The role of absorption on the propagation of electromagnetic waves in disordered multilayer structures

Solid State Communications,

Pergamon

003%1098(95)00392-4 THE ROLE OF ABSORPTION

Foundation

Vol. 95, No. 12, pp. 855-857, 1995 Elsevier Science Ltd Printed in Great Britain 00381098/95 $9.50+.00

ON THE PROPAGATION OF ELECTROMAGNETIC MULTILAYER STRUCTURES

WAVES IN DISORDERED

A. Kondilis and E.N. Economou for Research and Technology, P.O. Box 1527,711lO Heraklion, Crete, Greece Physics Department, University of Crete, Crete, Greece

(Received

17 April 1995; accepted 24 May 1995 by F. Yndurain)

The role of absorption on the propagation of light in disordered binary multilayer structures is examined. Both localization (disorder) and dissipation (absorption) affect the wave propagation. The numerical calculations performed indicate that the inverse of the joint extinction length is a linear function of the imaginary part of the complex refractive index in a wide range of absorption values. An analytical expression derived in the context of an approximate method agrees well with the exact numerical results. Keywords

: A. disordered systems

D. order-disorder

effects

well and nonabsorbing barrier layers. Thus absorption is introduced in the system through the imaginary part k, of the comnlex refractive index of the well. This is a realistic assumption since in practice the well material is much more dissipative than the one of the barrier. On the basis of our results l/ej varies in such a way as to preserve, over a wide range of absorption values, a linear dependence on k,: l/ej = l/e + b k, (2) where b is a function of, generally, all the parameters characterizing the system in question except k,. We note that the behavior expressed by eqn. (2) is valid for oblique incidence as well. However, in this case we must replace k, with its generalized form:

Recently, there have been several studies’-5 concerning light propagation through multilayer (ML) structures in the presence of disorder. Absorption is usually neglected in the theoretical calculation in order to simplify the problem and achieve a clear understanding of the effects caused by disorder. In this case the quantity of prime interest is the localization length, e, i.e. the extinction length of the exponentially4 decaying wave amplitude. Simplifications relating to the dimensionality (one dimension) of the problem make it feasible to obtain approximate analytical expressions5 for the localization length and its variance, in good agreement with accurate numerical results. In treating the problem of light localization in disordered ML’s, the starting uoint is usuallv a binarv periodic ML into.which disorde; is introduced in the form of either a randomly fluctuating dielectric constant’-3 or variable layer thicknesss-5. It is well known that the frequency spectrum of a periodic ML exhibits regions of free propagation, photonic bands, interrupted by photonic gaps. For frequencies inside a gap, the wave does not propagate, since Bragg reflections cause the wave amplitude to decay exponentially with depth. If absorption is present, the wave amplitude decreases exponentially with

K,=

Im ((Gi-

sin’t1)‘12)

where n, is the complex refractive index of the well and 8 is the angle of incidence. The joint extinction length @jis defined7 as: l/ej = -&it+ /N (4) where tN is the transmission amplitude of a ML consisting of N cells. Brackets denote ensemble averaging and @jis expressed in units of the average cell thickness, o= + , where D, (Db) is the well (barrier) thickness. For simplicity we introduce disorder only in the thickness of the well layers assuming the barrier layers’ thickness fixed. In all calculations we adjust the magnitude of the ensemble used and the number of cells N as to make sure that , and consequently I/!,, is evaluated with an accuracy of 3% or better. The density of states (DOS)8 of the periodic system that is formed by putting disorder and absorption equal to zero is shown in Fig. 1. The computation assumes normal incidence. The DOS is nlotted versus freauencv. the latter in form of the phase shift q=O.lDb/C, where c is the wave velocity inside the barrier layer. The DOS is a periodic function of cp with period Z. One observes regions of finite DOS, the photonic bands, separated by regions of zero DOS, the photonic gaps. There is a single gap located at the centre of each

depth even in the bands. The characteristic length, e,, of this exponential attenuation in a periodic ML is a measure of absorption as long as the frequency belongs to a band. Inside gaps e, is due to both absorption and destructive Bragg reflections and consequently remains finite even for zero absorption. In recent publication&4 the joint effect of disorder and absorption is considered. On the basis of numerical results, for frequencies outside gaps, a simple relationship is suggested reminding Matthiessen’s rule6 for noninteracting scattering events. It relates the joint extinction length !i in the presence of both the disorder and absorption to the localization length I! and the absorption length e,,. It reads:

1

lh?j= l/e + l/e, (1) In this paper on the basis of numerical calculations, we show that the validity of eqn. (1) is very limited. We have studied a binary disordered ML consisting of absorbing

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Fig. 1: Density of states per cell thickness, versus phase shift of the wave across the barrier layer, for a binary periodic ML. The ratio of the well refractive index to that of the barrier, equals c,=2. The optical paths n,D, (m=w, b) in the two types of layers are equal. Arrows indicate the locations at which the curves of Fig. 2 are calculated.

zone. For our purposes it is sufficient to restrict the presentation of results within the first zone (O>l), one observes a common behavior in all three curves of Fig. 2. They all vary according to the equation: [,=[a. Such a behavior is qualitatively understood because as absorption grows up, a uoint is reached at which the whole effect nracticallv takes place within the first absorbing layer near&the free surface of the ML. Then obviously !$a. As absorption decreases there is a smooth variation of the curves till the zero absorption (i.e. k,=O) point is reached. Its abscissa value is zero in the case of curves A and B since outside the gap the extinction length in a neriodic ML, in the absence of absorvtion. is infinite. inside the gap, even if absorption is zero, this length is finite as apparent by the abscissa of the lowest end point of curve C. The ordinate of the zero absorption point, for all curves of Fig. 1, has a finite value that equals the inverse of the disorder (or gap) localization length in each case. Since for l/&O, l/!j#O, and for I&>>1 1/$+1/e,, the relation l/ej vs l/t!, cannot be rigorously linear. However, the closer

Fig. 2 : Joint extinction constant plotted versus absorption constant. Symbols correspond to exact numerical data. Solid linesare a guide to the eye. The complex well refractive index n,=4-ik, and the barrier index nh=2. The variable thickness of the well layers obeys a Gaussian distribution with standard deviation o,=1/6 in units of average cell thickness. The optical path inside a barrier layer equals the path in the average-thickness well layer. As for the phase shift across any barrier layer: (p=O.2 J-C(deep in the band; curve A), (p=O.61 JC(band edge; curve B) and (p=OS n; (gap; curve C).

the point (l/e,,

l/t!,) at zero absorption

0), the closer the eqn. (1) is a good of such a behavior Another way

is to the origin (0,

curve l/!j vs l/e, is to a straight line and approximation. A representative example is illustrated in Fig. 2 by curve A. of presenting the numerical data of Fig. 2

is illustrated in Fig. 3 where l/ej is plotted vs k,. The latter being a measure of net absorption in all cases, is advantageous to l/e, which includes, if in a gap, the effect of Bragg reflections as well. We observe an improved linear dependence of l/ej versus k, as compared to Fig. 2. The pronounced curvature at low absorption, observed in the l/ej versus I/I!, graph of Fig. 2 is absent here. Inside the band (left hand side graph) the linear dependence is very clear. Inside the gap (right hand side graph) the curve slightly deviates from a straight line. Such deviation is observed in the band as well if we extend calculations to larger k, values. We have performed several calculations for small and large disorder at different angles of incidence. One must keep in mind that a measure of disorder is the standard deviation om of the phase shifts of the random thickness well layers. The general trend observed is that eqn. (2) (modified appropriately through eqn. (3) at oblique incidence) becomes more successful with increasing disorder. For o+
PROPAGA~ON

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where k~ is the light wavenumber in vacuum, D, and Db stand for the well and barrier variable layer thicknesses respectively X,=exp

0

2

1

3

4.l

Fig. 3: Joint extinction constant plotted versus the imaginary part of the complex well refractive index. Symbols and parameter values are identical to those of Fig. 2. The solid lines are the result of the computations performed by using eqn. (5). a range in its absorption coefficient that extends from 0 to about IO6 cm-l. This range, in a-Si:H, extends far beyond its absorption edge, the most important and thereby intensively investigated topic of this specific material. The dependence of l/L?1on abso~tion can be derived analytically as well in the context of an approximation method. The method consists in expanding enIt,& defined above, in a series of Fresnel reflection coefficients (FRC’s, associated by definition with a single interface) and keeping terms as high as second order. The products of FRc’s for increasingly higher order terms tend to zero since their moduli are always smaller than one. Besides, the higher the order the higher the number of random phase shifts that enter the calculation. Then one expects a decreasing contribution in the ensemble averaged expansion of enltNl with increasing term order. The method is described in detail in ref. 5 where the excellent agreement between the approximate analytical expression derived for e and the accurate numerical results leaves no doubt about its validity. In the present paper this approach is extended as to take into account besides the disorder the absorption as well. By assuming that the layer thicknesses are statistically independent we obtain through _ (4): l/~j = ~~~)

+ ~kb~b} i- Re (3 (‘~~~~~~~”

) (5)

and ro is the FRC at a well-barrier

interface.

(-Zi G,,,) where 6 w is the complex phase shift”’ ’ ’ of

&e wave inside a well layer (G = k~ [g-sin2 6]2’2 D,]. Xb is given by -a similar extiression for the barrier. N%e that Eq. (5) is valid for oblique incidence as well. Details on this derivation along with a scrutiny of its successes and failures will be presented in another publication. Solid lines in Fig, 3 are the result of the analytical expression (5) applied for nonabsorbing fixed thickness barrier layers and zero angle of incidence. The good agreement with exact numerical data is evident. Relation (5) holds well even inside the gap as shown in case C. We expect a lesser success of (5) in a gap, when disorder and absorption is low. The reason is that for low absorotion and disorder. and, especially inside the gap, the Multiple reflections; neglected in deriving eqn. (5), become more important. On the basis of the success of relation (5) in interpreting the numer$+l results, we proceed to obt& the analytical behavior of ej in the high and low absorption limits. The low limit, that reveals the linear dependence of l/ej on k, is obtained by keeping only first order terms in k,. The proportionality factor of k, is a complicated function of wavelength, disorder and other parameters. It takes a simple form if disorder is high, i.e. if the variance of the well phase shifts is much larger than unity. Then we obtain:

(6) where ro is the FRC in the absence of absorption. In the high absorption limit we reach the following equation: l/e1 = b k, + 2 sin2 ri

(7)

The harmonic term is the con~ibution of the nonabsorbing, fixed thickness, barrier layers. The first term is due to wave dissipation inside the absorbing well layers. The validity of eqn’s (6) and (7) has been tested with success in many cases. Through eqn. (5) one can obtain e, by reducing disorder to zero. In the high absorption limit it is easily proved that e, obeys the same equation as $ does. Thus the already known behavior according to which, at high absorption, !j=&+ is corroborated ~~ytically as welf.

REFERENCES 1. P. Sheng, B. White, Z.Q. Zhang and G. Pap~icolaou, Scntteriraa and L~~ali~ati5n of Classical Waves in Random bedia, p. 563. World Scientific, Signapore (1990). 2. C. Martijn de Sterke and R.C. McPhedran, Phys. Rev. B47, 7780 (1993). 3. A.R. McGurn, K.T. Christensen. F.M. Miiller and A.A. Maradudin, Phys. Rev. B47, 13 120 (1993). 4. A. Kondilis and P. Tzanetakis. Phvs. < Rev. B46. 15426 (1992). 5. A. Kondilis and P. Tzanetakis, J. Opt. Sot. Am. All, 1661 (1994).

6. N.W. Ashcroft and N.D. Mermin, Solid State Physics, p. 3’23. BRW, Philadelphia (1976). 7. One might adopt other definitions as well (see ref. 5). However the one used here is more appropriate. 8. The density of states is calculated using Bloch’s theorem, in a way analogous to that of the onedimensional electronic case of the Kronig-Penney model. 9. Z. Knittl. Ootics of thin films. , 1 v. 186. Wilev.i New York (19?6).* 10.B. Harbecke, Appl. Phys. B39, 165 (1986).