1. Introduction

Interconnects are a critical issue for information processing devices and systems [1], and optical interconnects have been thoroughly investigated for their ability to overcome the limitations of their electrical counterparts [2, 3]. Nanophotonics requires yet another type of interconnect since it is based on local electromagnetic interactions between a few nanometer-size particles, such as quantum dots (QDs), via optical near fields, which in turn allows device integration at densities beyond the diffraction limit [4]. For the interconnects in such high-density nanophotonic devices, far- and near-field conversion devices are required, as shown in Fig. 1(a); for instance, plasmon waveguides [5, 6] and nano-dot couplers [7] have been successfully demonstrated. However, since many of such conversion devices are needed for all nanophotonic devices distributed within a sub-wavelength scale region, such interconnection is another serious bottleneck for system integration. In this paper, we propose an interconnection method based on both far- and near-field interactions for data broadcasting purposes, meaning that multiple nanophotonic devices are supplied with the same input data.

2. Broadcast interconnects

We first describe the application of data-broadcasting interconnects to nanophotonic circuits and their physical principles. Data broadcasting is a fundamental functionality required for computing operations such as matrix-vector products [8, 9] and switching operations, such as those used in a broadcast-and-select architecture [10], where multiple functional blocks require the same input data, as schematically shown in Fig. 1(a). For example, consider a matrix-vector multiplication given by v = As, where v = (v ₁,⋯,v_m), s = (s ₁,⋯,s_n), and A is an m × n matrix. Here, to compute every v_j from the input data s, broadcast interconnects are required if each v_j. is calculated in distinct processing hardware. Optics is in fact well suited to such broadcast operations in the form of simple imaging optics [8, 9] or in optical waveguide couplers thanks to the nature of optical wave propagation. However, the integration density is physically limited by the diffraction limit, which leads to bulky system configurations.

 

Fig. 1. (a) Interconnections from macro-scale external systems to sub-wavelength-scale nanophotonic systems. (b) Broadcast interconnects.

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3. Operating principle

The physical operating principle of broadcasting is described here. An irreversible energy transfer between neighboring QDs is possible via local optical near-field interactions and intrasublevel relaxation, as schematically shown in Fig. 2(c) [4, 11–13]. This unique feature enables nonlinear functions, such as optical switching [14], convergence of energy [15], and data summation [16], as well as extremely high integration density. From the viewpoint of interconnects, however, it places stringent requirements on individual addressability since the devices are arrayed on the sub-wavelength scale, as mentioned in the introduction. However, data broadcasting allows another interconnection scheme. Suppose that arrays of nanophotonic circuit blocks, such as the nanophotonic switches described later, are distributed within an area whose size is comparable to the wavelength, as shown in Fig. 2(a). Here, for broadcasting, multiple input QDs simultaneously accept identical input data carried by diffraction-limited far-field light by tuning their optical frequency so that the light is coupled to energy sublevels that are associated with the input of each block, while not affecting the output and internal operations of each block, as illustrated in Fig. 2(b) and described in more detail below. In a frequency multiplexing sense, this interconnection method is similar to multi-wavelength chip-scale interconnects [17]. Known methods require a physical space comparable to the number of diffraction-limited input channels, due to wavelength demultiplexing, whereas in our proposed scheme, the device arrays are integrated on the sub-wavelength scale, and multiple frequencies are multiplexed in the far-field light supplied to the device.

Here we explain the far- and near-field coupling mentioned above based on a model assuming CuCl QDs, which are also employed in experiments described below. The potential barrier of CuCl QDs in an NaCl crystal can be regarded as infinitely high, and the energy eigenvalues for the quantized exciton energy level (n_x,n_y,n_z) in a CuCl QD with side of length L are given by

where E_B is the energy of the bulk Z ₃ exciton, M is the mass of the exciton for the motion of the center of mass (C. M. ), a_B is its Bohr radius, n_x, n_y, and n_z are quantum numbers for the C. M. motion (n_x, n_y, n_z =1,2,3,⋯), and a = L-a_B corresponds to an effective side length taking into account so-called dead layer correction [18]. According to Eq. (1), there exists a resonance between the quantized exciton energy sublevel of quantum number (1, 1, 1) for the QD with effective side length a and that of quantum number (2, 1, 1) for the QD with effective side length √2a. (For simplicity, we refer to the QDs with effective side lengths a and √2a as “QD a” and “QD √2a”, respectively.) Energy transfer between QD a and QD √2a occurs via optical near fields [11–13]. It should be noted here that the optical selection rule tells us that the exciton energy levels with even quantum numbers do not contribute to the optical transition, meaning that the electric dipole transitions between the ground state and the states with even quantum numbers are prohibited for far-field light [18, 19]. Optical near-fields, on the other hands, allow excitation of such states having even quantum numbers thanks to the local steep variation of electric field, as illustrated in Fig. 2(c). (Details are shown in Ref [4].) In this paper, we call exciton states with even quantum numbers “dipole-forbidden” states (or equivalently, dipole-forbidden energy sublevels) and states with odd quantum numbers “dipole-allowed” states (or equivalently, dipole-allowed energy sublevels). Far-field light can couple to dipole-allowed energy sublevels but cannot couple to dipole-forbidden energy sublevels.

In order to easily understand the principle of the broadcast interconnects shown later, here we introduce a frequency-and-quantum-dot-size diagram in Fig. 2(d), where the horizontal axis shows QD size and the vertical axis shows energy level or its equivalent optical frequency. The 3-digit sets in the diagram are the quantum numbers of the exciton states. In the example shown in Fig. 2(d), the (1, 1, 1)-level of the QD a and the (2, 1, 1)-level of the QD √2a exist at the same vertical position.

We note that the input energy level for the QDs, that is, the (1, 1, 1)-level, can also couple to the far-field excitation. We utilized this fact for data broadcasting. One of the design restrictions is that energy-sublevels for input channels do not overlap with those for output channels. Also, if there are QDs internally used for near-field coupling, dipole-allowed energy sublevels for those QDs cannot be used for input channels since the inputs are provided by far-field light, which may lead to misbehavior of internal near-field interactions if resonant levels exist. Therefore, frequency partitioning among the input, internal, and output channels is important. The frequencies used for broadcasting, denoted by Ω_i = {ω¯ _i,1,ω¯ _i,2,⋯,ω¯_i,A}, should be distinct values and should not overlap with the output channel frequencies Ω_o = {ω¯ _o,1,ω¯ _o,2,⋯,ω¯_i,B}. A and B indicate the number of frequencies used for input and output channels, respectively. Also, there will be frequencies needed for internal device operations that are not used for either input or output (discussed later in the sum of product example), denoted by Ω_n = {ω¯ _n,1,ω¯ _n,2,⋯,ω¯_n,C}, where C is the number of those frequencies. In other words, the design criteria for global data broadcasting is to exclusively assign input, output, and internal frequencies, Ω_i, Ω_o, and Ω_n, respectively. Figure 3 illustrates two examples of frequency partitioning using frequency-and-quantum-dot-size diagrams.

 

Fig. 2. (a) Broadcast-type interconnects to nanophotonic device arrays. (b) Near-field interaction between quantum dots for internal functions. (c) Far-field excitation for identical data input (broadcast) to nanophotonic devices within a diffraction-limit-sized area. (d) Frequency-and-quantum-dot-size diagram.

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In the example shown in Fig. 3(a), we used a nanophotonic switch (2-input AND gate) composed of three QDs with a size ratio of 1: √2: 2. The two input channels are assigned to QD a and QD 2a, and the output appears from QD √2a. The underlining principles are the irreversible signal transfer via resonant energy levels, which was described earlier. When the (1, 1, 1)-level of the QD 2a is occupied (state filling), an exciton in QD a will be transferred to the (1,1,1)-level of QD √2a and dissipated. Therefore, the output from QD √2a is at a high level. On the other hand, when the (1, 1, 1)-level of the QD 2a is not occupied, an exciton in QD a will be transferred to the (1,1,1)-level of QD 2a and dissipated, and so the output from the QD √2a is at a low level. The details of the switching principle are shown in Ref [14]. Here, multiple input dots QD a and QD 2a can accept identical input data via far-field light for broadcasting purposes.

 

Fig. 3. Frequency partitioning among external and internal channels, and examples in (a) multiple implementations of 3-dot nanophotonic switches, and (b) 4-dot configuration for sum of products.

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Adding more optical switches for different channels means adding different size dots, for instance, by multiplying the scale of the QDs by a constant while maintaining the ratio 1:√2:2, such as a QD trio of 2√2a, 4a, and √2a, so that the corresponding far-field resonant frequencies do not overlap with the other channels. More dense integration is also possible by appropriately configuring the size of the QDs. As an example, consider a QD whose size is $\sqrt{\frac{4}{3}}a$. The (1, 1, 1)-level in this QD $\sqrt{\frac{4}{3}}a$ can couple to the far-field excitation. It should be noted that this particular energy level is equal to the (2, 2, 1)-level in QD 2a, which is an already-used input QD; however, the far-field excitation in this particular energy level cannot couple to QD 2a since the (2, 2, 1)-level in QD 2a is a dipole-forbidden energy sublevel. Therefore, a QD trio composed of QDs of size $\sqrt{\frac{4}{3}}a$, $\sqrt{\frac{8}{3}}a$, and $\sqrt{\frac{16}{3}}a$ can make up another optical switch, while not interfering with other channels even though all of the input light is irradiated in the same area.

Another situation where an internally used frequency exists is a sum of products operation. A simplified example is shown in Fig. 3(b). The QD a and QD 2a operate on two inputs and their product appears in the (1, 1, 1)-level in QD √2a, which is further coupled to the sublevel (4,2,2) in QD 4a. The QD 4a is the output dot. Here, the QD √2a is internally used; thus any frequency that could couple to QD √2a cannot be used for other input channels.

We now briefly discuss the operation speed and the number of channels. (i) Speed: The operation speed of such systems involving optical near-field interactions is typically on the order of nanoseconds if the system involves a radiative dissipation process at the output QD. The internal operation speed is determined by optical near-field interactions between neighboring QDs, which is typically around one hundred picoseconds [4]. (ii) Total number of channels available: This depends on many factors, such as inhomogeneous and homogeneous broadening of the QDs, the linewidth of the input light, the configuration of the system (or the configuration of the QDs, such as the examples shown in Fig. 3), and so forth. Therefore, as an ultimate limiting factor, here we theoretically deal with the maximum size of the QD based on Eq. (1). The energy difference between the (2, 1, 1)-level and the (1, 1, 1)-level, or the intrasublevel relaxation, is derived from Eq. (1) as

which decreases as the QD becomes larger, but it should be larger than the intrasublevel relaxation constant in order to resolve those two energy levels. In the case of CuCl QDs, assuming M=2.3m_e [18] and an intrasublevel relaxation constant of ħΓ with Γ^-1 around 1 ps [4], the maximum QD size is around 30 nm. This will act as a limitation in the design of the system.

4. Experiment

To verify the broadcasting method, we performed the following experiments using CuCl QDs in an NaCl matrix at a temperature of 22 K. To operate a 3-dot nanophotonic switch (2-input AND gate), we irradiated at most two input light beams (IN1 and IN2). The experimental setup is summarized in Fig. 4(a). The input light IN1 and IN2 were assigned to 325 nm and 384.7 nm, which were coupled to QD a and QD 2a, respectively. They were irradiated over the entire sample (global irradiation) via far-field light. The spatial intensity distribution of the output signal, at 382.6 nm, which was obtained from QD √2a, was measured by scanning a near-field fiber probe. There were several nanophotonic switches (or AND gates) within an approximately 1 μm × 1 μm area, as schematically shown in Fig. 4(b). As described in Sec. 3, the output of the AND gate should appear only when both of the inputs are supplied. In Fig. 5(a), when only IN1 was applied to the sample, the output of the AND gate was ZERO (low-level), whereas in Fig. 5(b), when both inputs were irradiated, the output was ONE (high-level). Note the regions marked by ■, ●, and ◆. In those three regions, the output signal levels were respectively low and high in Figs. 5(a) and (b), which indicates that multiple AND gates were integrated at densities beyond the scale of the globally irradiated input beam area. That is to say, broadcast interconnects to nanophotonic switch arrays were accomplished by diffraction-limited far-field light.

 

Fig. 4. (a) Experimental setup. (b) Three nanophotonic switches (3-dot AND gates) are distributed within 1 μm × 1 μm area in the sample.

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The following is a description of the sample used in terms of the number of switches observed, namely, three, as demonstrated above. The CuCl QDs were inhomogeneously distributed in an NaCl matrix by using the Bridgman method [20], which was also employed in previous demonstrations [14–16]. Absorption and emission spectra of the sample measured at 5 K are shown in Fig. 5(c) by the thin and thick lines, respectively. The vertical dotted lines in the figure show photon energies that correspond to the (1, 1, 1)-levels of QDs of different size, as specified units of monolayers (ML), where 1 ML is equal to 0.2705 nm. Clear absorption and photoluminescence peaks were observed at a photon energy corresponding to a QD size of 17 ML, and thus, the output energy level was set to 3.24 eV for the output QD √2a, as shown above. Accordingly, the size of QD 2a, which couples to IN2, corresponded to a photon energy of 3.22 eV. The average distance between QDs was estimated at around 20 nm since the CuCl concentration was 1 mol% and the average dot size was around 4 nm. Therefore, the number of QDs in the observed volume of 1 μm × 1 μm × 30 nm was approximately 3,000. From the emission spectrum, it was estimated that QD √2a (3.24 eV) and QD 2a (3.22 eV) respectively occupied approximately 4% and 1% of all QDs, and hence the numbers of QD √2a and QD 2a in the scanned sample volume were 120 and 30, respectively. Here, if we virtually divide the entire sample into unit volumes of 30 nm × 30 nm × 30 nm, called a voxel, which is the volume occupied by a single 3-dot nanophotonic switch, there are in total 900 such voxels. Since the other input, IN1, was assigned to 325 nm, it excited all QDs in the sample, and hence there was at least one QD that functioned as IN1 in a voxel. Therefore, the expected number of 3-dot nanophotonic switches (i.e., the expected number of voxels in which an appropriate-size QD trio resides simultaneously) within a volume of 1 μm × 1 μm × 30 nm will be approximately (120/900) × (30/900) × 900 (=4), which is in good agreement with the experimental observation in Figs. 5(a) and (b).

5. Summary

In summary, broadcast interconnects for nanophotonic devices have been proposed and experimentally demonstrated using far- and near-field interactions. Combining this broadcasting mechanism with switching [14] and summation [16] will allow nano-scale integration of optical parallel processing devices, which have conventionally resulted in bulky systems.

 

Fig. 5. Experimental results. (a, b) Spatial intensity distribution of the output of 3-dot AND gates. (a) Output level: low (1 AND 0 = 0), and (b) output level: high (1 AND 1 = 1). (c) Absorption and photoluminescence spectrum of the sample.

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