Joural of Physics: Coferece Series PAPER OPEN ACCESS Aalytical solutios for multi-wave trasfer matrices i layered structures To cite this article: Yu N Belyayev 018 J Phys: Cof Ser 109 01008 View the article olie for updates ad ehacemets This cotet was dowloaded from IP address 14851383 o 19/01/019 at 15:50
IOP Publishig IOP Cof Series: Joural of Physics: Cof Series 134567890 109 (018) 01008 doi :101088/174-6596/109/1/01008 Aalytical solutios for multi-wave trasfer matrices i layered structures Yu N Belyayev Departmet of Mathematical Modellig ad Cyberetics, Syktyvkar State Uiversity, 55 Oktyabrskii pr, Syktyvkar, 167001, Russia E-mail: ybelyayev@mailru Abstract The matrix approach to the calculatio of -beam diffractio i a periodic structure is discussed The possibility of represetig N-layered trasfer matrix T N through the pricipal miors of the matrix T is ivestigated The method of computig iteger powers of a matrix by meas of symmetric polyomials of the latter is developed New aalytic formulas for iteger powers of matrices are represeted These solutios are demostrated usig examples of matrices of the sixth order 1 Itroductio The physical properties of may atural ad artificial objects are well described withi the framework of oe-dimesioal periodic models [1, ] Examples of such structures that have foud wide applicatio are optical thi films [3], liquid crystals [4], superlattices [5], photoic crystals [6] Matrix approaches have become oe of the mai oes i the calculatios of wave propagatios i such media The problem of calculatig the wave trasmissio through N- layered structure is reduced to fidig so-called trasfer matrix T (or characteristic matrix [7], or scatterig matrix [8], or propagator matrix [9]) of oe separate layer ad matrix T N for the whole structure I this regard, of great iterest are aalytical solutios for iteger powers of matrices The Baker-Vadermode formula [10] represets the solutio of this problem i terms of matrix eigevalues A remarkable result for uimodular secod-order matrices was obtaied by Abelés [11] i the study of iterferece optical films I his paper the elemets of the characteristic matrix M N have bee expressed through the trace of the matrix M usig the Chebyshev polyomials of the secod kid Similar formulas for the iteger powers of secod order matrices of geeral form ad certai matrices of the third ad fourth orders were foud i [1] I this paper, we preset ew aalytical solutios for iteger powers of th order matrices Their proofs are based o the properties of symmetric polyomials of th order [13] Polyomial approach to the calculatio of iteger powers of matrices Accordig to oe of the corollaries of the Cayley-Hamilto theorem [14], iteger power j of ay matrix A ca be expressed as a liear combiatio of the first powers A 0 I, A,, A 1 : 1 A j = C jh A h, j 0 (1) h=0 Cotet from this work may be used uder the terms of the Creative Commos Attributio 30 licece Ay further distributio of this work must maitai attributio to the author(s) ad the title of the work, joural citatio ad DOI Published uder licece by IOP Publishig Ltd 1
IOP Publishig IOP Cof Series: Joural of Physics: Cof Series 134567890 109 (018) 01008 doi :101088/174-6596/109/1/01008 We use the followig represetatio of the coordiates C jh [13]: C j h = h p h+g B j 1 g (), h = 0, 1,, 1, j = 0, 1,,, () g=0 where p m, m = 1,,, are coefficiets of characteristic equatio λ = i=1 p iλ i of matrix A = a gh ad fuctios B g () are defied by recurrece relatios B g () = 0, 0 g ; B 1 () = 1; B g () = p j B g j (), g (3) Two represetatios of the coefficiets p m, amely p 1 = g=1 λ g, p = l j λ lλ j,, p = ( 1) 1 j=1 λ j ad p 1 = g=1 a gg, p = a ii a ij j>i a ji a jj,, p = ( 1) 1 det A give respectively two iterpretatio of the fuctios B g (): symmetric polyomials of th order [13] (symmetric with respect to the eigevalues λ j, j = 1,, ) ad polyomials of pricipal miors of the matrix A 1 Expressio of A N i terms of symmetric polyomials T h e o r e m 1 Symmetric polyomials of th order are expressed i terms of the eigevalues λ i, i = 1,,, of the matrix by the formulas 1 λ 1 λ1 λ1 1 1 λ 1 λ 1 λ g 1 B g () = g, = 1 λ λ 1 λ λ λ 1 λ 1 j=1 1 λ λ, g = 1 λ λ λ g λ g, (4) where g is ay iteger, if all values λ i 0, ad g 0, if some eigevalue λ i = 0 P r o o f The properties of the determiats make it obvious that the formulas (4) correspod to the defiitio (3) of polyomials B g () for g = 0, 1,, 1 Suppose that the equality (4) holds for g = j + 1,, j 1, j, where j 1 We show that the it will also true (ie fuctios (4) satisfy the defiitio (3)) for g = j + 1 Ideed, accordig to characteristic equatio, λ j+1 = p 1 λ j + p λ j 1 + + p λ j +1 Therefore B j+1 () = 1 1 λ 1 λ 3 1 λ λ 3 1 λ λ 3 1 λ λ 1 p 1 λ j 1 + p λ j 1 1 + + p λ j +1 1 p 1 λ j + p λ j 1 + + p λ j +1 p 1 λ j + p λ j 1 + + p λ j +1 = 1 p h j+1 h = p h B j+1 h (), λ which agrees with the recurrece formulas (3) Note that substitutios of the expressios (4) ito the formulas () ad (1) lead to oe more proof of the Baker-Vadermode formula A more explicit form of symmetric polyomials B g () follows from (4) if we use the value of the Vadermode determiat = j>i (λ j λ i ) For example, suppose that the characteristic equatio of a sixth-order matrix has the form: λ 6 p λ 4 p 4 λ p 6 = 0 I this case, the eigevalues are pairwise equal to each other with a precisio up to the sig, say h=1 h=1
IOP Publishig IOP Cof Series: Joural of Physics: Cof Series 134567890 109 (018) 01008 doi :101088/174-6596/109/1/01008 λ 1 = λ 4, λ = λ 5, λ 3 = λ 6 These eigevalues are expressed i the radicals i terms of the coefficiets of the characteristic equatio I tur, the coefficiets of the characteristic equatio p, p 4, p 6 are expressed i terms of the eigevalues λ 1, λ, λ 3 by the formulas: p = λ 1 +λ +λ 3, p 4 = λ 1 λ + λ 1 λ 3 + λ λ 3, p 6 = λ 1 λ λ 3 I this case, the symmetric polyomials of sixth order are expressed through the matrix eigevalues as: B j (6) = 1 + ( 1)j 1 Λ j 1 Λ 4, Λ i = λ i 1(λ 3 λ ) λ i (λ 3 λ 1) + λ i 3(λ λ 1) (5) I accordace with (1) { A j p 6 B j (6)A + [p 4 B j (6) + p 6 B j 4 (6)]A 3 + B j (6)A 5, if j = 7, 9, 11,, = p 6 B j 1 (6)I + [p 4 B j 1 (6) + p 6 B j 3 (6)]A + B j+1 (6)A 4, if j = 6, 8, 10,, where the polyomials B g (6) are defied by formulas (5) ad I is idetity matrix Represetatio of iteger powers of a matrix by meas of the coefficiets of its characteristic equatio T h e o r e m If oly two of the coefficiets of the characteristic equatio p h ad p l, l = h, are ozero, the the polyomials B g () are expressed by the formulas: (6) B g () = { ( pl ) j 1 U j 1 (b), if g = h(j 1) + 1, 0, if g h(j 1) + 1, j = 1,,, (7) where U j (b) = si[(j + 1) arccos b] 1 b, b = p h p l (8) P r o o f It s obvious that h(j 1) + 1 1 Therefore, it follows from (7) that B g () = 0, if g, which agrees with (3) If g = 1, the, by formulas (7) ad (8), we fid B 1 () = U 0 (b) = 1 This also satisfies defiitio (3) Fially, if g, the, accordig to the defiitio (3) ad the coditio of the theorem, the fuctio (7) must be a solutio of equatio B g () = p h B g h () + p l B g l (), g (9) It s easy to check Let g h(j 1) + 1 The g h ad g h ot equal to h(j 1) + 1 I this case formulas (7) give values B g () = B g h () = B g l () = 0, which satisfy equatio (9) At last, if g = h(j 1) + 1, the substitutio of (7) ito (9) gives equality U j 1 (b) = bu j (b) U j 3 (b), (10) whose idetity is established by elemetary trasformatios usig the fuctios (8) Thus, the fuctios (7) are solutios of equatios (9), which was to be proved If 1 b 1, the fuctios U j (b) with idexes j 0 are orthogoal Chebyshev polyomials of the secod kid For istace for sixth order matrix A with p 1 = p = 0, p 3 0, p 4 = p 5 = 0, p 6 0, we fid from (1), () ad (7) A 6+i+3k = p 6 B 5+3k (6)A i + B 8+3k (6)A i+3, i = 0, 1,, k = 0, 1, (11) where B g (6) = (det A) g 5 6 si [((g ) arccos b)/3]/ 1 b, b = p 3 /( det A) for g = 5, 8, 11, ad B g (6) = 0 for g 5, 8, 11, 3
IOP Publishig IOP Cof Series: Joural of Physics: Cof Series 134567890 109 (018) 01008 doi :101088/174-6596/109/1/01008 E x a m p l e Trasfer matrix T of elastic waves i a crystallie layer of thickess d is expressed by the formula T = exp(w d), where sixth-order matrix W = w ij is determied [15] from the system of equatios of motio ad Hooke s law For may geometries of elastic waves scatterig by a cubic crystal, the symmetric polyomials B g (6) of the matrix W ad matrices W j satisfy relatios (5) ad (6) respectively For these cases, the aalytical solutio for the trasfer matrix has the form (W d) j T = exp(w d) = j! j=1 = 5 l=0 W l 3 g=1 { τ gl cosh(λg d), if l = 0,, 4, Λ 4 sih(λ g d), if l = 1, 3, 5, (1) where τ 10 = λ λ 3 (λ 3 λ ), τ 0 = λ 1 λ 3 (λ 1 λ 3 ), τ 30 = λ 1 λ (λ λ 1 ), τ g1 = τ g0 /λ g, τ 1 = λ 4 λ4 3, τ = λ 4 3 λ4 1, τ 3 = λ 4 1 λ4, τ g3 = τ g /λ g, τ 14 = λ 3 λ, τ 4 = λ 1 λ 3, τ 34 = λ λ 1, τ g5 = τ g4 /λ g I the case whe the matrix W has the form (11), a aalytic solutio for the elastic wave trasfer matrix ca be foud i a similar way For sufficietly small thickess d the series i=1 (W d)j /(j!) is well approximated by the sum of the first two or three terms, ie T = I + W d + W d / Such defiitio of the matrix T, if max w ij d 1, provides calculatio of matrix T elemets with a relative error of less tha 6(max w ij d ) 3 This allows us to cosider a homogeeous layer of thickess Nd as N-layered periodic structure ad calculate its trasfer matrix T N i accordace with the formulas (1), (), (3) It is easy to show that i this case the coefficiets p m ad the polyomials of pricipal miors up to terms of the secod order of smalless i max w ij d are expressed by the formulas p m = ( 1) m 1 6!(1 + mx)/[m!(6 m)!], m = 1,, 6; B g (6) = c g [1 + (g 5)x], g 5, where x = 6 i=1 w iid/6 ad the coefficiets c g are determied from the recurrece relatios: c g = 0, g 4; c 5 = 1; c g = 6 j=1 c g j6!/[j!(6 j)!], g 5 3 Coclusio As the examples of two- ad four-beam scatterig of electromagetic waves i layered media show, aalytical formulas for trasfer matrices facilitate a theoretical study of the features of the correspodig diffractio spectra Such formulas as (6), (11), (1) play o less importat role i the verificatio of umerical algorithms for calculatig matrices T N with large N, sice roud-off errors durig matrix calculatios deped strogly o the umber of matrix multiplicatios Refereces [1] Elachi C 1976 Waves i active ad passive periodic structures Proc IEEE 64 1666 1698 [] Karpov S Yu, Stolyarov S N 1993 Propagatio ad trasformatio of electromagetic waves i oedimesioal periodic structures Physics-Uspekhi 36 1 [3] Kittl Z 1975 Optics of thi films (Lodo: J Wiley) [4] Belyakov V A 199 Diffractio Optics of Complex-Structured Periodic Media (New York: Spriger) [5] Herma M 1986 Semicoductor Superlattices (Berli: Akademie) [6] Sakoda K 001 Optical Properties of Photoic Crystals (Berli: Spriger) [7] Bor M, Wolf E 1964 Priciples of optics (Oxford: Pergamo press) [8] Cowley J M 1975 Diffractio Physics (New York: N-HPubl Comp) [9] Brekhovskikh L M, Godi O A 1990 Acoustics of layered media (Berli: Spriger-Verlag) [10] Baker H F 1903 O the itegratio of liear differetial equatio Proc Lodo Math Soc 35 333 378 [11] Abelés F 1950 Recherehes sur la propagatio des odes electromagetiques siusoidals das les milieux stratifiés Applicatio aux coches mices A Phys 5 596 640 706 78 [1] Belyayev Yu N 015 Characteristic matrices of layered periodic structures Proc It Cof Days o Diffractio 015 May 5 9 St Petersburg Russia 4 48 [13] Belyayev Yu N 013 O the calculatio of fuctios of matrices Mathematical Notes 94 177 184 [14] Gatmacher F R 1959 The Theory of Matrices vol 1 (New York: Chelsea) [15] Belyayev Yu N 018 Calculatio of the six-beam diffractio i layered media usig polyomials of pricipal miors Joural of theoretical ad computatioal acoustics 6 1850017-1 1850017-1 4