# Effect of Fabry-Perot resonances in disordered one-dimensional array of alternating dielectric bi-layers.

код для вставкиСкачатьAnn. Phys. (Berlin) 18, No. 12, 887 – 890 (2009) / DOI 10.1002/andp.200910388 Effect of Fabry-Perot resonances in disordered one-dimensional array of alternating dielectric bi-layers G. A. Luna-Acosta1,∗ and N. M. Makarov2 1 2 Instituto de Fı́sica, Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla, Pue., 72570, México Instituto de Ciencias, Universidad Autónoma de Puebla, Priv. 17 Norte No 3417, Col. San Miguel Hueyotlipan, Puebla, Pue., 72050, México Received 29 September 2009, accepted 1 October 2009 Published online 11 December 2009 Key words Disordered systems, Fabry-Perot resonances, Anderson localization, photonic arrays. PACS 72.15.Rn, 42.25.Bs, 42.79.Qs We study numerically and analytically the role of Fabry-Perot resonances in the transmission through a one-dimensional finite array formed by two alternating dielectric slabs. The disorder consists in varying randomly the width of one type of layers while keeping constant the width of the other type. Our numerical simulations show that localization is strongly inhibited in a wide neighborhood of the Fabry-Perot resonances. Comparison of our numerical results with an analytical expression for the average transmission, derived for weak disorder and finite number of cells, reveals that such expression works well even for medium disorder up to a certain frequency. Our results are valid for photonic and phononic one-dimensional disordered crystals, as well as for semiconductor superlattices. c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Effects of disorder in periodic lattices have been intensely studied over several decades. Anderson Localization is the most important phenomenon that predicts exponentially small transmission through onedimensional long enough structures with any amount of disorder [1]. More recently, there have appeared interesting ideas for defeating localization, such as the random dimer model [2] or more general, by involving long range correlations in random potentials (see, for example, [3– 5]). The interest in this studies is both, academic and practical. From the practical side, it is advantageous to design random structures that produce enhanced suppression of localization in a given frequency or energy interval. Another, quite natural, way of inhibiting localization is expected to occur when the system allows the existence of Fabry-Perot resonances. The purpose of this paper is to verify this idea and to show that Fabry-Perot resonances do suppress localization not only at some discrete resonance set of frequencies/energies but within some wide neighborhood of them. This phenomena does not appear in the most commonly studied systems, such as disordered Kronig-Penney models with delta-like potentials. In the next section we define the model and describe the method of calculating the spectral transmission. In Sect. 3, we present our numerical results, comparing them with the analytical expression recently derived [6] for the average transmission, and draw conclusions. 2 The model and its transfer matrix We consider an array formed by two alternating dielectric slabs with refractive indices na and nb placed in an electromagnetic metallic-wall waveguide of constant width w and height h. The layers with the ∗ Corresponding author E-mail: gluna@sirio.ifuap.buap.mx, Phone: +52 222 229 5610, Fax: +52 222 229 5611 c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 888 G. A. Luna-Acosta and N. M. Makarov: Fabry-Perot resonances in disordered 1D systems refractive index na (nb ), referred as the a-layers (b-layers), have lengths denoted, respectively, by da (n) and db (n). The positional disorder in our model consists in randomly varying lengths of only one type of layer, say the a-layer, such that da (n) = da + ση(n), da (n) = da , (1) db (n) = db . Here σ is the r.m.s deviation of da (n) and σ 2 its variance. The random sequence η(n) is assumed to be uncorrelated, i.e., η(n)η(n ) = δnn , η(n) = 0. The angular brackets . . . stand for the statistical average over different realizations of the randomly layered structure, that is periodic on average with period d = da + db . We treat here only the case of lowest TE mode of frequency ν, where the electric field E is given by Ey = sin(πz/w)Ψ(x), Ex = Ez = 0. Within every a- or b-layer, the factor Ψ(x) obeys the 1D Helmholtz equation, with wave numbers 2π 2 2 na,b ν − (c/2w)2 . (2) ka,b = c The transfer matrix equation for the array of N cells with or without positional disorder can be presented as A+ N +1 A− N +1 = Q̂ N A+ 1 A− 1 (3) , where the transfer matrix Q̂N of the system is the product of the transfer matrices for each (a, b) cell, Q̂N = Q̂(N )Q̂(N − 1)..Q̂(n)...Q̂(2)Q̂(1). (4) ± th Here A± site. n and Bn are the complex amplitudes of the forward/backward traveling wave at the n The transfer matrix Q̂(n) of the nth (a, b) elementary cell has the following elements, Q11 (n) = [cos(kb db ) + iα+ sin(kb db )] exp[ika da (n)] = Q∗22 (n); (5a) Q∗12 (n) = iα− sin(kb db ) exp[ika da (n)] = Q21 (n). (5b) Here the asterisk stands for the complex conjugation and 1 ka kb α± = ± , α2+ − α2− = 1. 2 kb ka (6) The determinant of Q̂(n) is equal to unit, det Q̂(n) = 1. Note that the transfer matrix Q̂(n) differs from cell to cell only in the phase factor exp[ika da (n)]. In our following numerical simulations we have A− N +1 = 0. Thus the transmittance of N cells is given by + 2 N −2 TN ≡ |A+ N +1 /A1 | = |Q11 | (7) In the case of no disorder, σ = 0, the unperturbed transfer matrix Q̂(0) is described by Eq. (5) with da (n) replaced by the constant length da . The transmission through N identical cells is expressed in closed form as (see, e.g., [7]) (0) TN −1 (0) sin(N κd) 2 = 1 + Q12 , sin(κd) c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (8) www.ann-phys.org Ann. Phys. (Berlin) 18, No. 12 (2009) 889 Table 1 Parameter values of random displacement. Here d = da + db = 2da = 9.0 cms case very weak weak medium strong strongest √ σ/d = / 3 0.29 × 10−2 1.73 × 10−2 5.77 × 10−2 17.3 × 10−2 28.86 × 10−2 (cms) 0.5 × 10−2 3.0 × 10−2 10.0 × 10−2 30.0 × 10−2 50.0 × 10−2 (σ/d)2 8.33 × 10−5 3.00 × 10−4 3.33 × 10−3 3.00 × 10−2 8.33 × 10−2 where κ is the Bloch wave number defined by the conventional dispersion equation cos(κd) = cos(ka da ) cos(kb db ) − α+ sin(ka da ) sin(kb db ). (9) Expression (8) indicates that for the case without disorder the transmission is perfect (T (0)N = 1) for (0) all N cells when Q12 = −iα− sin(kb db ) = 0 or when sin(N κd)/ sin(κd) = 0. The former occurs at the Fabry-Perot resonances kb db = mπ in the b-layers. The latter produces N − 1 oscillations in each spectral band, associated with the total system length N d. 3 Results and conclusions Figure 1 shows transmission spectra, obtained using Eqs. (7) and (4), for representative values of random disorder for the case of 21 cells and Fig. 2 for the case of 420 cells. Table 1 lists the amount of disorder ≡| dan − da |max /d, (10) which is the relative value of the maximum deviation of da (n) from the average da . In Figs. 1 and 2 we have also plotted: 1) the frequency position of the Fabry-Perot resonance (dashed vertical lines) and 2) the average transmission (thick curves) derived analytically in [6], −1 |T N |th ≡ exp(−lloc ) = exp[−(σ/d)2 F (ν)N ], 1 0.8 TN 0.6 0.4 0.2 a) 8 1 0.8 0.6 0.4 0.2 8.5 9 9.5 10 10.5 0 10 c) −1 9 9.5 10 10.5 d) −2 −3 8 9 10 10 0 8 8.5 9 9.5 10 10.5 9 9.5 10 10.5 0 10 e) −1 −1 10 f) 10 −2 −2 10 10 −3 10 8.5 10 −3 TN 8 −1 −2 10 (11) b) 10 10 10 α2− sin2 (kb db ) . 2 sin2 (κd) 0 10 10 TN F (ν) = (ka d)2 −3 8 8.5 9 9.5 ν (GHz) www.ann-phys.org 10 10.5 10 8 8.5 ν( GHz) Fig. 1 (online colour at: www.annphys.org) Transmission spectra (thin solid lines) for 21 cells: (a) no disorder, (b) very weak, (c) weak, (d) medium, (e) strong, and (f) strongest disorder (see table). The theoretical average transmission is plotted in thick dotted lines. The dashed line marks the position of the Fabry-Perot resonance in the shown frequency range. c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 890 G. A. Luna-Acosta and N. M. Makarov: Fabry-Perot resonances in disordered 1D systems 1 0.8 TN 0.6 0.4 0.2 a) 1 0.8 0.6 0.4 0.2 8 8.5 9 9.5 10 10.5 0 10 10 c) −2 8.5 9 9.5 10 10.5 10 d) 8 8.5 9 9.5 10 10.5 9 9.5 10 10.5 0 10 e) −1 f) −1 10 −2 −2 10 10 −3 10 10.5 −3 8 10 N 10 −2 0 T 9.5 10 −3 10 9 10 10 10 8.5 −1 10 N 8 0 −1 T b) −3 8 8.5 9 9.5 10 10.5 10 ν (GHz) 8 8.5 Fig. 2 (online colour at: www.annphys.org) Transmission spectra (thin solid lines) for 420 cells: (a) no disorder, (b) very weak, (c) weak, (d) medium, (e) strong, and (f) strongest disorder. The theoretical average transmission is plotted in thick dotted lines. The dashed line marks the position of the Fabry-Perot resonance in the shown frequency range. ν (GHz) Inspection of these figures allows to conclude that: i) transmission is almost perfect in a frequency neighborhood containing a Fabry-Perot resonance, whereas the other bands tend to disappear with disorder; 2) for a fixed amount of disorder, the transmission is larger, the larger the array is, 3) the analytical expression (11) describes correctly the average transmission for weak disorder even for an array with as few as 20 cells, and 4) this expression works fine for larger arrays even for medium disorder up to a certain frequency. Acknowledgements We acknowledge partial support from CONACyT, convenio P51458. References [1] P. W. Anderson, Phys. Rev. 109, 1492 (1958). [2] A. Sanchez and f. Adame, J. Phys. A: Math. Gen 27, 3725 (1994). [3] U. Kuhl, F. M. Izrailev, A. A. Krokhin, and H.-J. Stöckmann, Appl. Phys. Lett. 77, 663 (2000); F. M. Izrailev, A. A. Krokhin, and S. Ulloa, Phys. Rev. B 63, 041102 (2001); F. M. Izrailev and N. M. Makarov, J. Phys. A 38, 10613 (2005); U. Kuhl, F. M. Izrailev, and A. A. Krokhin, Phys. Rev. Lett. 100, 126402 (2008). [4] F. A. B. F. de Moura and M. L. Lyra, Phys. Rev. Lett. 84, 199 (2000); J. M. Luck, Phys. Rev. B 39, 5834 (1989). [5] P. Carpena, P. Bernaola-Galvan, P. Ch. Ivanov, and E. Stanley, Nature 418, 955 (2002). [6] G. A. Luna-Acosta, F. M. Izrailev, N. M. Makarov, U. Kuhl, and H.-J. Stöckmann, Phys. Rev. G 80, 114112 (2009). [7] D. J. Griffiths and C. A. Steinke, Am. J. Phys. 69, 137 (2000). c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org

1/--страниц