to the following set of finite-differential equations: r ( k) R ( k), (5) T ( k) t ( k) T ( k) t ( k) T ( ) with the initial condition k R ( k) R( k) r( k), T ( k) t ( k) T ( k) t ( k) T ( k) (4) (6) T ( k), R ( k) 0. (7) 0 0 ote that the initial condition corresonds to the system without layers, i.e. to the free sace motion. It is easy to see that with resect to quantities T ( k ), R( k) T( k) the set of equations (5), (6) is linear. In general case of an arbitrary non-regular structure, when the layers of the system and the distances between them differ, the set of equations (5), (6) can be solved only numerically. As it will be shown below on the base of equations (5), (6) analytical exressions for transmission and reflection amlitudes can be derived. 4. The transfer matrix elements as functions of a slab border oint It is interesting to aly the result (5) (7) to the roblem of determination of scattering amlitudes for a single layer with an arbitrary but continuously changing from oint to oint ermittivity. We will follow to the method develoed in the aers [0-], where the scattering amlitudes as functions of the slab borders are considered (see Fig. ). For simlicity we will consider a slab with the initial oint at x 0. Let us identify a slab having width y with a - layered structure and a slab of width y y with -layered structure where the last layer is absent. The last layer of the structure will be considered as a layer with small width y. In accordance with the above mentioned Eq. (5), (6) one can consider ( ) ( y) T k, R ( k) R ( k) ( y) and ( y y), ( y y), (8) T ( k) T ( k) T ( k) where ( y) tky (, ). ote that in accordance with Eq. () when y x the slab width equals zero and the magnitude of y x corresonds to the system whole width. To obtain transmission and reflection amlitudes for a thin layer of a width the well-known formulas corresonding to the uniform slab [7]: y we will use 94

6 Armenian Journal of Physics, 0, vol. 4, issue q ( y) k ( ) ex{ iky} cos[ q( y) y] i ( ) sin[ q( y) y], (9) t k kq y r ( k) qy ( ) k i exikysin[ q( y) y], (0) t ( k) kq( y) where qy ( ) k ( y). Considering qy ( ) y and taking into account (), it is easy to get t ( k) iu( x) y k r ( k) iu( x). () t ( k) k, y exiky ex{ikx} r( k, y)ex{ ikx} y y (x) y t( k, y)ex{ ikx} y Fig.. Transmission and reflection amlitudes as functions of the slab right border. Substituting Eq. (8) and Eq. () in the set (5), (6) one can found the following set of equations for determination of the transfer matrix elements of a slab with a continually changing ermittivity: d( x) iv( x) iv( x) ( x) ( x)ex{ i kx}, () dx k k with initial conditions d( x) iv( x) iv( x) ( x) ( x)ex{ i kx}, (3) dx k k ( x ), x ( ) 0. (4) Equations for the transfer matrix elements ( x), ( x) ( ( x) t( k, x), ( x) rkx (, ) tkx (, )) are obtained from Eq. () and Eq. (3) by changing k by k : d( x) iv( x) iv( x) ( x) ( x)ex{ i kx}, (5) dx k k with initial conditions d iv( x) iv( x) ( x) ( x) ( x)ex{ i kx}, (6) dx k k ( x ), x ( ) 0. (7) It is easy to see that the airs of quantities ( x), ( x) and ( x), ( x) satisfy the same set of equations but initial conditions for them are different. ote that in case of a nonabsorbing slab 95

7 Armenian Journal of Physics, 0, vol. 4, issue ( u (x) is a real function) the set of equations (), (3) are the comlex conjugate with resect to the set (5), (6). So we have shown that the roblem of determination of scattering amlitudes for an absorbing slab with an arbitrary continually changing (x) is reduced to Cauchy roblem for a set of linear differential equations. It is imortant to mention that the set of equations (), (3) (or (5), (6)) are very useful for a numerical integration but these equations are not alicable for analytical calculations. By using equations (), (3) and (5) it is ossible to show that the roblem of determination of transfer matrix elements can be formulated as an initial tye of roblem for the wave equation (): ex{ ik( x x)} i d( x) d( x) ( x) ( x) ik ( x) k dx dx, (8) ex{ ik( x x)} i d( x) d( x) ( x) ( x) ik ( x) k dx dx, (9) where ( ) ( ) satisfy the wave equation (): x x with the initial conditions d ( x),, (30) dx, k V( x), 0 d ( x ) dx 0 and ( x) 0, d( x) dx. (3) It is imortant to mention that due to the fact that the equation (30) contains only for ( x), ( x) can be obtained from (8), (9) by changing k by k k the exressions. ote that determination of the transfer matrix elements in the form of the initial roblem (8) (3) is useful both for analytical and numerical calculations. This result is a generalization of the corresonding formulas of the aer [3] obtained for media without dissiation. 5. Scattering on an ideal transfer structure Let us consider a layered structure, when the layers are identical and equidistantly located. At this ste of consideration, we do not assume that a single layer structure must be uniform or that we exlicitly know the deendence of the transfer matrix elements on the frequency of an incident radiation and the arameters of a scattering layer. For the considered layered structure the ermittivity is characterized by the following deendence: V( x) V ( x), (3) n where V ( x ) relates to the n-th layer of the system, which is resented by means of the function n V ( ), defining the otical roerties of the first layer, in accordance with the following relation x n 96

8 where L is the system eriod. ( ) ( ( ) ) n Armenian Journal of Physics, 0, vol. 4, issue V x V x n L, n,, (33) Denoting the border oints of the first layer as y, y ( y y ), one can write down u( x) ( x) ( x y) ( y x), (34) c where ( x) ( x) i ( x) is the ermittivity of the first layer and ( x) is the ste function. ote that the ossible value of the eriod L should be considered more than a width magnitude of a single layer ( L( y y) ), otherwise an overlaing of neighbor layers will take lace. Let us consider a relation between the transfer matrix elements of the system s n -th layer and the transfer matrix elements of its first layer. In accordance with the last equation of the section, one can write down,, ex{ i kl( n)}, ex{ i kl( n )}, (35) n n n where for simlicity we denoted the transfer matrix elements of the first layer as,,,. Using Eq. (35), the set of equations (5), (6) can be written as rk ( ) R ( ) ex ( ) k i kl n, (36) T ( k) t( k) T ( k) t( k) T ( k) with initial conditions R( k) R( k) r( k) exi kl( n) T ( k) t( k) T ( k) t( k) T ( k) n (37) T ( k), R ( k) 0. (38) 0 0 ote that here the quantities t (k) and r (k) are transmission and reflection amlitudes of the structure s first layer: t( k), rk ( ) tk ( ). The set of equations (5), (6) can be solved by means of different methods and its solution has the form ex{ ikl} ex{ ikl} sin( L) ( ) ex{ ikl} cos( L) T ( ) ( ) sin( ), k t k t k L (39) R ( k) rk ( ) sin( L) ex{ ikl( )}, T ( k) t( k) sin( L) (40) where we introduced the following notation: ex{ ikl} ex{ ikl} cosl. (4) tk ( ) t( k) 97

9 Armenian Journal of Physics, 0, vol. 4, issue For case of nonabsorbing media (as it was mentioned above, for that case the equality * t( k) t ( k) takes lace) the result (39) (4) is reduced to the corresonding formulas of the aer []. ote that the quantity can take either real or urely imagine value since in this case the right side of the equality (4) is a real quantity: cos L Reex{ ikl} t( k). However, in the more general case of absorbing media has a comlex value and as a function of k it is an even function ( ( k) ( k) ). The obtained formulas (39) (4) exress the exlicit deendences of the system scattering amlitudes T, R on the number of layers and the scattering amlitudes t, r of the element of the system structure. Desite the fact that Eqs. (39) (4) are analytical exressions for their investigation there is a necessity in additional discussions. 6. Photonic crystal with a two-layered structure element Below we will examine the obtained result (39)-(4) for a hotonic crystal with a two-layered structure element. Let us suose that the structure element consists of two homogeneous and contacting with each other layers of widths a, a, so that the eriod of the hotonic crystal is L a a. To determine the transmission and reflection amlitudes of a two-layered structure t and r (see Eq. (39)-(4)) we will use the formulas (5), (6): () () r ( k) r ( k), (4) () () () () tk ( ) t ( k) t ( k) t ( k) t ( k) () () r( k) r ( k) r ( k), (43) () () () () t( k) t ( k) t ( k) t ( k) t ( k) where ( j) t, ( j) r ( j, ) are the transmission and reflection amlitudes of the first and second layers of the structure elements, corresondingly: q k ex{ } cos[ ] sin[ ] j ika ( j) j q ja j i q ja j t ( k) kqj ( j) r ( k) j ( j) t ( k) kqj, ( j, ) (44) q k i ex{ ikb }sin[ q a ], ( j, ) (45) j j j where qj k j, k c and, are the ermittivities of the layers. In Eq. (45) b j is the coordinate of the middle oint of the structure element layer. So, if the initial oint of the first structure element coincides with a coordinate origin then b a, b a a. Substituting Eqs. (44), (45) in Eq. (4) from Eq. (4) one can write q q cosl cos qa cos q a sin q a sin q a, (46) qq 98

10 Armenian Journal of Physics, 0, vol. 4, issue which is the well-known exression for the quasi-wave number that determines the relationshi between the wave daming in a suerlattice and daming for a single layer forming this structure (see, for examle, [7]). Below we will aly the result (39) (46) for a secific structure, namely, we consider a onedimensional hotonic crystal on the base of nanocomosite: metal nanoarticle dielectric matrix. ote that the comosite media with nanoarticles of noble metals are of great ractical interest in the develoment of various otical devices [3, 4]. It is clear that the otical roerties of such media are determined by lasma oscillations in the nanoarticles and the roerties of a transarent matrix. We calculate the transmission, reflection and absortion coefficients of a one-dimensional hotonic crystal consisting of nanocomosite metal nanoarticles which are randomly distributed in a transarent matrix. To determine the ermittivity of the nanocomosite ( ) we use the Maxwell Garnett formula: mix ( ) d m( ) d f ( ) ( ), (47) mix d m d where f is the relative volume occuied by nanoarticles, ( ) is the ermittivity of the metal material of the nanoarticle, d is the dielectric constant of the matrix, and is the frequency of radiation. m mix Fig.. Deendences of ( ), ( ) on the dimensionless frequency. mix mix Using Eq. (47) one can write mix ( ) d m( )( f) d( f ). (48) ( )( f ) ( f ) m As it follows from Eq. (48), when f (the whole volume of the matrix is occuied by metal nanoarticles) we have mix m, when f 0 (in the volume of the matrix there are no metal d 99

11 Armenian Journal of Physics, 0, vol. 4, issue nanoarticles) mix d. The dielectric constant of the metal nanoarticles is determined in accordance with the Drude model: where 0 is a constant, (see [5]) 0 5, 9 ev, 0.0 ev. m( ) 0, (49) ( i ) is the lasma frequency, the relaxation constant. In the case of silver Absortion Reflection Transmission / Fig. 3. Deendences of transmission, reflection and absortion coefficients on the dimensionless frequency. Further, in all numerical calculations we consider a nanocomosite based on silver. In the Fig. we resented the deendences of mix( ), mix( ) on the dimensionless frequency. As it can be seen from the figure, the both curves have a resonant character. In Fig. 3 we lotted the deendences of transmission, reflection and absortion coefficients on the dimensionless frequency. The calculation was made for a hotonic crystal consisting of 6 cells, each of which contains two layers a nanocomosite layer of width a with ermittivity ( ) and a vacuum layer of width a. The filling factor f was taken as f 0.. For erforming of corresonding mix 00

12 Armenian Journal of Physics, 0, vol. 4, issue calculations the sizes of the layers were considered in units of the lasma wavelength ( c ). The width of the layers were chosen to be equal to each other and the eriod of the structure to be equal to ( La a, a a and L ). As it can be seen from the uer curve of Fig. 3, for the region of frequencies being under consideration there are two bands of non-transarency and with increasing of a frequency value the bandwidth decreases. We also note that there is no frequency value where the transmission coefficient takes a value being equal to unity. On this base one can conclude that when a eriodic absorbing media occuies the whole sace, then in that system any wave rocess is doomed to attenuation. As it follows from the middle and lower curves of Fig. 3, the reason of existing a wave non-transarency band may serve as a reflection and absortion. 7. Conclusion In this aer on the base of the generalized transfer matrix method the roblem of electromagnetic wave roagation through an arbitrary eriodic one-dimensional absorbing media was considered. It is shown that for an arbitrary layered structure the roblem of determination of scattering amlitudes is reduced to solution of some set of recurrent equations. This result was alied to a hotonic crystal with an arbitrary form of a structure element. Analytical exressions for the transmission and reflection amlitudes as functions of the number of hotonic crystal cells are found. The obtained exressions were considered for a model system reresenting a one-dimensional eriodic array of alternating layers of vacuum and a metal characterized by the comlex frequencydeendent dielectric function. It was shown that in an occuied whole sace eriodic system a harmonic time wave rocess cannot exist. It means that any wave rocess always attenuates. REFERECES. E.Yablonovitch, Phys. Rev. Lett., 58, 059 (987).. S.John, Phys. Rev. Lett. 58, 486 (987). 3. C.M.Soukoulis, Photonic Band Gas and Localization, Plenum, J.D.Joannooulos, R.D.Meade, J..Winn, Photonic Crystals, Princeton University Press, C.M.Soukoulis, Photonic Bandga Materials, Kluwer, J.D.Joannooulos, P.R.Villeneuve, S.Fan, ature 386, 43 (997). 7. C.M.Cornelius, J.P.Dowling, Phys. Rev. A, 59, 4736 (999). 8. H.V.Baghdasaryan, T.M.Knyazyan, T.H.Baghdasaryan, B.Witzigmann, F.Roemer, Oto- Electronics Review, 8, 438 (00). 9. D.F.Sievenier, M.E.Sickmiller, E.Yablonovitch, Phys. Rev. Lett., 76, 480 (996). 0. J.B.Pendry, A.J.Holden, W.J.Steward, I.Youngs, Phys. Rev. Lett., 76, 4773 (996).. A.J.Ward, J.B.Pendry, W.J.Stewart, J. Phys.: Condens. Matter, 7, 7 (995). 0

13 Armenian Journal of Physics, 0, vol. 4, issue. M.Scalora, M.J.Bloemer, A.S.Pethel, J.P.Dowling, C.M.Bowden, A.S.Manka, J. Al. Phys., 83, 377 (998). 3. M.J.Bloemer, M.Scalora, Al. Phys. Lett., 7, 676 (998). 4. J.Yu, Y.Shen, X.Liu, R.Fu, J Zi, Z.Zhu, J. Phys.: Condens. Matter, 6, 5 (004). 5. V.Kuzmiak, A.A.Maradudin, Phys. Rev. B, 55, 747 (997). 6. H.Contoanagos, E.Yablonovitch,.G.Alexooulos, J. Ot. Soc. Am. A, 6, 94 (999). 7. P.Yeh, Otical Waves in Layered Media, Wiley, A.Zh.Khachatrian, Journal of Physics, 3, 78 (00). 9. A.Zh.Khachatrian, Electromagnetic waves and Electronic Systems, 5, 49 (00). 0. D.M.Sedrakian, A.Zh.Khachatrian, Physics Letters A, 65, 94 (000).. D.M.Sedrakian, A.Zh.Khachatrian, Annalen der Physik,, 503 (00).. D.M.Sedrakian, A.H.Gevorgyan, A.Zh.Khachatrian, Otics Communications 9, 35 (00). 3. Z.Wang, C.T.Chan, W.Zhang, Phys. Rev. B, 64, 308 (00). 4. G.Gantzounis,.Stefanou, Y.J.Yannoaas, J. Phys.: Condens. Matter, 7, 79 (005). 5. P.B.Johnson, R.W.Christy, Phys. Rev. B, 6, 4370 (97). 0