We study the energy density of shaped waves inside a quasi-1D disordered waveguide. We find that the spatial energy density of optimally shaped waves, when expanded in the complete set of eigenfunctions of the diffusion equation, is well described by considering only a few of the lowest eigenfunctions. Taking into account only the fundamental eigenfunction, the total internal energy inside the sample is underestimated by only 2%. The spatial distribution of the shaped energy density is very similar to the fundamental eigenfunction, up to a cosine distance of about 0.01. We obtain the energy density of transmission eigenchannels inside the sample by numerical simulation of the scattering matrix. Computing the transmission-averaged energy density over all transmission channels yields the ensemble averaged energy density of shaped waves. From the averaged energy density, we reconstruct its spatial distribution using the eigenfunctions of the diffusion equation. The results of our study have exciting applications in controlled biomedical imaging, efficient light harvesting in solar cells, enhanced energy conversion in solid-state lighting, and low threshold random lasers.
© 2016 Optical Society of America
There are numerous scattering media that are either of natural origin - such as biological tissue, fog, cloud, and teeth - or man-made - such as paint, diffuser glass, and phosphor inclusions in white LEDs. These materials multiply scatter waves in all directions, due to the random spatial variations of the refractive indices. Random interference occurs between the multiply scattered waves that is well-known as speckles . Upon averaging over realizations of scatterers, the speckles average out and the resulting average energy density is well described by diffusion theory [2–4].
Recently, novel wave-shaping methods such as feedback-based wavefront shaping [5–11], time reversal [12–14], phase conjugation [15–17], and transmission matrix-based control [18–20] have demonstrated the control of transmission and reflection through scattering media. These wave-shaping methods remarkably enhance the intensity in a single speckle as well as the total reflected and transmitted intensity. The wave-shaping methods have led the way for exciting applications such as non-invasive biomedical imaging [21–23], advanced optics [24–29], and cryptography and secure communication [30,31].
Although these methods are useful tools in controlling the intensity at the interfaces of a scattering medium, the energy density distribution of the shaped waves inside the scattering medium is still unknown. Here, we refer to shaped waves to mean perfect phase conjugation of the transmitted waves, which is equivalent to shaping to optimally focus light to a single speckle spot [7, 11]. In the case of light, the knowledge of the energy density distribution is important for applications such as enhanced energy conversion in white LEDs [32–36], efficient light harvesting in solar cells [37–39], high yield random laser [40, 41], and controlled illumination in biomedical imaging . To date, only numerical calculations of scalar waves [43–45] and a single-realization elastic wave experiment  have addressed the energy density distribution of shaped waves inside two-dimensional (2D) scattering media. However, none of these studies provide an analytical model for the energy density distribution. For high-transmission channels, Davy et al  described the energy density using a parabolic function. In , our team reported the first measurement of the total energy density of light inside a three-dimensional (3D) scattering medium and reported an enhanced total energy density for shaped waves. In that paper, the experimental result was interpreted with a model that describes the total energy density of shaped waves using only the fundamental eigenfunction of the diffusion equation. The model agreed well with experimental observation. However, the much wanted full description of the spatial energy density for shaped waves needs a more fundamental study.
In this paper, we examine the spatial energy density of shaped waves inside a scattering medium. Our starting point is the mathematical theorem that the eigenfunctions of the diffusion equation form a complete set [48–50]. Therefore, this set can describe any function defined on the sample support, which is specified by the two boundaries. As an example, the complete set of diffusion eigenfunctions can accurately describe the spatial energy density as well as any speckle inside the sample. The fact that the diffusion eigenfunctions can describe any function inside the medium implies in no way a connection to the diffusion approximation. However, such a connection can be inferred if it turns out that the required number of eigenfunctions is small. In this paper, we will demonstrate that expanding the spatial energy density using the complete set of diffusion eigenfunctions is enlightening because only a few diffusion eigenfunctions suffice to reconstruct the energy density.
We obtained the spatial energy density inside a quasi-1D disordered waveguide by numerical calculation of concatenated scattering matrices [51–53]. A quasi-1D disordered waveguide provides a tractable platform to study the essential physics. We calculated the spatial energy density W(z) of all transmission channels. By calculating the transmission-averaged energy density for all transmission channels, we retrieved the energy density of shaped waves [7, 11]. We obtained the intensity transmission coefficients of all channels, and we henceforth refer to τ as the transmission of a specific channel.
We show that only the first seven diffusion eigenfunctions suffice to reconstruct the spatial energy density distribution of the shaped waves. Taking into account only the fundamental eigenfunction of the diffusion equation, the total internal energy inside the sample is underestimated by only 2%. The spatial energy density is very similar to the fundamental eigenfunction, with a cosine distance of 0.01. Furthermore, we are able to reconstruct the energy density distribution of both high- and low-transmission channels, using a few M eigenfunctions of the diffusion equation, e.g. M = 16 for channels with transmission τ = 0.1.
It is fascinating to see that the energy densities of shaped waves and transmission channels, which are due to interference effects, can be truly described using a few diffusion eigensolutions. The reconstruction of the distribution of the energy densities that we found here can be extended to three dimensional (3D) scattering samples. The reconstruction also applies to all forms of waves such as light, sound, elastic waves and quantum waves.
2.1. Eigenfunctions of the diffusion equation54]. Anticipating the fact that the directions (x,y) are decoupled from the z-direction we decompose the energy density as
Since we are considering a slab that is a quasi-1D scattering medium, we assume statistical translational invariance in the perpendicular directions (x,y). The sample boundaries are at z = 0 and z = L, where L is the sample thickness. This symmetry allows us to solve W⊥(x,y,t) by using its two-dimensional spatial Fourier transform W⊥(q⊥,t). Solving the (x,y)-part of Eq. (1) and using the initial condition t = 0 gives
Next we solve the remaining 1D equation
We will turn Eq. (4) into a Sturm-Liouville eigenvalue problem by introducing the Ansatz W(z,t) = e−γtW(z)
As is well-known from linear algebra  solutions of Sturm-Liouville equations are orthogonal and form a complete set. To solve Eq. (5), we apply the Neumann’s boundary conditions, which are given asEqs. 7 and 8 will lead to a discrete set of eigenvalues Γm. The general eigenfunction for eigenvalue Γm can be cast in the form
As outlined in detail in  the wavevector κm and phase ηm will be determined from solving Eq. (5) with the boundary conditions. The parameter Am takes care of the normalization of the eigenfunctions, such that
For the exact solution of Eq. (11), see Appendix. The first ten wavevectors obtained from the solution are shown in Table 1 for a particular set of boundary conditions. In Fig. 1, we plot the first three eigenfunctions using Eq. (9) with the same set of boundary conditions. The fundamental diffusion eigenfunction m = 1 is a half-period sine function, which has positive values throughout the sample. Therefore, the m = 1 eigenfunction can describe a specific energy density inside a scattering medium, since the energy density is positive. On the other hand, the m = 2 and m = 3 diffusion eigenfunctions (and other higher eigenfunctions not shown) have both positive and negative values within the sample boundary. Since a negative energy density is non-physical, diffusion eigenfunctions m = 2 and m = 3 cannot individually describe energy density inside a scattering medium. However, a sum over these diffusion eigensolutions that yields positive values throughout the sample can describe a physical energy density inside the medium. The weight of the fundamental eigenfunction takes care of the sum to be positive everywhere.
2.2. Reconstructing the energy density with eigenfunctions of the diffusion equation
The diffusion equation is of the Sturm-Liouville type so its eigenfunctions form a complete and orthogonal set . Therefore, any distribution of energy density within the domain of the sample boundaries can be decomposed into a sum over a finite number M of eigenfunctions for a specific set of coefficients. Such a decomposition is only useful if the number of needed modes is small. We express a certain ensemble averaged energy density Ws(z) inside the scattering medium in terms of the normalized eigenfunctions Wm(z)Eq. (14) into Eq. (15) to obtain
We use the orthonormality of functions Wp(z) and Wm(z), which is expressed as
Therefore, Eq. (16) becomes
We now can write the decomposition (14) as
In this paper, the spatial energy density Ws(z) that we reconstruct is that of shaped waves Wsw(z). We also reconstruct the spatial energy density of transmission eigenchannels Wn(z, τ). The spatial energy densities are obtained from numerical simulation and the details of the simulation are described in Section 3. The simulation results are compared to the reconstructed energy density Wre(z) in Section 4.
In order to quantify the overlap between the reconstructed function Wre(z) and the numerical data Ws(z), we use the cosine distance COSD , which is defined as
Ns is the number of points in the simulation data. COSD varies between 1 and 0 and tends towards 0 as the reconstructed function fully describes the simulation data. As a figure-of-merit of a good reconstruction, we choose the eigenfunction that has COSD ≤ 10−4. At this COSD, the reconstructed function deviates by only 2% from the actual function. This level of accuracy is within what is achievable in a measurement with a realistic level of experimental noise.
3. Numerical samples and setup
3.1. Numerical setup
We perform simulations of transport of monochromatic scalar waves through a waveguide containing scatterers. The waveguide is modeled as a quasi-one-dimensional (quasi-1D) system possessing Nm = 100 transversal modes, with elastic scattering on bulk impurities. This is implemented as a 2D waveguide with 100 transversal modes, containing scatterers at random positions with a density of 0.2 scatterer per square wavelength. A schematic of the numerical sample is shown in Fig. 2. In order to calculate the energy density inside the waveguide we employ a recursive S-matrix formalism, where the waveguide is first divided into sub-wavelength slices, each containing a maximum of one scatterer. For each slice we define the S-matrix in a mode representation as follows ,51]. This numerically stable procedure yields the S-matrix of the entire waveguide and of subsections of it. From the transmission matrix we extract the singular values and vectors corresponding to all channels by singular value decomposition. The energy density inside the waveguide, for a given incident field, is calculated as follows: For a given axial coordinate zc the S matrices S1,S2 of the waveguide sections 0 < z < zc and zc < z < L respectively are calculated. The field in the z = zc plane is then found by calculating recurrent scattering diagrams:
3.2. Sample characterization
In order to characterize the numerical samples, we plot in Fig. 3(a) the probability distribution P(τ) as a function of the transmission τ. The probability distribution P(τ) obtained from the simulation is bi-modal: there is a high probability for transmitting channels with transmission τ close to zero and close to one. In Fig. 3(a), we compare the Dorokhov-Mello-Pereyra-Kumar (DMPK) distribution  with our numerical result. The probability distribution of transmission for a scattering medium is expected to follow the DMPK distribution. Indeed, the DMPK distribution agrees well with our numerical result. Furthermore, in order to confirm that our numerical samples are in the diffusive regime, we plot in Fig. 3(b) the equally-weighted ensemble averaged energy density versus reduced sample depth. The ensemble averaged energy density shows a linear decrease from the front surface towards the end surface of the sample, in agreement with the prediction of diffusion theory for a diffusive sample, see Fig. 3(b).
4. Results and discussions
4.1. Energy density of shaped wavesEq. (24) implies that wavefront shaping couples an energy fraction of exactly τn into a transmission eigenchannel n. Using Eq. (24), we obtained the energy density of the shaped waves from the transmission-weighted average over the energy density of all transmission channels. We show in Fig. 4(a) the ensemble averaged energy density distribution of shaped waves. The energy density distribution of the shaped waves increases from the interfaces of the sample and is maximal at z = 4.9ℓ, which is about 20% off the center of the sample (z = 6.1ℓ). The ensemble averaged energy density of wavefront-shaped light was first obtain in  by solving the Maxwell equation using the finite-difference time-domain (FDTD) method. Despite the different numerical calculation methods, our numerical result is closely similar to the one obtained by Choi et al . The exact peak position of the energy density obtained in  is difficult to estimate due to the noise in the data, however, the peak position is close to the center of the sample.
We reconstructed the distribution of the energy density of shaped waves from eigenfunctions of the diffusion equation. From the reconstruction, we obtain the cosine distance COSD, which we plot in Fig. 4. In case of shaped waves, a summation over the first M = 7 eigenfunctions is remarkably sufficient (COSD ≤ 10−4) to reconstruct the energy density, see Fig. 4(a). Only the fundamental diffusion eigenfunction m = 1 has COSD ≈ 10−2 and is not sufficient to describe the energy density profile. The fundamental eigensolution peaks at exactly the center of the sample, in contrast to the energy density of shaped waves. To put the number M = 7 in perspective, we performed the same reconstruction on the profile of 50 speckle spots within the same sample domain. We find that over 300 eigenfunctions are required to obtain a COSD ≤ 10−4, indicating that the diffusion eigenfunctions are not a suitable basis for a random speckle. In contrast, the low number of eigenfunctions required to represent the shaped waves implies that the solution of the diffusion equation is close to the ensemble averaged energy density of shaped waves.
We further investigated the contribution of the eigenfunctions to the total energy density integrated along the depth of the sample relative to the contribution of the first 100 eigenfunctions. As a reference, we choose the first 100 eigenfunctions because the higher order eigenfunctions have a very negligible contribution. In Fig. 5, we plot the contributions. For shaped waves, the m = 1 contributes 98% of the total energy density inside the sample. Remarkably, our finding about the strong contribution of m = 1 to the total internal energy density agrees very well with the heuristic model in  that wavefront shaping predominantly couples light to the fundamental diffusion eigensolution. The contribution of m = 2 is negligible compared to m = 1 since m = 2 has negative energy density in almost half of the sample depth, which cancels the positive energy density (see Fig. 1). A similar effect also happens for all other even index eigenfunction.
4.2. Energy density of transmission channels
The ensemble averaged energy density of open transmitting eigenchannels with a transmission τ in the range 0.99 < τ < 1 is shown in Fig. 6. The energy density shows a symmetric profile, which peaks in the middle of the sample. Our numerical result is similar to the one obtained by Davy et al . Interestingly, only the fundamental diffusion eigenfunction m = 1 is sufficient to reconstruct the energy density of the open channel, see Fig. 4(a). The fundamental diffusion eigenfunction m = 1 is expected to describe the open eigenchannels for the following reasons: Firstly, as the open eigenchannel has the highest individual transmission, the fundamental eigenfunction (m = 1) as well contributes most to the total transmission as shown in . Secondly, the fundamental eigenfunction is the only physical solution with a positive energy density along the sample depth z, see Fig. 1 and therefore this eigenfunction must have the largest weight in the expansion. Interestingly, a simplified mathematical model by Davy et al  described the energy density of open channels as a parabolic function. We note that the deviations from a parabolic shape are obvious only near the edges, and the shape of the energy density curve agrees better with the cosine shape of the fundamental eigenfunction, see Fig. 6(a). Indeed, the COSD for the parabola is 10−3 compared to 10−5 for the fundamental eigenfunction.
We also use the eigenfunctions of the diffusion equation to reconstruct the ensemble averaged energy densities of low-transmission channels obtained from simulation. In Figs. 6(b – d), we show the ensemble averaged energy densities for transmission eigenchannels with transmissions τ = 0.5, 0.10 and 0.01. For the eigenchannel with transmission τ = 0.5, only M = 7 eigenfunctions of the diffusion equation are sufficient to reconstruct its energy density, M = 16 eigenfunctions for τ = 0.1, and M = 33 is required for a closed channel τ = 0.01. As the transmission τ decreases, the number of eigenfunctions sufficient to reconstruct the energy density increases. The increase in the number M of sufficient eigenfunctions is because the asymmetry of the distribution of energy density increases as the transmission τ reduces, and likewise, the asymmetry of the eigenfunctions increases with the eigensolution index. Therefore, the higher index eigenfunctions contribute more to an asymmetric function. The contribution of the eigenfunctions with index m = 1, 2, 3 and 4 relative to a summation over the first 100 eigenfunctions is shown in Fig. 7. The relative contribution of the fundamental eigenfunction of the diffusion equation m = 1, which is a symmetric function, increases as the transmission increases, while the relative contributions of the greater index eigenfunctions, which are asymmetric functions, are higher for low-transmission channels. In Ref. , Davy et al. obtained the energy density of the transmission eigenchannels using a numerical method that is similar to ours. The authors describe the profile of the transmission eigenchannels with the solution of an implicit equation. In contrast, our approach provides accurate heuristic analytical expressions that describe the profile of the transmission eigenchannels, which constitutes a complementary approach.
4.3. Effect of sample thickness
Here, we investigate the effect of the sample thickness on the number of eigenfunctions sufficient to reconstruct the energy density of shaped waves. We therefore perform simulations of the energy density on different samples with a range of sample thicknesses: L = 5ℓ, 12.1ℓ, 18.8ℓ and 39.5ℓ. We show in Fig. 8 the COSD versus the eigensolution index for the 4 samples. Interestingly, all the samples have a convergence to a value of COSD ≤ 10−4 at the seventh eigensolution. There is a slight deviation for the thickest sample L = 39.5ℓ that is probably due to artifacts from the simulation due to the large number of waveguide slices. This convergence of the COSD shows that the summation over the first 7 eigenfunctions of the diffusion equation is sufficient to describe the energy density of shaped waves, irrespective of the sample thickness.
In the previous sections, we have seen that the diffusion fundamental eigenfunction of the diffusion equation m = 1 dominates the energy density of high-transmission channels. We investigate if this result holds for the different samples thicknesses. In Fig. 8, we show the coefficient of the m = 1 mode relative to the sum of the first 100 eigenfunctions versus the transmission for the 4 different thicknesses. The relative contribution of the m = 1 mode is similar for all samples, irrespective of the thickness and this is quite remarkable. The fundamental diffusion eigenfunction dominates the energy density of high-transmission channels for all sample thicknesses.
We have shown that only a few eigenfunctions of the diffusion equation suffice to accurately reconstruct the distribution of the shaped energy density inside a quasi-1D scattering waveguides. In particular, the fundamental eigenfunction of the diffusion equation is very similar to the distribution of energy density of shaped waves. To reconstruct the distribution of open channels only the diffusion fundamental eigenfunction m = 1 is sufficient. In addition, we have shown that a few diffusion eigenfunctions reconstruct the energy density of low-transmission eigenchannels, to a great precision. The diffusion eigenfunctions are very efficient to reconstruct these energy densities that are within the boundaries of the sample. This reconstruction is enlightening because even energy densities that are as a result of wave phenomena – shaped waves and transmission eigenchannels – can be described efficiently with diffusion eigenfunctions. The diffusion eigenfunctions are capable of this reconstruction since they are a complete set of functions. Our results are relevant for applications that require the precise knowledge of distribution of the energy density inside scattering media. Such applications include efficient light harvesting in solar cells especially in near infrared where silicon has a low absorption [37–39]; enhanced energy conversion in white LEDs, which serves to reduce the quantity of expensive phosphor [32–36]; low threshold and higher output yield of random lasers [40, 41]; as well as in homogeneous excitation of probes in biological tissues .
A. Method of calculating the wave vectors of diffusion eigenfunctions
In Eq. (11), we show an implicit equation, which defines the allowed wave vectors of the diffusion eigenfunctions. The equation is again stated here:
Our goal is to find the wave vectors κrm that fulfills the condition
In Fig. 9, we plot the two functions f1 and f2. The condition in Eq. (29) is fulfilled by the wave vectors at the intersection of the two functions f1 and f2, as indicated in Fig. 9. We found the wave vectors by dividing function f into domains with size π/L. In most of the domains, only one root is expected, except the domain where the function f2 has a divergence, where two roots appear. In each domain, we use a Simple Bisection method to search the domain for wave vectors that fulfill the desired condition. Each domain is divided into two equal parts. The function f is first evaluated for the left part of the domain and then checked if there is any root present. If there is no root present, the left domain is further divided into smaller subdomains and each subdomains is searched for the root. We divide the domains until the size of the subdomain is 10−12. If there is no root in all the subdomains of the left domain, then, the right domain is searched. This way, all the desired roots are found iteratively.
We thank Pepijn Pinkse, Henri Thyrrestrup, Hasan Yilmaz, and Hui Cao for useful discussions. This project is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie” (FOM) FOM-program “Stirring of light!”, which is part of the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek” (NWO). We acknowledge NWO-Vici, DARPA, ERC grants 279248, 678919, and STW.
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