document: during the last decades , a new type of artificial materials , the so - called left - handed metamaterials ( lh ) , have attracted a great deal of attention . 
   they present negative indices of refraction for some wavelengths @xcite , with considerable applications in modern optics and microelectronics @xcite . 
   metamaterials can resolve images beyond the diffraction limit @xcite , act as an electromagnetic cloak @xcite , enhance the quantum interference @xcite or yield to slow light propagation @xcite . 
   regarding the localization length in disordered systems , the presence of negative refraction in one - dimensional ( 1d ) disordered metamaterials strongly suppresses anderson localization @xcite . as a consequence , 
   an unusual behavior of the localization length @xmath0 at long - wavelengths @xmath5 has been observed . 
   et al . _ reported a sixth power dependence of @xmath0 with @xmath5 under refractive - index disorder @xcite instead of the well - known quadratic asymptotic behavior @xmath10 @xcite . 
   recently , mogilevtsev _ et al . 
   _ @xcite have also found a suppression of anderson localization of light in 1d disordered metamaterials combining oblique incidence and dispersion while torres - herrera _ et al . 
   _ @xcite have developed a fourth order perturbation theory to resolve the problem of non - conventional anderson localization in bilayered periodic - on - average structures . 
   the effects of polarization and oblique incidence on light propagation in disordered metamaterials were also studied in ref . 
   @xcite .    in this article , we calculate numerically the localization length of light @xmath0 for a one - dimensional arrangement of layers with index of refraction @xmath1 and thickness @xmath2 alternating with layers of index of refraction @xmath3 and thickness @xmath4 . in order to introduce disorder in our system 
   , we change the position of the layer boundaries with respect to the periodic values maintaining the same values of the refraction indices @xmath1 and @xmath3 . 
   this is the case of positional disorder , in contrast to the compositional disorder where there exist fluctuations of the index of refraction @xcite . 
   two structures will be analyzed in detail : homogeneous stacks ( h ) , composed entirely by the traditional right - handed materials ( rh ) with positive indices of refraction , and mixed stacks ( m ) with alternating layers of left- and right- handed materials . 
   for the sake of simplicity , the optical path in both layers will be the same , that is , the condition @xmath9 is satisfied in most of the work . 
   these periodic - on - average bilayered photonic systems have already been studied analytically by izrailev _ 
   these authors have developed a perturbative theory up to second order in the disorder to derive an analytical expression for the localization length for both h and m stacks . in our case , we have obtained two equations for the localization length @xmath0 as a function of the wavelength @xmath5 from our numerical results . for h stacks , a quadratic dependence of @xmath0 for long - wavelengths 
   is found , as previously reported in the literature . on the other hand , the localization length saturates for lower values of @xmath5 . 
   an exhaustive study of @xmath0 in the allowed and forbidden bands ( gaps ) of weakly disordered systems will be carried out . 
   we will show that the localization length is modulated by the corresponding bands and this modulation decreases as the disorder increases . for low - disordered m stacks and wavelengths of several orders of magnitude greater than the grating period @xmath11 
   , the localization length @xmath0 depends linearly on @xmath5 with a slope inversely proportional to the modulus of the reflection amplitude between alternating layers . 
   the plan of the work is as follows . in sec . 
   ii we carry out an exhaustive description of our one - dimensional disordered system and the numerical method used in our localization length calculations . 
   a detailed analysis of @xmath0 in the allowed bands and gaps of homogeneous stacks is performed in sec . 
   iii where a practical expression for the localization length as a function of @xmath5 and the disorder is derived . in sec . 
   iv we calculate @xmath0 for mixed stacks of alternating lh and rh layers . 
   a linear dependence of the localization length at long - wavelengths is found for low - disordered m stacks . 
   finally , we summarize our results in sec .  v. 
   let us consider a one - dimensional arrangement of layers with index of refraction @xmath1 alternating with layers of index of refraction @xmath3 . 
   the width of each one is the sum of a fixed length @xmath12 for @xmath13 and a random contribution of zero mean and a given amplitude . 
   the wave - numbers in layers of both types are @xmath14 , where @xmath15 is the frequency and @xmath16 the vacuum speed of light . 
   as previously mentioned , the grating period of our system @xmath17 is defined as the sum of the average thicknesses @xmath2 and @xmath4 of the two types of layers , that is , @xmath18 . 
   we have introduced the optical path condition @xmath19 for simplicity ( in the case of left - handed layers @xmath20 , so the absolute value has been written to consider these type of materials ) . without disorder , each layer would be limited by two boundaries @xmath21 and @xmath22 where @xmath23 is the total number of boundaries . 
   the periodic part of the system considered is schematically represented in fig . 
   [ fig1 ] .     and thickness @xmath2 alternating with layers of index of refraction @xmath3 and thickness @xmath4 . 
   the grating period is @xmath11 . ]    in the presence of disorder , the position of the corresponding boundaries are @xmath24 except for the first and the last boundary , so as to maintain the same total length @xmath25 . 
   the parameters @xmath26 are zero - mean independent random numbers within the interval @xmath27 $ ] . throughout all our calculations 
   , we have chosen values of the disorder parameter @xmath28 less than @xmath2 and @xmath4 . 
   for each @xmath25 , we calculate the transmission coefficient of our structure @xmath29 and average its logarithm , @xmath30 , over 800 disorder configurations . 
   then , we obtain numerically the localization length @xmath0 via a linear regression of @xmath30 @xcite @xmath31 here , the angular brackets @xmath32 stand for averaging over the disorder . 
   we choose 6 values of the total length @xmath25 to perform the linear regression of eq.([linearregr ] ) . 
   the localization length @xmath0 is evaluated as a function of the disorder parameter @xmath28 and the frequency of the incident photon @xmath15 . 
   we calculate the transmission coefficient of our system via the characteristic determinant method , firstly introduced by aronov _ 
   this is an exact and non perturbative method that provides the information contained in the green function of the whole system . in our case 
   , the characteristic determinant @xmath33 can be written as @xcite @xmath34 where the index @xmath35 runs from 1 to @xmath23 and the coefficients @xmath36 and @xmath37 can be written as @xmath38 and @xmath39 the parameters @xmath40 , which are the reflection amplitudes between media @xmath41 and @xmath35 , are given by @xmath42 where @xmath43 corresponds to the impedance of layer @xmath35 and can be be expressed for normal incidence in terms of its dielectric permittivity @xmath44 and magnetic permeability @xmath45 @xmath46    the quantity @xmath47 entering eqs . 
   ( [ aj ] ) and ( [ bj ] ) is a phase term @xcite @xmath48.\ ] ] here @xmath49 is the wave - number in a layer with boundaries @xmath50 and @xmath51 . 
   this recurrence relation facilitates the numerical computation of the determinant . 
   the initial conditions are the following @xmath52 the transmission coefficient of our structure @xmath29 is given in terms of the determinant @xmath53 by @xmath54 
   before dealing with mixed stacks , we present results for low - disordered homogeneous systems with underlying periodicity , which has not been previously studied . in this section 
   we perform a detailed analysis of the localization length @xmath0 in the allowed bands and in the forbidden gaps of disordered h stacks as a function of the disorder @xmath28 , the incident wavelength @xmath5 and the reflection coefficient between alternating layers @xmath55 . 
   as it is well known , in the absence of disorder the transmission spectrum of right - handed systems presents allowed and forbidden bands whose position can be easily determined via the following dispersion relation obtained from the bloch - floquet theorem @xcite @xmath56 where @xmath57 is the block wave - vector . when the modulus of the right - hand side of eq . 
   ( [ beta ] ) is greater than 1 , @xmath57 has to be taken as imaginary . 
   this situation corresponds to a forbidden band . 
   taking into account the condition @xmath7 , eq . 
   ( [ beta ] ) reduces to @xmath58 on the other hand , when @xmath59 is equal to unity , the incident frequency @xmath15 is located at the center of the @xmath60-th allowed band , @xmath61 . 
   after some algebra , we obtain from eq . 
   ( [ betasimpl ] ) @xmath62    let us first consider a periodic h stack formed by 50 layers of length @xmath63 52.92 nm and index of refraction @xmath64 1.58 alternating with 49 layers of length @xmath65 39.38 nm and @xmath66 2.12 . 
   the total size of our structure is 4.57 @xmath67 m and the reflection coefficient between alternating layers 0.05259 . 
   fig.[fig2](a ) represents the transmission coefficient @xmath29 as a function of the frequency @xmath15 to illustrate its behavior . 
   also shown are the center of each allowed band calculated via eq.([omegacenter ] ) . 
   there are 99 peaks in each band so they can hardly been resolved on the scale used . 
   moreover , in fig.[fig2](b ) the parameter @xmath59 is plotted versus the frequency @xmath15 for this periodic system . the first gap and the first allowed band have been shown for a better comprehension .     and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the homogeneous periodic system described in the text ( 99 layers ) . 
   ]    a systematic numerical simulation of a realistic system with 50000 layers has been carried out . 
   the parameters are the same as in the previous example . in fig . 
   [ fig3 ] we represent the localization length @xmath0 versus the wavelength @xmath5 for different values of the disorder parameter @xmath28 ( shown in the legend of the figure ) . 
   the dashed line corresponds to  total disorder  , that is , an arrangement of layers with random boundaries and alternating indices of refraction @xmath1 and @xmath3 . several features are evident in the figure . for long - wavelengths 
   , one observes a quadratic asymptotic behavior , as can be compared with the dotted line @xcite . 
   an in - deep numerical analysis of the coefficient characterizing this dependence has been performed . to this aim , 
   20 different h stacks were considered and the following expression for the localization length was found @xmath68 where @xmath69 is the optical path across one grating period @xmath17 . 
   all the lengths in eq.([longlochinf ] ) are expressed in units of @xmath17 . in the opposite limit of short @xmath5 , 
   the localization length @xmath0 saturates to a constant value @xcite . 
   our numerical results have shown that this constant is proportional to the inverse of the reflection coefficient between alternating layers @xmath70 , that is , @xmath71    izrailev _ et al . _ @xcite have developed a perturbative theory up to second order in the disorder to calculate analytically the localization length in both homogeneous and mixed stacks . 
   this model is quite general and is valid for both quarter stack medium ( mainly considered in our work ) and systems with different optical widths . 
   assuming uncorrelated disorder and random perturbations with the same amplitude in both layers ( the main considerations in our numerical calculations ) one can easily derive the following analytical expression for @xmath0 at long - wavelengths from izrailev s formulation @xmath72 for similar values of the layer impedances @xmath73 , the first term in eq.([izragen ] ) can be approximated by @xmath74 and @xmath75 , so eq.([izragen ] ) reduces to @xmath76 which is similar to our numerical expression eq . 
   ( [ longlochinf ] ) . 
   the randomness only affects partially the periodicity of the system , which manifests in the existence of bands and gaps . 
   the localization length depends on the position in the band and on the disorder . 
   the modulation of @xmath0 by the bands can be clearly appreciated in fig . 
   these results are consistent with other published works on this topic @xcite . 
   recently , mogilevtsev _ et al . 
   _ @xcite have reported that the photonic gaps of the corresponding periodic structure are not completely destroyed by the presence of disorder while luna - acosta _ et al . 
   _ @xcite have shown that the resonance bands survive even for relatively strong disorder and large number of cells .     versus the wavelength @xmath5 for different values of the disorder parameter @xmath28 . 
   the h stack corresponds to the arrangement represented in fig . 
   [ fig2 ] but now 50000 layers have been considered . the dashed line stands for the  total disorder  case . all lengths are expressed in units of the grating period @xmath17 . ]    having a close look into the first gap in fig . 
   [ fig3 ] , one observes that the localization length is practically independent of the disorder @xmath28 . in order to visualize this effect , fig.[fig4 ] 
   represents ( a ) the first and ( b ) the second gaps depicted in fig . 
   [ fig3 ] . as mentioned , the dependence of @xmath0 with the disorder is almost negligible in the first gap . when the wavelength is similar to the grating period @xmath17 , the influence of the disorder is greater , as can be easily deduced from simple inspection of fig . 
   [ fig4](b ) .     versus the wavelength @xmath5 for ( a ) the first and ( b ) the second gaps depicted in fig .  [ fig3 ] . 
   ]    let us now focus on the allowed bands and study in detail the behavior of the localization length in these regions . to this aim , 
   three - dimensional ( 3d ) graphs of @xmath0 versus the wavelength @xmath5 and the disorder @xmath28 have been plotted in fig.[fig5 ] for ( a ) the first and ( b ) the third allowed bands ( see again fig . 
   [ fig2 ] ) . 
   all this magnitudes have been normalized to the grating period @xmath17 . 
   the localization length @xmath0 is enhanced in a small region around the center of each allowed band . 
   a similar result was found by hernndez - herrejn _ et al . 
   _ @xcite who obtained a resonant effect of @xmath0 close to the band center in the kronig - penney model with weak compositional and positional disorder . 
   this increase in the localization length is due to emergence of the fabry 
   perot resonances associated with multiple reflections inside the layers from the interfaces @xcite . in particular , for homogeneous quarter stack systems , the fabry 
   perot resonances arise exactly in the middle of each allowed band where @xmath57 vanishes @xcite . 
   the saturation of @xmath0 for short - wavelengths is also appreciated in these 3d images .     versus the wavelength @xmath5 and the disorder @xmath28 for ( a ) the first and ( b ) the third allowed bands . 
   the h stack is the same as in fig . 
   all lengths are expressed in units of the grating period @xmath17.,scaledwidth=47.0% ]    up to now , h stacks with the same optical path in layers of both types have been considered , that is , arrangements verifying the condition @xmath9 in the absence of disorder . as a consequence , 
   the transmission spectrum @xmath29 of the corresponding periodic system presented a symmetric distribution of allowed bands and gaps ( as previously shown in fig . 
   [ fig2 ] ) . 
   what happens in the case of a non - symmetric band distribution , that is , when the condition @xmath9 is not satisfied ? to answer this question , we have plotted the transmission coefficient @xmath29 ( fig . 
   [ fig6](a ) ) and the parameter @xmath77 ( fig . 
   [ fig6](b ) ) versus the frequency @xmath15 for a periodic h stack formed by 50 layers of length @xmath63 52.92 nm and index of refraction @xmath64 1.58 alternating with 49 layers of length @xmath65 28.80 nm and @xmath66 2.12 . 
   note that the condition @xmath7 is no longer held , so the band structure is asymmetric . 
   accordingly , the localization length @xmath0 shown in fig.[fig6](c ) presents an irregular form in the allowed and forbidden bands . as in the symmetric case , no band modulation exists for high disorders and the quadratic asymptotic behavior for long - wavelengths is also verified . 
   moreover , the peaks in the localization length due to fabry  perot resonances still can be appreciated , although they are no longer in the center of the bands @xcite . 
   a total number of 50000 layers was considered in our localization length calculations .     and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the asymmetric periodic h stack described in the main text ( 99 layers ) and ( c ) the corresponding localization length @xmath0 versus the wavelength @xmath5 for different disorder parameters @xmath28 ( 50000 layers ) . 
   once analyzed in detail the behavior of the localization length @xmath0 for homogeneous systems , let us now deal with m stacks composed of alternating lh and rh layers .    in our numerical calculations 
   we have considered a periodic m stack formed by 50 layers of length @xmath63 52.92 nm and index of refraction @xmath64 -1.58 alternating with 49 layers of length @xmath65 39.38 nm and @xmath66 2.12 . 
   again , the condition @xmath19 has been imposed . 
   note that this arrangement has similar parameters than the one depicted in sec . 
   iii , but now @xmath1 is negative . 
   this change of sign results in a severe modification of the transmission coefficient @xmath29 , as we will show immediately . for this the periodic system , fig . 
   [ fig7 ] represents ( a ) the transmission coefficient @xmath29 and ( b ) the parameter @xmath59 versus the frequency @xmath15 of the incident light . unlike the h stack case , 
   no allowed bands exist and practically the entire transmission spectrum is formed by gaps 
   . a set of periodically distributed lorentzian resonances is found instead . 
   the position of the center of each resonance is given by eq . 
   ( [ omegacenter ] ) , that is , the center of the allowed bands in homogeneous systems .     and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the mixed periodic system described in the text ( 99 layers ) . ]    in respect to the localization length , positional disorder was introduced as explained in sec . 
   as previously considered , the total number of layers in our numerical calculations was 50000 and the number of disordered configurations to average the logarithm of the transmission coefficient was 800 . the result is shown in fig . 
   [ fig8 ] where the localization length @xmath0 is represented versus the wavelength @xmath5 for different values of the disorder parameter @xmath28 . 
   the dashed line corresponds to the  total disorder  case . 
   again , for long - wavelengths a quadratic asymptotic behavior of @xmath0 is found , but now a region where the localization length is proportional to @xmath5 exists . 
   we will turn to this point in the next figure to quantify the slope of this linear dependence . 
   as it is noticed , the lorentzian resonances associated with multiple reflections in the layers modulate the shape of @xmath0 and this modulation decreases as the disorder increases . 
   moreover , the saturation of the localization length for low - wavelengths can also be appreciated . as in the h stack case , the constant where @xmath0 saturates is proportional to the inverse of the reflection coefficient between alternating layers @xmath70 .     versus the wavelength @xmath5 for different disorder parameters @xmath28 . 
   the m stack corresponds to the one represented in fig . 
   [ fig7 ] but here 50000 layers have been considered . 
   the dashed line stands for the  total disorder  case . ] 
   the linear dependence of @xmath0 with the wavelength @xmath5 has been exhaustively studied by our group to find a simple analytical expression for the localization length in this region . 
   more than 30 different m stacks have been simulated and we have arrived at the following empirical equation @xmath78 where @xmath0 , @xmath5 and @xmath79 are expressed in units of the grating period @xmath17 . in fig . 
   [ fig9 ] , our numerical calculations of the slope @xmath80 versus @xmath81 have been plotted for several values of @xmath79 , triangles ( 1.25 ) , squares ( 3.25 ) and circles ( 7.55 ) . 
   the solid lines correspond to the results obtained via eq . 
   ( [ longlocm ] ) . 
   one notices a good degree of validity for a wide range of @xmath81 values .     versus @xmath81 for several values of @xmath79 ( expressed in units of the grating period @xmath17 ) . 
   the solid lines correspond to the results obtained via eq.([longlocm ] ) . ]    finally , let us now consider an asymmetrical m stack where the condition @xmath9 is no longer satisfied . in fig.[fig10 
   ] we have represented ( a ) the transmission coefficient @xmath29 and ( b ) the parameter @xmath59 versus the frequency @xmath15 for a periodic m stack formed by 50 layers of length of length @xmath63 52.92 nm and index of refraction @xmath64 -1.58 alternating with 49 layers of length @xmath65 28.80 nm and @xmath66 2.12 . note the strong difference between this transmission spectrum and the symmetrical one ( see fig . 
   [ fig7](a ) ) where a set of periodically distributed lorenztian resonances exists . despite this fact , the localization length @xmath0 shown in fig.[fig10](c ) presents a region of linear dependence with the wavelength , as in the symmetric case . 
   however , eq.([longlocm ] ) can not be used to evaluate the localization length in this region .     and ( b ) the parameter @xmath59 versus the frequency @xmath15 for the asymmetric periodic m stack described in the main text ( 99 layers ) and ( c ) the corresponding localization length @xmath0 versus the wavelength @xmath5 for different disorder parameters @xmath28 ( 50000 layers ) . 
   we have analyzed numerically the localization length of light @xmath0 for homogeneous and mixed stacks of layers with index of refraction @xmath82 and thickness @xmath2 alternating with layers of index of refraction @xmath83 and thickness @xmath4 . 
   the positions of the layer boundaries have been randomly shifted with respect to ordered periodic values . 
   the refraction indices @xmath1 and @xmath3 present no disorder .    for h stacks , the parabolic behavior of the localization length in the limit of long - wavelengths , previously found in purely disordered systems @xcite , has been recovered . 
   on the other hand , the localization length @xmath0 saturates for very low values of @xmath5 . 
   the transmission bands modulate the localization length @xmath0 and this modulation decreases with increasing disorder . 
   moreover , the localization length is practically independent of the disorder @xmath28 at the first gap , that is , it has a very low tendency in this region . 
   we have also characterized @xmath0 in terms of the reflection coefficient of alternating layers @xmath70 and the optical path across one grating period @xmath79 . 
   eq.([longlochinf ] ) has been proved to be valid for a wide range of @xmath70 values , that is , from transparent to opaque h stacks . 
   it has also been shown ( see fig .  [ fig5 ] ) that the localization length @xmath0 is enhanced at the center of each allowed band .    when left - handed metamaterials are introduced in our system , 
   the localization length behavior presents some differences with respect to the traditional stacks , formed exclusively by right - handed materials . for low - disordered m stacks and wavelengths of several orders of magnitude greater than the grating period @xmath17 
   , the localization length @xmath0 depends linearly on @xmath5 with a slope inversely proportional to the modulus of the reflection amplitude between alternating layers @xmath81 ( see eq . 
   ( [ longlocm ] ) ) . 
   as in the h case , @xmath0 saturates for low - wavelengths , being this saturation constant proportional to the inverse of @xmath70 . 
   if we take into account losses , there is an absorption term whose absorption length @xmath84 is @xcite @xmath85 where @xmath86 is an absorption coefficient . 
   the inverse of the total decay length is the sum of the inverse of the localization length @xmath0 plus the inverse of the absorption length @xmath87 . 
   note that @xmath84 is proportional to @xmath5 , so , for low - disordered m stacks and weak absorption metamaterials , the final expression for the localization length @xmath0 in the linear region can be written as @xmath88    in the case of both homogeneous and mixed stacks with non - symmetric band distribution , that is , when the condition @xmath89 is not satisfied , the localization length @xmath0 presents an irregular form in all the transmission spectrum . 
   these changes in @xmath0 are more sensitives in mixed stacks than in homogeneous structures . 
   a. yariv and p. yeh , optical waves in crystals , propagation and control of laser radiation ( wiley , new york , 1984 ) ; o. del barco , m. ortuo and v. gasparian , phys . 
   a * 74 * , 032104 ( 2006 ) ; o. del barco and m. ortuo , phys . 
   a * 81 * , 023833 ( 2010 ) .
summary: we have analyzed numerically the localization length of light @xmath0 for nearly periodic arrangements of homogeneous stacks ( formed exclusively by right - handed materials ) and mixed stacks ( with alternating right and left - handed metamaterials ) . 
   layers with index of refraction @xmath1 and thickness @xmath2 alternate with layers of index of refraction @xmath3 and thickness @xmath4 . 
   positional disorder has been considered by shifting randomly the positions of the layer boundaries with respect to periodic values . for homogeneous stacks , we have shown that the localization length is modulated by the corresponding bands and that @xmath0 is enhanced at the center of each allowed band . in the limit of long - wavelengths @xmath5 , 
   the parabolic behavior previously found in purely disordered systems is recovered , whereas for @xmath6 a saturation is reached . in the case of nearly periodic mixed stacks with the condition @xmath7 , instead of bands there is a periodic arrangement of lorenztian resonances , which again reflects itself in the behavior of the localization length . for wavelengths of several orders of magnitude greater than @xmath8 , 
   the localization length @xmath0 depends linearly on @xmath5 with a slope inversely proportional to the modulus of the reflection amplitude between alternating layers . when the condition @xmath9 is no longer satisfied , the transmission spectrum is very irregular and this considerably affects the localization length .