1. Introduction

Recent advances in the field of optical near-field interactions between sub-wavelength-size objects allow the design of optical devices and systems at densities beyond those conventionally limited by the diffraction limit of light [1, 2]. This higher integration density, however, is only one of benefits of optical near-fields over electronics. Another benefit is that optical near-fields are physically based on the flow of excitation, not electron transfer. Based on these features, theoretical frameworks for signal transfer via optical near-fields have been proposed, such as angular spectrum representations [3] or the virtual photon model [4]. Another aspect that could be exploited in devices and systems is the physical hierarchy of optical near-fields, meaning that the system exhibits different physical behavior at different scales [5]. More concretely, a certain nanostructure may appear differently depending on how we observe it, for example, depending on the size of a fiber probe used to measure it. In this paper, we demonstrate one of the simplest forms of such a hierarchy in optical near-fields and associate it with an information hierarchy.

Hierarchy of information can be found, for example, in terms of its meaning, quality, or level of importance, such as detailed and abstract information, high-resolution and low-resolution information, high-security and less-critical information, and so forth. One of the most simple and fundamental structures in such a hierarchy that we will discuss in this paper is one where several data items in a “detailed” layer are categorized as a single data item in an “abstract” layer. Such a hierarchy can be considered as an additional degree-of-freedom in conventional data multiplexing technologies used to record multiple independent data items in a recording medium [6, 7].

This paper is organized as follows. In Section 2, we discuss the physical hierarchy in optical near-fields considered in this paper. Section 3 discusses a logical model applied to two-mode optical memory retrieval. Section 4 shows numerical simulations and experimental results. Section 5 concludes the paper.

2. A scale-dependent hierarchy in optical near-fields

The hierarchy we consider in this paper is the scale-dependency of optical near-field interactions. We begin with a physical model of optical near-fields based on dipole-dipole interactions [8]. Suppose that a probe, which is modeled by a sphere of radius r _P, is placed close to a sample to be observed, which is modeled as a sphere of radius r _S. Figure 1(a) shows three different sizes for the probe and the sample. When they are illuminated by incident light whose electric field is E ₀, electric dipole moments are induced in both the probe and the samples; these moments are respectively denoted by p _P=α_P E ₀ and p _S=α_S E ₀. The electric dipole moment induced in the sample, p _S, then generates an electric field, which changes the electric dipole moment in the probe by an amount Δp _P=Δα_P E ₀. Similarly, p _P changes the electric dipole moment in the sample by Δp _S=Δα_S E ₀. These electromagnetic interactions are called dipole-dipole interactions. The scattering intensity induced by these electric dipole moments is given by

where Δα =Δα_S =Δα_P [8]. The second term in Eq. (1) shows the intensity of the scattered light generated by the dipole-dipole interactions, containing the information of interest, which is the relative difference between the probe and the sample. The first term in Eq. (1) is the background signal for the measurement. Therefore, the ratio of the second term to the first term of Eq. (1) corresponds to a signal contrast, which will be maximized when the sizes of the probe and the sample are the same (r _P=r _S), as shown in Fig. 1(b). (Detailed derivation is found in reference [8].) Accordingly, the spatial resolution varies depending on the sizes of the probe and the sample. Figure 1(c) shows normalized signal intensity profiles corresponding to three cases where the sum r _S+r _P is respectively D, 3D, and 6D (D=80 nm). Thus, we can see a scale-dependent physical hierarchy in this framework, where a small probe, say r _P=D/2, can nicely resolve objects with a comparable resolution, whereas a large probe, say r _P=3D/2, cannot resolve detailed structure but it can resolve structure with a resolution comparable to the probe size. Therefore, although a large diameter probe cannot detect smaller-scale structure, it could detect certain features that are associated with a nanostructure; we discuss this aspect in more detail below.

 

Fig. 1. A simple spatial hierarchy observed in optical near-fields. (a) Dipole-dipole interaction. (b) Signal contrast as a function of the ratio of the radius of the sample and the probe. (c) Spatial resolution varies depending on the scale of the sample and the probe. (d) Array of dipole moments distributed on sub-wavelength scale and its observation in near-mode and far-mode.

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We consider the observation of an array of small particles distributed with sub-wavelength pitch using a probe whose diameter is substantially larger than that of observed particles. As discussed earlier, the spatial distribution of the particles can be obtained by using a fiber probe of comparable size. (We call this situation the “near-mode”.) On the other hand, the distribution cannot be resolved by a large fiber probe. (We call this situation the “far-mode”.) This far-mode is approximately equivalent to a case where we observe an array of dipole moments distributed with sub-wavelength scale in a far-field. Here, as schematically shown in Fig. 1(d), each of the dipole moments is assumed to be induced in a small particle centered on a circle on the xy plane, and the minimum distance between adjacent particles is b. The electric field at position r in Fig. 1(d) is given by

where ω is the operating frequency, k is the wave number, and s_θ represents the position of a dipole specified by θ (where θ = 1, 2, …). The existence of the dipole at position sθ is indicated by a near-code, which will be introduced later in Section 3, that is

If we assume that b<<λ<<r, which corresponds to the far-mode, Eq. (2) is simplified to

which means that the electric field intensity at position r is proportional to the number of dipole moments given by the near-code. When the near-code consists of n bits, the number of dipole moments can range from zero to n. Therefore, n+1 different signal levels appear in the signal obtained in far-mode. On the other hand, the total number of bit combinations in the near-code is 2ⁿ, which are resolvable in the near-mode.

3. A logical model for hierarchical memory systems

We focus on the differences in the number of signal levels and combinations discussed in Section 2 using a simple digital coding scheme.

A two-layer hierarchical memory based on concepts that we call far-codes and near-codes is described in this section. The far-code depends on an array of bits distributed within a certain area and is determined logically to be either ZERO or ONE. Consider an (N+1)-bit digital code, where N is an even number. Now, let the far-code be defined depending on the number of ONEs (or ZEROs) contained in the (N+1)-bit digital code according to

 

Fig. 2. Example of logical model for the near-code and far-code. (a) Original 8-bit information is coded differently in the near-code depending on its corresponding far-code, which is either ZERO or ONE. (b) Example of the near-codes and far-codes. (c) Using such encoding and decoding principles, nanoparticle arrangements associated with different levels of the information hierarchy.

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The (N+1) digits provide a total of 2^N+1 possible permutations, or codes. Here, we note that half of them, namely 2^N permutations, have more than N/2+1 ONEs among the (N+1) digits (i.e., far-code = 1), and the other half, also 2^N permutations, have more than N/2+1 ZEROs (i.e., far-code = 0). In other words, 2^N different codes could be assigned to two (N+1)-bit digital sequences so that their corresponding far-codes are ZERO and ONE, respectively. We call this (N+1)-bit code a near-code.

Example near-codes are listed in Fig. 2 for N = 8. The correspondence between 2^N original codes and the (N+1)-bit near-codes is arbitrary. Therefore, we use a table-lookup when decoding an (N+1)-bit near-code to obtain the original code. The example near-codes shown in Fig. 2(a) are listed in ascending order, but other lookup-tables or mappings are also possible.

Figure 2(b) schematically demonstrates example codes in which a 9-bit near-code is represented as a 3×3 array of circles, where black and white mean ONE and ZERO, respectively. Here, if the number of ONEs in the near-code is larger than five, then the far-code is ONE, otherwise the far-code is ZERO. Suppose, for example, that the far-code stores text data and the near-code stores 256-level (8-bit) image data. Consider a situation where the far-code should represent an ASCII code for “A”, whose binary sequence is “0100001”. Here, we assume that the gray levels of the first two pixels, which will be coded in the near-code, are the same value. (Here, they are “92”.) However, the first two far-codes are different (ZERO followed by ONE). Referring to the rule shown in Fig. 2(a), and noticing that the first far-code is ZERO and the near-code should represent “92”, the first near-code should be “001101010”. In the same way, the second near-code is “110001011”, which represents “92”, whereas its corresponding far-code is ONE. Based on such an encoding and decoding scheme, the near-codes and far-codes could be respectively associated with image and text data for multimedia applications, or with high-security and less-critical verifications for security applications, as schematically shown in Fig. 2(c).

4. Simulations and experiments

Simulations were performed to verify the principles of hierarchical access to memory devices by assuming ideal isotropic metal particles to see how the scattering light varies depending on the number of particles in the far-code by using a Finite Difference Time Domain (FDTD)-based simulator (Poynting for Optics, a product of Fujitsu, Japan). Here we assume that 80-nm-diameter particles are distributed around a 200-nm-radius circle with a constant angular interval, as shown in Fig. 3(a). Seven particles at most can be arranged in the ring. The wavelength used in the simulation was 600 nm. The solid squares in Fig. 3(b) show calculated scattering cross-sections as a function of the number of particles in the ring. The scattering cross-section increases nearly linearly with the number of particles in the ring. This result is expected from the physical model described in Section 2.

In order to experimentally demonstrate this behavior, we measured far-mode intensity distributions of an array of Au particles, each with a diameter around 80 nm, distributed on a SiO₂ substrate in a 200-nm-radius circle. The inset of Fig. 1(a) shows an SEM image of one such particle distribution. These particles were fabricated by a liftoff technique using electron-beam (EB) lithography with a Cr buffer layer. Multiple groups of Au particles were separated by 2 μm. In the SEM image shown in Fig. 3(d), the values indicate the number of particles within each group. In order to illuminate all Au particles in each group and collect the scattered light from them, we used a near-field optical microscope in an illumination collection setup with an apertured fiber probe having a diameter (D) of 500 nm, as shown in Fig. 3(c). The light source used was a laser diode with an operating wavelength of 680 nm. The distance between the substrate and the probe was maintained at 750 nm. Figure 3(e) is an intensity pattern captured by the probe, from which the information in the far-mode is retrieved. The solid circles in Fig. 3(b) indicate the peak intensity of each section, which increased substantially linearly. The signal level difference in the far-mode is small when the difference in the number of particles is small, and therefore, a better way of resolving signals in the far-mode will be required. However, the experimental results shown here demonstrate that this is one fundamental architecture that can exploit the physical hierarchy.

 

Fig. 3. (a) Each section consists of small particles. (b, square marks) Scattering cross-sections are calculated depending on the number of particles in each section. (b, circular marks) Peak-intensity of each section in intensity profile shown in (e). (c) Experimental setup. (d) SEM picture of an Au particle array. (e) Intensity pattern captured for the far-code.

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Fig. 4. Scattering cross-section as a function of number of elements forming donut.

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Concerning the size and the pitch of the particles used in the simulation and experiments, the electric field enhancement due to closely placed neighboring particles is negligible [9, 10]; this is one condition to ensure that a particular far-mode signal level does not significantly differ depending on the distribution of the particles so long as the number of particles is the same, as shown by the solid squares in Fig. 3(b). Such electric field enhancement, however, could provide another degree-of-freedom in the design of hierarchical systems. For example, when we choose a donut-shape, instead of a point particle, for each of the bits in near-code, the far-mode signals could show a step-like response instead of a nearly linear response, according to the number of elements forming the donut. Fig. 4 shows scattering cross-sections obtained by FDTD simulations assuming an 80-nm-wide (W), 200-nm-radius (R) donut shape. In this example, we arrange the combination of elements forming the donut so that the signal increases significantly when the number of elements forming the donut is larger than five. This indicates that the digitization procedure, for instance required in Eq. (5), can be physically realized by engineering the shape of each bit on the nanometer scale.

5. Summary

Optical near-field interactions are observed differently depending on the physical scale of the observation. Noticing that information also has an inherent hierarchy, we present a hierarchical memory access scheme in which the so-called near-mode is used to read detailed dipole distributions, whereas the far-mode is used to detect features within a region-of-interest. Simulations and experimental results using metal nanoparticles are also shown. With a simple coding scheme, different physical accesses to the media are associated with different hierarchical information. Further studies on hierarchies in optical near-fields, regarding their fundamental theory [3], experimental realization, applications, and so forth, are currently underway.