While a three-dimensional (3D) scattering medium is from the outset opaque, such a medium sustains intriguing transport channels with near-unity transmission that are pursued for fundamental reasons and for applications in solid-state lighting, random lasers, solar cells, and biomedical optics. Here, we study the 3D spatially resolved distribution of the energy density of light in a 3D scattering medium upon the excitation of highly transmitting channels. The coupling into these channels is excited by spatially shaping the incident optical wavefronts to a focus on the back surface. To probe the local energy density, we excite isolated fluorescent nanospheres distributed inside the medium. From the spatial fluorescent intensity pattern we obtain the position of each nanosphere, while the total fluorescent intensity gauges the energy density. Our 3D spatially resolved measurements reveal that the differential fluorescent enhancement changes with depth, up to at the back surface of the medium, and the enhancement reveals a strong peak versus transverse position. We successfully interpret our results with a newly developed 3D model without adjustable parameters that considers the time-reversed diffusion starting from a point source at the back surface.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The interference of multiple scattered waves in complex media holds much fascinating physics such as coherent backscattering, Anderson localization, and mesoscopic correlations [1–4]. Transport through complex media is described by so-called channels that are eigenmodes of the transmission matrix . Remarkably, open transmission channels with near-unity transmission are predicted to perfectly transmit a properly designed incident field even if the medium is optically thick . It has recently been demonstrated that light is sent preferentially into a combination of open and highly transmitting channels by the spatial shaping of the incident wavefronts [7–9]. This development has led to tightly focused transmitted light (henceforth referred to as “optimized light”) [10–13], focusing inside a scattering medium [14,15], enhanced optical transport through a scattering medium [7,8,16–18], sending an image through a scattering medium , and imaging inside a scattering medium [20–22].
In contrast, only a few studies address the energy density of optimized light that plays a central role in diverse applications of light–matter interactions, such as solid-state lighting [23,24], random lasers , solar cells , biomedical optics , or control of fluorescent proteins . In the absence of wavefront control, the ensemble-averaged energy density depends linearly on depth in the medium . The critical questioning is how can the energy density be controlled by exciting open channels, and what is the resulting three-dimensional (3D) energy density. In particular, the 3D energy density profile of shaped light has not been experimentally studied to date. Due to the inherent opacity, direct optical imaging cannot be used to probe the 3D energy density profile. In Ref. , it was shown that spontaneous emission of embedded fluorescent nanoparticles does report the energy density, and it was observed that the depth-integrated global energy density is increased by wavefront shaping but the 3D profile could not be resolved. Several studies on low-dimensional systems [30–36] indicate that the energy density versus position has a maximum near the center of the sample, while the transverse dependence was not addressed. Thus, to investigate how the 3D local optical energy density is controlled by wavefront shaping, a local 3D -resolved measurement is called for.
In this work, we investigate the 3D local spatially resolved energy density in a 3D scattering medium, with optimized incident light. Figure 1 illustrates our experiment: using a spatial light modulator (SLM), we shape the incident green light to a focus at the back surface of a disordered ensemble of ZnO nanoparticles, a procedure that is known to enhance the coupling of light into highly transmitting channels [7,30,31,34]. The resulting energy density is probed locally by fluorescent nanospheres. The density of the nanospheres is so low that only one of them is present in the illuminated volume. Wavefront shaping increases the local energy density by an enhancement factor that we denote as . Consequently, the fluorescence emission of a nanosphere, which is proportional to the local energy density at its location, is enhanced by the same factor. We performed measurements on several nanospheres inside a sample, and for each individual sphere we measured two key parameters, namely, the nanosphere location and the differential fluorescence enhancement . Here the fidelity quantifies the overlap between the experimentally generated wavefront and the perfect wavefront that optimally couples light to the target position .
2. MAIN OBSERVATION
Figures 2 and 3 show our main results: the measured differential fluorescence enhancement versus depth and transverse position, respectively, in scattering samples with thicknesses of and . In Fig. 2, increases with depth from front to back, and increases up to 16 and 26 with thickness . In the hypothetical situation where control of the incident light does not systematically change the internal energy density we would find . The data deviate strongly from this condition, which shows that the energy density is strongly controlled. We propose a 3D model without free parameters that describes the data in Fig. 2 very well.
To verify the 3D character of , we translate the sample transversely along the axis at constant depth. Figure 3 shows the differential fluorescence enhancement versus the transverse displacement relative to the optical axis . For both samples, reveals clear maxima, revealing the effect of the optimized focus . Due to cylindrical symmetry in the transverse plane, similar behavior occurs versus both and , thus, scanning the coordinate [or even combinations () or (), etc.] is completely equivalent. The surface map in Fig. 3 shows the cylinder-symmetric transverse distribution with the measured data points. The observed strong dependence on the transverse coordinate is also well described by our 3D model, while it is not explained at all by previous 1D diffusion models that are necessarily independent of [30–36].
3. EXPERIMENTAL METHODS
The samples were prepared by spray painting a suspension of ZnO nanoparticles and a low concentration of fluorescent particles on a glass slide (details in Supplement 1). After evaporation, we obtained a dense ensemble of strongly scattering ZnO nanoparticles, with a transport mean-free path of . The thickness of the sample was controlled by the spraying time, and we made 8 μm and 16 μm thick samples. Since the lateral extent of our samples (3 mm) is much greater than the thickness and since the thickness is much greater than the mean-free path, the photon transport in the samples is truly 3D.
B. Determination of the Depth of the Nanoparticle
The locations of the fluorescent nanospheres are a priori unknown since the nanospheres end up at random positions. To determine the locations , we first scanned the sample to find isolated fluorescent spheres and then recorded the diffuse fluorescent spot at the back surface of the sample [see Fig. 4(a)]. We performed a Fourier transformation of the fluorescent spot [see Fig. 4(b)] and filtered high-frequency noise. We model the nanosphere as a point source in the 3D diffusion equation  and fit the solution in Fourier space to the Fourier transform of the fluorescence spot with the nanosphere depth as the only adjustable parameter; see Figs. 4(c)–4(d). For the particular fluorescent nanosphere shown in Fig. 4, the depth is . To determine the -error bar, we performed 100 measurements on a single nanosphere. Since the measured fluorescent intensity varies for the 100 measurements, the mean and standard deviation over the 100 measurements give and -error bar, respectively. The variation in the -error bars from each nanosphere in Figs. 2 and 3 is probably because different nanospheres reveal different intensities (e.g., due to different doping or bleaching).
C. Wavefront Shaping and Fidelity
Next, we performed wavefront shaping experiments with the optical axis of the system centered on a nanosphere at coordinates . We obtained a feedback signal for the wavefront shaping optimization from an area of , which is smaller than the speckle area . The output beam diameter is estimated to be about 56 μm for an unoptimized incident wavefront, corresponding to order of magnitude transmission channels. We used the piecewise sequential algorithm to find the optimized incident wavefront [7,10], with input degrees of freedom on the spatial light modulator, as discussed in Supplement 1.
Ideally, a perfectly shaped wavefront is the phase conjugate of the wavefront originating from a point source located at the target position . A real wavefront in an experiment inevitably differs from a perfect wavefront due to finite resolution, temporal decoherence, modulation noise, and spatial extent of the generated field [7,10,39]. The deviation of the wavefront from the ideal one due to all these effects can be represented in a single measure: the fidelity (that is the same as in Refs. [7,29]). Experimentally, the fidelity is gauged as , where is the refractive index of the substrate, the intensity for the optimized wavefront, and the total transmitted intensity with an unoptimized reference wavefront [7,29]. Since a real wavefront is the superposition of the perfectly shaped wavefront that controls the energy density and a random error wavefront , the energy density due to a real incident wavefront is necessarily a linear combination of the perfectly optimized energy density and a diffusive unoptimized energy density
[The energy densities in Eq. (1) are ensemble averaged; see Supplement 1]. By probing the fluorescent spheres at different positions, we obtain the local energy density enhancement defined as . Equation (1) leads to a linear dependence of the energy density enhancement on fidelity:40].
D. Controlling Fidelity
To determine from Eq. (2), it is necessary to control the fidelity. Therefore, we systematically perturbed the optimized wavefront by adding to each pixel a random phase noise. The perturbed optimized pattern is expressed as5. For each perturbed phase we collected a fluorescence image and reference fluorescent images , each with a random incident wavefront. ( and are integrated over all camera pixels within the fluorescent peak). We determined experimentally the fluorescence enhancement from the ratio of and the average . We repeated the wavefront shaping and fidelity scanning procedure times on each nanosphere to obtain an ensemble average.
The measured collection of fluorescence enhancement data points versus fidelity is shown in Fig. 6 for one fluorescent nanosphere. While the data show inevitable variations, which are primarily due to a low-signal-to-noise ratio from the intensity of single nanosphere (see Supplement 1), the fluorescence enhancement clearly increases with , to an average of at the maximum fidelity, as confirmed by the rebinned data. From the linear dependence between and with unity axis intercept [see Eq. (2)], we obtain the slope that is directly obtained from the data, without any extrapolation. The procedure above was done both versus depth and versus transverse position, and all resulting differential fluorescence enhancements are shown in Figs. 2 and 3.
4. THEORY AND DISCUSSION
To model the 3D energy density of optimized light, we consider the optimized target to be a point source of diffuse light, as shown in Fig. 1. The 3D energy density of the point source is described by the 3D diffusion equation [1,14] (for details, see Supplement 1). Light from the point source diffuses in a cone from the back surface to the front surface, preferentially via open channels. While the time reverse (or phase conjugate) of the light transmitted to the front surface describes light traveling to the target point at the back surface, part of the light injected at the front surface contributes to a background, notably in the space outside the optimized focus (see Fig. 1). This background originates from the fact that open channels do not form a complete basis and hence cannot compose an ideal background-free focus [7,31]. Therefore, at perfect fidelity we describe the optimized energy density as a sum of two components:41,42]. For open channels, it has recently been shown that the energy density profile along tracks the fundamental mode of the diffusion equation [29,34,36]. To obtain , we normalize and map its dependence onto the spatial profile of the fundamental diffusion mode (see Supplement 1). Similarly, we describe by mapping the fundamental diffusion mode onto a Gaussian profile with a constant width along . The amplitudes of and are fixed by the total transmitted intensity.
In our experiments, a fluorescent nanosphere at position is excited by the local energy density in the case of optimized light (modeled above) or for unoptimized incident light. We describe as a product of the solution of the 1D diffusion equation (versus ) and a Gaussian [in . Figure 7(a) shows the energy density of optimized light at various depths calculated using Eq. (4) (see Supplement 1). The energy density first spreads until about and converges to a focus at the back surface of the sample. These energy densities also serve to interpret transverse scans as in Fig. 3: the scan in Fig. 3(a) taken near in the sample has a width of about 3 μm, which agrees well with the calculated results in Fig. 7(a) (middle panel).
Figure 7(b) shows the -integrated energy densities , , and as functions of position . The figure reveals that matches with , both having a peak close to the center of the sample and decreasing toward the sample surfaces. The agreement of the two functions is expected since is translationally invariant along . In addition, we find that both and are enhanced compared to the unoptimized density , as shown earlier . Figure 7(c) shows the energy densities of optimized and unoptimized light on the optical axis at . increases steadily and has a maximum close to the back surface of the sample. The peak close to the back surface of the sample is attributed to the position of the point source at in the solution of the diffusion equation. In contrast, increases only slightly until around and then decreases toward the back surface of the sample. To the best of our knowledge, this is the first description of the energy density of wavefront-shaped light in 3D.
From the ratio of and , we obtain differential fluorescence enhancement [see Eqs. (S11)–(S14)] that is plotted as a function of depth in Fig. 2. For both samples, our 3D model shows that increases steadily as increases to the back surface of the sample, in excellent agreement with the experimental data. The steady increase is mainly due to the focusing of the energy density of optimized light , as shown in Fig. 7(a). Figures 2(a) and 2(b) also show that the twice thicker sample has about twice greater differential fluorescence enhancement . This effect was observed for many fluorescent particles inside the scattering medium . We attribute the dependence of the fluorescence enhancement on sample thickness to the fluorescent enhancement being determined by the ratio of optimized and unoptimized intensities, the latter decreasing linearly versus depth; see Fig. 7(b). Both agreements show that the intensity enhancement observed on the back surface is associated with the 3D enhancement of the local energy density in the bulk of the scattering medium.
By exciting highly transmitting channels in a 3D scattering medium by wavefront shaping, we observe that the local energy density is considerably enhanced. The enhancement increases toward the back surface of the sample and has a maximum along the transverse direction, revealing the effect of the optimized focus. A 3D model without adjustable parameters successfully describes the experimental data. Our results thus offer new insights on the 3D redistribution of the energy density in 3D scattering media, which is extremely useful to enhance the efficiency of energy conversion in systems such as random lasers, solar cells, and white LEDs. For white LEDs, wavefront shaping could serve to control the color temperature by optimizing for warm or cold white light. Our results also pertain to wavefront shaping of classical waves, such as acoustic and pressure waves , and to quantum waves, such as electrons in nanostructures.
Stichting voor Fundamenteel Onderzoek der Materie (FOM); Stichting voor de Technische Wetenschappen (STW); Defense Advanced Research Projects Agency (DARPA); Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).
We thank Cornelis Harteveld for technical help and Jeremy Baumberg, Sylvain Gigan, Pepijn Pinkse, Stefan Rotter, Tristan Tentrup, Wilbert IJzerman, and Floris Zwanenburg for discussions. This work 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), and we acknowledge support by NWO-Vici, DARPA, NWO-TTW, and Rubicon fellowship.
See Supplement 1 for supporting content.
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