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

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Ann. 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
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