In a strongly scattering medium where Anderson localization takes place, constructive interference of local non-propagating waves dominate over the incoherent addition of propagating waves. This results in the disappearance of propagating waves within the medium, which significantly attenuates energy transmission. In this numerical study performed in the optical regime, we systematically found resonance modes, called eigenchannels, of a 2-D Anderson localized system that allow for the near-perfect energy transmission. We observed that the internal field distribution of these eigenchannels exhibit dense clustering of localized modes. This strongly suggests that the clustered resonance modes facilitate long-range energy flow of local waves. Our study explicitly elucidates the interplay between wave localization and transmission enhancement in the Anderson localization regime.
© 2012 OSA
The interference of multiply scattered waves in the disordered media leads to interesting phenomena such as the Anderson localization [1–4] and the existence of open eigenchannels [5, 6]. Unlike in the diffusion regime where transmission decreases linearly over the thickness, it decreases exponentially when the Anderson localization occurs. In the strongly scattering media, constructively interfering local waves dominate over the incoherently interfering propagating waves, which undermines the propagation of waves within the medium . On the contrary, there exist so-called open eigenchannels that allow perfect transmission through a disordered medium. The well-controlled coherent combination of input channels induces constructive interference throughout the medium and subsequently at the exit of the medium. Recent experimental studies performed in the optical regime proved the existence of open eigenchannels . And in our recent study, we experimentally demonstrated that the injection of light into high-transmission eigenchannels enhances energy transmission . The random matrix theory [5, 8] predicts that open eigenchannels exist regardless of the degree of disorder. This implies that they will exist even in the Anderson localization regime. In fact, the existence of the high-transmission eigenchannels has been observed in a recent study performed in the microwave regime for the disordered wave guide exhibiting Anderson localization . We investigate this rather intriguing combination of wave localization and transmission enhancement, both originated from the constructive interference, to study the way open eigenchannels manifest themselves in the Anderson localization regime.
In our previous study, we introduced the procedure of finding open eigenchannels for a medium in the diffusion regime and visualized their field distribution within the medium . It was observed that open eigenchannels store more energy within the medium than that of the plane wave illumination and the internal intensity is enhanced across the entire sample. As to be shown in the following, however, the propagating waves that used to be prevalent in the diffusion regime disappear in the Anderson localization regime. Therefore, the mechanism of the transmission enhancement is expected to be dramatically different in the strongly scattering medium from that in the weakly scattering medium. We note that there have been studies performed for a 1D disordered medium that suggest the mechanism of the transmission enhancement in the strongly scattering medium [11, 12]. According to these studies, the propagation of energy can be mediated by the coupling of neighboring localized modes, which forms nonlocalized modes called necklace states.
In this Letter, we investigate the internal field distribution of open eigenchannels for the Anderson localized systems. In the optical regime, we numerically prepare a highly disordered 2D medium whose thickness exceeds the localization length of the medium. Using the finite-difference time-domain (FDTD) method, we compute electromagnetic wave propagation through the medium and construct a transmission matrix, t. We observed that the eigenvalues of the matrix tt+, which are the transmittance of the corresponding eigenchannels, reaches to almost unity, thereby finding the open eigenchannels of the medium. Moreover, we compute the field distribution of open eigenchannels within the medium to find the mechanism of transmission enhancement. We found that clustering of localized modes mediates the coupling of waves such that the energy can efficiently propagate toward the output plane.
2. Preparation of disordered media exhibiting Anderson localization
For a disordered medium, we numerically prepare a 2D medium made of absorption-free dielectric square particles with a side length of 200 nm (Fig. 1(a) ). The width of the medium in the x-direction (transverse direction) and the thickness in the z-direction are 130 μm and 8 μm, respectively. The particles are randomly distributed in the vacuum with a fill factor of 30.0 ± 0.5%. The incident light is a monochromatic plane wave of wavelength λ = 600 nm and the polarization of the light is perpendicular to the x-z plane. The numerical cell size is set as 10 nm, which is small enough compared with the wavelength to ensure the accuracy of the computation. The propagation of the wave is computed until the steady state is established. To obtain the complex electric field both within and outside the medium, a phase-shifting interferometry method is used . For each position, we record the electric field four times with an interval of one-quarter of an optical oscillation and process them to acquire the amplitude and phase of the wave.
In order to control the degree of disorder, we vary the refractive index, np, of the particles constituting the medium. To check whether the medium is in the strong localization regime or not, we survey the relationship between the transmittance and the thickness of the medium. In the Anderson localization regime, the transmittance of the incident wave decreases exponentially with the increase of the thickness. Therefore, the localization length, , is defined by the equation, , where T and L stand for the average transmittance and thickness, respectively. We calculate the transmittance of the disordered medium of a given np at various thicknesses (Fig. 1(b)). The medium of larger np has a steeper decrease in transmittance with increasing thickness. From the fitting, the localization length corresponding to np = 2.5 (squares) and 1.6 (circles) are determined to be 3.46 ± 0.40 and 14.5 ± 0.91, respectively. We note that the transmittance can be better fitted by the modified Ohm’s law  rather than the exponential function for np = 1.6 medium at small thickness because the medium is in the diffusion regime. But at large thickness where localization takes place, the exponential fit was accurate enough to determine the localization length. In order to set up two distinctive settings of the disordered media, one in the strong localization regime and the other in the weak localization regime, we set the thickness of the medium to be studied in the following as 8 μm, such that it is larger than the localization length of the np = 2.5 medium and smaller than that of the medium with np = 1.6 .
For the two 8 μm-thick disordered media of np = 1.6 and 2.5, we obtain the spatial intensity map of the electromagnetic wave within each disordered medium (Figs. 2(a) and 2(b)). The illumination source is a plane wave normally incident to the input surface. As predicted in the theory , we observe that internal intensity decreases exponentially for the np = 2.5 medium while it decreases linearly for the np = 1.6 medium. In particular, prominent intensity-enhanced spots exist for np = 2.5 while the intensity is relatively uniformly distributed for np = 1.6. These results suggest that the np = 2.5 medium is in the strong localization regime while the np = 1.6 medium is in the weak localization regime. Another interesting consequence of the wave localization is the change in the statistics of the internal intensity distribution. Past studies have concerned the statistical properties of the speckle pattern only for the transmitted field [16, 17]. Specifically, the statistics of the speckle intensity is expected to deviate from Rayleigh’s law in the localization regime, which was indeed the case for np = 2.5 medium. Here we can also analyze the speckle pattern within the medium. Specifically, we calculate the speckledness, which is the ratio between the average of the intensity squared and the square of the average intensity, for both np = 1.6 and np = 2.5 media. It turns out that the speckledness is 5.8 for the strongly scattering medium and 2.1 for the weakly scattering medium. This means that the variance of the intensity distribution for the np = 2.5 medium is about 4.4 times larger than that of the np = 1.6 medium. Therefore, we can infer that the wave localization increases the heterogeneity of the internal intensity distribution.
In addition to the statistics of internal intensity, we characterize the properties of internal field and derive a clear condition to distinguish wave localization from diffusion. Since the FDTD method computes the complex amplitude of the field within the medium, we can take the discrete 2D Fourier transform of the internal field with respect to the spatial coordinates x and z for the field within the medium. This provides us with an angular spectrum of the internal waves in the spatial frequency coordinates [18, 19]. Figures 2(c) and 2(d) are the field amplitude in k-space for np = 1.6 and 2.5, respectively. A bright ring appears in the weakly scattering medium (Fig. 2(c)). The presence of the ring indicates that waves propagate in all different directions. The radius of the ring, (1.2 ± 0.5) k0, corresponds to the magnitude of the effective wave vector inside the medium. This agrees well with the effective wave number, 1.27k0, estimated by the effective medium theory . It is larger than k0, the wave number in the free space, because of the high index particles. In the strongly scattering medium (Fig. 2(d)), on the other hand, the ring pattern disappears. This indicates that the propagating waves with pronounced oscillations of electric field over the extended space could be dominated by the localized waves with an abrupt amplitude variation over the wavelength scale. The spatial Fourier transform of the localized waves such as evanescent waves typically exhibits broadened spectra due to the abrupt decay of the field amplitude. This observation agrees well with the theory of Anderson localization in which the locally interfering waves dominate over the incoherently interfering propagating waves. Our analysis could possibly provide a direct observation of the disappearance of propagating waves within the medium and the prevalence of the local non-propagating waves.
3. Construction of a transmission matrix
For the np = 2.5 medium, we next search for its transmission eigenchannels. At first, a transmission matrix H is constructed for the disordered medium whose element connects the plane waves of various wave vectors at the input plane to those at the output planes. Specifically, the complex amplitude, , of the plane wave at the output plane is determined by with the complex amplitude of the plane wave at the input plane. Here, and are the wave numbers of the plane waves at the input and output planes, respectively, along x-direction. The number of basis needed to form a complete set of channels is determined by the recording width and the wavelength λ. The plane waves whose kx satisfies the conditions, kx = 2mπ/W and -k0 < kx < k0 with an integer, are independent input channels. Although the width of the medium used in the computation is 130 μm, we choose a width of W = 90 μm at the output side of the medium for the sampling (black dashed lines in Fig. 1(a)). This is to avoid diffracted waves from the edges. The integer m satisfying the above condition ranges from −149 to 149 such that the number of channels in the basis is 299.
For each of the plane waves in the basis set, we compute its propagation and then record the transmitted wave along the sampling line (dashed line in Fig. 1(a)). In order to use the wave vector space as output channels, we convert the measured electric field on the recording line into k-space representation through the discrete Fourier transform. By repeating this procedure for all the incident channels, we construct a transmission matrix whose dimensions are 299 × 299 (Fig. 3(a) ). The phase part of the transmission matrix is acquired simultaneously (not shown in the figure).
4. Internal field distribution of transmission eigenchannels
We next obtain the transmission eigenchannels of the medium from the transmission matrix. The singular value decomposition (SVD) is applied to the transmission matrix, H.Figure 3(b) shows the plot of singular values after arranging them in descending order. The largest value is 0.90 which is quite close to unity. In terms of transmittance, which is the square of the singular values, the eigenchannel of the largest singular value is about 50 times higher than the average transmittance of the medium, 0.017. The transmittance of the largest singular value is smaller than unity partly because the medium is in the open slab geometry. Some of the energy entering the incident plane may not reach the detector due to the strong scattering in the medium. This may corrupt the solidarity of the transmission matrix. Also, the total number of channels is not large enough such that the probability of finding an eigenchannel of unity transmission may not be sufficient. We confirmed that the highest transmittance of the eigenchannels increases when the width of the sample is increased. This suggests that transmission of eigenchannels can reach to unity even for the Anderson localized medium.
For each singular value, we obtain a corresponding eigenchannel in the form of the coefficients of incident channels. The coefficient is determined from the corresponding column of the unitary matrix V. By superposing incident channels of angular plane waves, the corresponding eigenchannel is constructed at the input plane. We then compute the propagation of the constructed eigenchannel and record the map of the field within the medium. In the Fig. 4 , the intensity maps of the first eigenchannel with maximum transmittance and the 23rd eigenchannel that has the same transmittance as the average transmittance are shown. The maximum intensity of the first eigenchannel (Fig. 4(a)) is enhanced by about 17,000 times in comparison with that of the plane wave illumination (Fig. 2(a)). The total stored energy within the medium is also enhanced in the open eigenchannel to about 18 times that of the plane wave illumination.
We found that there exists a striking difference between the Anderson localization regime and the diffusion regime. In the diffusion regime, the field energy of the open eigenchannel is spread over the entire transverse extent . In the Anderson localization regime, however, it is highly concentrated in space such that a dense cluster of localized modes appears within the disordered medium (Fig. 4(a)). Moreover, the clustering is found to exist only for the eigenchannels of high transmittance. We repeatedly observe this appearance of the cluster for other media whose spatial organizations of constituting particles are different but with the same particle size and index, and the fill factor. As the transmittance becomes low, the internal waves gradually spread along the transverse direction (Fig. 4(b)-4(e)). For the eigenchannels of sufficiently low transmission, the internal wave covers the entire transverse range. These suggest that the dense cluster be formed in the process of enhancing transmission through a disordered medium where non-propagating localized waves are dominant.
We quantify the degree of transverse confinement of the internal waves, the unique characteristic of the cluster to enhance transmission in the Anderson localization regime. For a fixed z, we calculate the standard deviation of the field distribution along the transverse direction by using the local intensity as a weighting Eq. (2).
We then define the average of the standard deviation for all z in the medium as the effective transverse width, Weff. Figure 5 shows the plot of Weff for the eigenchannels of the medium. The Weff is about 3 μm for the first eigenchannel, and gradually increases up to about the 70th eigenchannel and then saturates. The transverse width is thus in strong anti-correlation with the transmittance of the eigenchannels − the higher the transmittance, the narrower the width of the cluster. In the case of the weakly scattering medium, it is found that the effective transverse width is largely constant regardless of the eigenchannel index.
The existence of the dense cluster of internal modes in the Anderson localization regime can be explained as follows. According to Fig. 2(d), most waves exist in the form of non-propagating waves in the strongly scattering medium. In this case, the propagation of energy can mainly be mediated by the neighboring localized modes and by the coupling among them. This can be thought of as a kind of frustrated total internal reflection in which a neighboring pair of prisms induces the coupling of non-propagating evanescent waves and therefore enhances wave propagation. Studies in the 1D media also showed that neighboring of localized modes enhances the transmission [11, 12].
But it is rather insufficient to explain the perfect transmission by the simple neighboring of the localized modes. Since the coupling of neighboring modes will be imperfect, the field amplitude will be monotonically decreasing along the axial direction. According to our study, the internal field distribution exhibits maximum intensity around the middle of the sample. This suggests that many of the localized modes connecting the input and output planes are collectively coupled. In our search for the eigenchannels of high transmittance, the internal field is driven in such a way to transversely reinforce the coupling of neighboring localized modes. This results in the formation of the cluster of localized modes that exhibits dense coupling of internal waves along the transverse direction but encompassing the entire longitudinal extent of the medium. Our observation is somewhat related to the prediction made by Cherroret et al.  in which the transmitted wave of a focused illumination shows better spatial confinement in the Anderson localization regime than in the diffusion regime. High transmission eigenchannels constituting the focused illumination will more likely to survive to form a spatially confined output beam in the strongly scattering medium.
Finally, the square-shape particle used in the study exhibits weak resonance around np = 2.2 and 2.96 for the source wavelength of 600 nm. The strongly scattering medium (np = 2.5) considered in our study is in closer proximity to the single particle resonance than the weakly scattering medium (np = 1.6). Therefore, the increased scattering cross-section of individual particle at near resonance has partly contributed to the wave localization. However, our observations including the disappearance of effective wavenumber and spatial confinement of highly transmitting eigenchannels in the localization regime are rather independent of the single particle resonance. This suggests that our observation is mainly determined by the collective behavior of the entire medium.
We studied the interplay between wave localization in a strongly scattering medium and the open eigenchannels of the disordered medium which induce perfect transmission. From the direct investigation of the internal field in the medium, we proved that the non-propagating localized waves dominate over the propagating waves in the Anderson localization regime. This has resulted in a unique consequence that open eigenchannels of high transmittance are manifested as a dense neighboring of localized modes with strong transverse confinement. In this way, the localized waves in the extended space are tightly coupled among each other and energy flow is enhanced from input to the output planes of the medium. Our study explicitly elucidates the mechanism of transmission enhancement in the Anderson localization regime. Future studies may include the investigation of phase transition from diffusion to localization regimes in terms of the statistical properties of internal fields.
This research was supported by the Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Education, Science and Technology (MEST) (2011-0005018, 2011-0029807, and 2011-0016568), the National R&D Program for Cancer Control, the Ministry of Health & Welfare, South Korea (1120290), a Korean University Grant, and Nano Material Technology Development Program through the NRF of Korea funded by MEST (2012-0006205 and 2012-0006655).
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