## Abstract

We experimentally demonstrated the basic concept of modulatable optical near-field interactions by utilizing energy transfer between closely positioned resonant CdSe/ZnS quantum dot (QD) pairs dispersed on a flexible substrate. Modulation by physical flexion of the substrate changes the distances between quantum dots to control the magnitude of the coupling strength. The modulation capability was qualitatively confirmed as a change of the emission spectrum. We defined two kinds of modulatability for quantitative evaluation of the capability, and an evident difference was revealed between resonant and non-resonant QDs.

© 2011 OSA

## 1. Introduction

Nanophotonics, which utilizes the local interaction between nanometric materials via optical near-fields, has realized novel photonic devices, fabrication techniques, and systems which will help to meet the requirements of future optical technology [1]. Optical near fields can be viewed as the elementary surface excitations on nanometric materials, which can mediate the local interaction between closely spaced nanometric materials. The interaction potential is expressed by a Yukawa function, which represents the localization of the optical near-field energy around the nanometric particles [2]. Recently, several applications of nanometric, and energy-efficient optical functions have been actively developed by utilizing local energy transfer via optical near-fields and its subsequent dissipation [3–5]. These energy transfer in nanometric materials have been observed in various materials such as [6–10].

The operation of nanophotonic devices described above exhibits a *one-to-one* correspondence with respect to input signals, because the physical properties, size, shape, and alignment of the components for the characteristic optical near-field interactions are built into the substrate and spatially fixed. In order to realize a *one-to-many* correspondence with a single nanophotonic device, it is necessary to implement *modulatable* optical near-field interactions and associated *modulatable* optical functions. Here, we propose a novel concept of *Modulatable Nanophotonics* to realize such a system. It is realized by providing appropriate external controls to modulate the parameters of the components. In our concept, optical far-field retrieval of induced optical near-field interactions is achieved via modulation of the intensity, polarization, and spectrum of the subsequent irradiation. This is an important step toward further exploiting the possibilities of light–matter interactions on the nanometer scale [11].

In this paper, we demonstrate our concept by controlling the magnitude of the optical near-field coupling strength among multiple quantum dots (QDs) randomly dispersed on a flexible substrate. External control is provided by physical flexion of the substrate, whose response can be acquired as a change of the emission spectra in the optical far-field. In Section 2, we describe the basics of local energy transfer via optical near-fields and our experimental concept using QDs on a flexible substrate. Sections 3 and 4 show the concepts and results of numerical and experimental demonstrations, respectively. Quantitative evaluation of each result was performed based on specially defined figure-of-merits. We conclude in Section 5 with a brief summary.

## 2. Modulation of emission spectra of resonant quantum dot pairs

To demonstrate the modulation capability, we utilize *resonant* conditions between appropriate QD pairs on a physically flexible substrate. A conceptual diagram is shown in Fig. 1
. Closely positioned resonant QD pairs exhibit high-quantum-efficiency and selective energy transfer via induced optical near-fields between QDs. As a typical case of local energy transfer, here we consider the optical near-field interaction between the exciton of the lowest excited state *E*
_{1S} in a smaller QD and that of the second-lowest excited state *E*
_{2L} in a larger QD. These two states are electric dipole allowed and forbidden energy levels, respectively [12,13]. However, energy transfer is allowed via the optical near-field interactions in the case of *E*
_{1S} = *E*
_{2L} because of the steep gradient field due to the localized nature of the optical near-field. Thus, the excitation energy in the smaller QD is transferred to the larger QD via this optical near-field interaction. The important point is that energy can be successfully transferred from the smaller QD to the larger QD only if *E*
_{1S} and *E*
_{2L} are in a resonant condition. Transferred energy is immediately dissipated from *E*
_{2L} to the lowest excited state *E*
_{1L} of the larger QD, and a photon is then emitted. Experimental and theoretical results showing good agreement with each other have been discussed in detail in previous papers [6–9,14].

Here, we consider the case of similar QDs dispersed on a substrate having sufficient physical flexibility. Flexing the substrate can vary the distance between the QDs. Figure 2
shows schematic diagrams comparing the emission processes in resonant and non-resonant QD pairs. In the case of a resonant QDs pair, denoted QD_{S-R} and QD_{L-R} in Fig. 2(a), energy transfers occur when the QDs are sufficiently close, because the magnitude of the optical near-field interaction, i.e., the coupling strength between QD_{S-R} and QD_{L-R}, depends on the distance *r* between them, as represented by the Yukawa function:

*μ*and

*A*represent distribution of optical near-fields which determined by exciton energy and amount proportional to the dipole moment, respectively [2]. Thus, emission is preferentially obtained from

*E*

_{1L}of QD

_{L-R}. In contrast, if the separation between QD

_{S-R}and QD

_{L-R}is increased by flexion of the substrate in order to significantly decrease the coupling strength, both QD

_{S-R}and QD

_{L-R}emit independently. This means that the spectral intensities of QD

_{S-R}and QD

_{L-R,}as well as the relative spectral intensity ratio from them, can be modulated by the flexion, i.e., by modulating the coupling strength. Thus, the spectral intensity ratios from QD

_{S-R}and QD

_{L-R}with and without the flexion are evidently different, as shown in Fig. 2(b). On the other hand, in the case of non-resonant QDs, denoted QD

_{S-NR}and QD

_{L-NR}in Fig. 3(a) , energy transfer never occurs between the two. Each QD emits individually regardless of whether the substrate is flexed. In this case, only a change in spectral intensity that depends on the areal density of the QDs is obtained. Therefore, the spectral intensity ratios from QD

_{S-NR}and QD

_{L-NR}with and without flexion are equal, as shown in Fig. 3(b).

Here, we define *modulatability *
${M}_{\text{sp}}$ of the emission by using the spectral intensities $I\left({\lambda}_{\text{S}}\right)$ and $I\left({\lambda}_{\text{L}}\right)$ from the two QDs:

*I*and ${I}^{\prime}$ represent the peak emission intensities of the spectra without and with the flexion, respectively. Wavelengths of peak emission intensities of the spectra of QD

_{S}and QD

_{L}are represented as ${\lambda}_{\text{S}}$ and ${\lambda}_{\text{L}}$, which are the emissions from

*E*

_{1S}and

*E*

_{1L}, respectively. In the case of the resonant QDs, as shown in Fig. 2(b), the value of ${I}^{\prime}/I$ depends on the magnitude of the coupling strength, governed by the flexion, which means that the modulatability can take a non-zero value (${M}_{\text{sp}}\ne 0$). On the other hand, in the case of the non-resonant QDs, as shown in Fig. 3(b), the value of ${I}^{\prime}/I$ remains unchanged for all wavelengths because the area density of the QDs is homogeneously modulated by the flexion. Therefore, ${M}_{\text{sp}}$ is calculated to be nearly zero (${M}_{\text{sp}}\cong 0$). One remark regarding Eq. (2) is that singularity may be a problem when denominators therein result in zero. However, as described below in Sections 3 and 4, we can assume that the denominators are non-zero in realistic physical situations, which guarantees the effectiveness as the representation of Eq. (2).

## 3. Numerical demonstration

To numerically estimate the spectral intensity modulation, we assumed a calculation model consisting of four QDs configured as resonant QD pairs, i.e., two small QDs (QD_{S-A} and QD_{S-C}) and two large QDs (QD_{L-B} and QD_{L-D}), representing many QDs dispersed on the substrate, as shown in Fig. 4
. We simulated the temporal evolution of exciton populations of the relevant excited states in these QDs by quantum master equations [15]. The exciton populations correspond to the intensity of radiation from each QD, and their evolutions are modulated by modulating the magnitude of the optical near-field coupling strength between QDs, by flexion of the substrate. As previously reported in [16], the initial condition is set as a vacuum state, and the duration and the amplitude of the incident light are given by the Hamiltonian representing interactions between the incident light and the QDs. Here, the magnitude of the optical near-field interaction, i.e., the coupling strength, between the two QDs is denoted by the Yukawa function of Eq. (1).

In this model, the evolution of the exciton population in each QD is calculated for various distances *r*, which corresponds to different degrees of coupling, controlled by flexion, as shown by Fig. 4(a). Initial distances between each QD are set on a 2D *xy*-surface as ${r}_{\text{AB}}={r}_{\text{BD}}={r}_{\text{AC}}={r}_{\text{CD}}=100$ (nm), and the magnitudes of the coupling strengths ${U}_{\text{AB}}$, ${U}_{\text{BD}}$, ${U}_{\text{AC}}$, ${U}_{\text{CD}}$, ${U}_{\text{AD}}$, and ${U}_{\text{BC}}$ depend on these distances, as schematically shown by Fig. 4(b). In our model, the relaxation time constant ${\Gamma}^{-1}$ is set at 10 ps, and the radiation lifetimes of QD_{S} and QD_{L}, denoted as ${\gamma}^{-1}{}_{\text{S}}$ and ${\gamma}^{-1}{}_{\text{L}}$, are respectively set at 2.83 ns and 1 ns as a typical parameter set for the CdSe/ZnS QDs used for our previous experiments [17], which based on the pattern of conventional energy transfer models between CdSe/ZnS QDs. Based on these setup parameters and proportional relations $A\propto \mu \propto {L}^{3}$ in Eq. (1) [18], the diameter of the smaller QDs, *L*, is assumed to be approximately 50 nm, and that of the larger QDs, ${L}^{\prime}$, satisfies ${L}^{\prime}=\sqrt{2}L$. Based on theoretically assumed distributions of optical near-fields, which described by the Yukawa function [2], scale of optical near-fields has been propotional to inverse-square of size of the source. Therefore, sufficient energy transfer between QDs via optical near-fields is expected with such model.

First, we assumed *stretching* of the substrate in the *x* and *y* directions, as shown in Fig. 5(a)
, and calculated the exciton populations with various stretch lengths $\delta r$. In the case of the *x*-directional stretching, ${r}_{\text{AC}}$ and ${r}_{\text{BD}}$ are constant, as the first-order approximation, and ${r}_{\text{AB}}$, ${r}_{\text{CD}}$, ${r}_{\text{AD}}$ and ${r}_{\text{BC}}$ are varied. On the other hand, in the case of the *y*-directional stretching, ${r}_{\text{AB}}$ and ${r}_{\text{CD}}$ are constant, and ${r}_{\text{AC}}$, ${r}_{\text{BD}}$, ${r}_{\text{AD}}$ and ${r}_{\text{BC}}$ are varied. Figure 5(b) shows the calculated results. The horizontal axis represents the relative stretch length, which is defined as $\delta r/L$.

Evident changes in the exciton populations are obtained by the *x*-directional stretching, because the excitons always preferentially transfer from smaller QDs to larger QDs (from QD_{S-A} to QD_{L-D}). This result indicates that the coupling strength between the smaller QDs and the larger QDs is decreased by the *x*-directional stretching, and they emit independently.

Here we assume that the emission intensity from each QD is proportional to the exciton population at the corresponding QD energy level. Therefore, in the case of the *x*-directional stretching, as shown in Fig. 5(a), ${M}_{\text{sp}}$ is calculated as 9.26 at a stretch length of $\delta r=$500 nm. The population in the lowest excited state of the larger QDs finally approaches zero, and ${M}_{\text{sp}}$ is calculated as infinity. On the other hand, *y*-directional stretching gives ${M}_{\text{sp}}\cong 0$ at the same stretch length. Therefore, only the former model can be said to be a *modulatable nanophotonic system*.

Next, a shear model is assumed for the *x*-directional shift of QD_{S-C} and QD_{L-D}. Its schematic diagram is shown in Fig. 5(c). In the case of the model, ${r}_{\text{AB}}$ and ${r}_{\text{CD}}$ are constant, and ${r}_{\text{AC}}$ and ${r}_{\text{BD}}$ are increased. Figure 5(d) shows the results of calculations. By increasing the sheared length $\delta r$, which is defined by the shifted values of QD_{S-C} and QD_{L-D}, only the exciton population at QD_{L-B} increases, because the magnitude of energy transfer toward QD_{L-B} increases not only from QD_{S-A} but also from QD_{S-C}. In contrast, that to QD_{L-D} decreases. This is also because of the increase in the energy transferred from QD_{S-C} not only to QD_{L-D} but also to QD_{L-B}. As a result, the population at QD_{S-C} is decreased, whereas that at QD_{S-A} remains unchanged, because the main route of energy transfer is from QD_{S-A} to QD_{L-B}, regardless of the stretching. Here, by these contributions of the exciton populations at the smaller QDs (QD_{S-A} and QD_{S-C}) and the larger QDs (QD_{L-B} and QD_{L-D}), the calculated *M* in the shear model shows a non-zero value (${M}_{\text{sp}}$ = 0.11 for a sheared length of $\delta r=$100 nm).

## 4. Experimental demonstration

We used commercially available CdSe/ZnS spherical QDs (*Evident Technologies*) as a test specimen. QDs are uniformly dispersed in toluene solvents with 10 mg/mL of concentration. Their exciton energy transfer has already been studied [7,19,20], and *resonant* and *non-resonant* conditions via optical near-field interactions have been experimentally verified [17]. The same types of QDs were adopted as QD_{L-R} and QD_{L-NR}. Their peak absorption wavelength ${\lambda}_{\text{AB}}$ and peak emission wavelength ${\lambda}_{\text{EM}}$ were as given in their technical specifications from the manufacturer, 583 nm and 605 nm, respectively, the latter corresponding to the wavelength of emission from *E*
_{1L} in Fig. 1. The respective diameters *D* of the resonant pair, QD_{S-R} and QD_{L-R}, were assumed to be 8.2 nm and 8.7 nm, and those of the non-resonant pair, QD_{S-NR} and QD_{L-NR}, were assumed to be 7.7 nm and 8.7 nm. The QDs adopted as QD_{S-R} have ${\lambda}_{\text{AB}}$ = 523 nm and ${\lambda}_{\text{EM}}$ = 546 nm, and those adopted as QD_{S-NR} have ${\lambda}_{\text{AB}}$ = 565 nm and ${\lambda}_{\text{EM}}$ = 578 nm. These emission wavelengths ${\lambda}_{\text{EM}}$ correspond to the wavelengths of emission from *E*
_{1L}. As the flexible substrate, we used polydimethylsiloxane (PDMS), which is particularly known for its obvious rhelogical properties. We mixed 5 mL of each QD solution as resonant and non-resonant QD pairs. The mixed QD solutions were dispersed on a 2 cm × 2 cm square of PDMS substrate and allowed to dry naturally. In our experiments, although evident heterogeneity of the distributions was observed, the average *r* was assumed to be approximately 5–10 nm from the thickness of the ZnS shell and the length of the modified ligand to each QD.

Here we consider only 2-D flexion of the substrate. The QDs were excited by a He-Cd laser (wavelength 325 nm) with 5 mW/cm^{2} power density. In our experiment, as shown in Fig. 6
, the PDMS substrate was set on an aperture formed at the side of a vacuum desiccator and was flexed by evacuation. The flexion brings dispersed QDs close to each other, as represented by $\delta r<0$. The air pressure was decreased and fixed to ~0.07 MPa to achieve a 20% in-plane compression ratio of the substrate, which was geometrically determined from the size of the aperture and the depth of the flexed substrate.

We experimentally observed the emission spectra of the resonant and non-resonant QDs without and with flexion by using a spectrometer (JASCO; CT-25TP), as shown in Figs. 7(a) and (b) , respectively. The fractional ratio of the numbers of small QDs to large QDs was 1:1. The dashed curves in these figures represent Gaussian curves fitted to the measured spectral profiles. The differences in peak wavelength from those shown in the technical specifications depend on the variability of each particle's size and the inhomogeneous dispersion of the size distribution. In addition, because all peak wavelengths in Figs. 7 are shifted to red from 5 to 10 nm, the difference can also be explained by re-absorption effect in solid-state QDs on PDMS substrate.

As shown, the intensities of the spectra are increased as a whole because the numbers of QDs per unit area are increased by the flexion of each substrate. However, only an increase of the emission intensity from QD_{S-R} was suppressed. This is because the average distance between QD_{S-R} to QD_{L-R} is shorten by the flexion, which induces energy transfer from QD_{S-R} to QD_{L-R}. Similar behavior was predicted by our numerical demonstrations described above. For quantitative evaluation of this behavior, we evaluated the intensities at the spectral peak from the fitted curves and calculated the values of ${I}^{\prime}\left({\lambda}_{\text{L}}\right)/I\left({\lambda}_{\text{L}}\right)$ and ${I}^{\prime}\left({\lambda}_{\text{S}}\right)/I\left({\lambda}_{\text{S}}\right)$, as shown in Fig. 7. By taking the difference, the modulatabilities ${M}_{\text{sp}}$ obtained with the resonant and non-resonant QDs were 0.45 and 0.02, respectively. Therefore, as in the numerical demonstration in Sec. 3, it was confirmed that the resonant QDs showed a much larger ${M}_{\text{sp}}$ compared with that of the non-resonant QDs.

Similar experiments were conducted with other resonant samples with various fractional ratios (1:1, 2:1, and 3:1) and dilution rates (×1 and ×2). Such parameters control the average distance between dispersed QDs. The assumed average distances in each experimental setup and the calculated modulatabilities ${M}_{\text{sp}}$ were compared (Table 1 ). Each average distance was calculated from that of the previous experimental setup, which was assumed to be approximately 10 nm. In the cases where resonant QD pairs were used, regardless of the fractional ratios and dilution rates, ${M}_{\text{sp}}$ generally showed non-zero values. Moreover, by diluting the QD samples, ${M}_{\text{sp}}$ decreased. This is because the average distances between QDs were increased by the dilution, and the coupling strength decreased. In Table 1, it was revealed that one-to-many fractional ratios of QDs can show efficient energy transfer and subsequent emission. As previously reported in [21], in a mixture of the equivalent number of larger QDs and smaller QDs, not all of excitons in a smaller QD can be successfully transferred to a neighboring larger QD, because the radiation lifetime of a larger QD is limited as few nanoseconds. So that part of the excitons in smaller QDs must be decayed at each QD and results in loss in the conversion from the input to output. On the other hand, in a mixture of a few larger QDs and a lot of QDs, excitons in smaller QDs can be transferred to neighboring smaller QDs before being radiatively decayed till relaxation at a larger QD. This indicates that the latter case can effectively output radiated excitons at smaller QDs as relaxation at larger QDs.

The difference in the rates of change of each spectral peak intensity of the resonant QD sample corresponds to a difference in color tone of the emission from the sample. Our experimental results indicated that a change in the emitted color tone was successfully obtained by flexion of the substrate on which the resonant QDs were dispersed. From this viewpoint, we constructed a chromaticity diagram based on the color matching functions of each observed emission spectrum and defined another modulatability ${M}_{\text{ch}}$ to directly evaluate the amount of each modulation for several samples. The value of ${M}_{\text{ch}}$ is defined as a shifted distance between coordinates of each spectrum in the chromaticity diagram due to the modulation. Figure 8 shows chromaticity coordinates of previous experimental results and corresponding values of ${M}_{\text{ch}}$. As shown, the resonant QD sample revealed larger ${M}_{\text{ch}}$ (Fig. 8(a)), whereas the non-resonant QD sample revealed relatively small values of ${M}_{\text{ch}}$ (Fig. 8(b)). For comparison with ${M}_{\text{sp}}$ in Table 1, Table 2 shows a list of all values of ${M}_{\text{ch}}$ for the samples described in Table 1. A similar tendency of the modulatabilies ${M}_{\text{ch}}$ in Table 2 was obtained. Because we used similar types of QD pairs, in our experiment, similar shift vectors from the two coordinates were obtained. That is to say, we can evaluate each modulation, not only in terms of the amount but also the type, based on possible directions of each vector from before modulation to after modulation. Evaluation of the type is a fundamental issue for further discussions on applications utilizing their one-to-many correspondences.

## 5. Summary

In conclusion, we have described the basic concept of Modulatable Nanophotonics and numerically and experimentally demonstrated the concept by utilizing resonant and non-resonant QD pairs dispersed on flexible substrates. The modulatability was qualitatively evaluated by introducing modulatabilities ${M}_{\text{sp}}$ and ${M}_{\text{ch}}$. Resonant QD pairs exhibited unique modulatability of their emission spectra, which depends on controlling the magnitude of the coupling strength between the QDs via optical near-field interactions. Such modulatability cannot be obtained with non-resonant QD pairs.

From the viewpoint of information retrieval, the results of our demonstration indicate that this method can retrieve the effects of optical near-field interactions as a modulation of the optical far-field response. As we demonstrated in this paper, selecting appropriate resonant QD pairs can realize various *modulatabilities* of the emission spectrum based on nanophotonics. By developing the concept further and experimentally examining various implementations, our idea can be applied to a *modulatable multi-spectrum emitting element* whose emission spectrum can be freely switched by applying external modulation.

## Acknowledgments

This work was supported in part by a comprehensive program for personnel training and industry–academia collaboration based on projects funded by the New Energy and Industrial Technology Development Organization (NEDO), Japan, as well as the Global Center of Excellence (G-COE) “Secure-Life Electronics” and Special Coordination Funds for Promoting Science and Technology sponsored by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

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