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Nanophotonic Systems Based on Localized and Hierarchical Optical Near-Field Processes

  • Makoto Naruse
Reference work entry

Abstract

Nanophotonics offers ultrahigh-density system integration since it is based on local interactions between nanometer-scale matter via optical near-fields and is not constrained by the diffraction limit. In addition, it also gives qualitatively novel benefits over conventional optics and electronics. From a system architectural perspective, nanophotonics drastically changes the fundamental design rules of functional optical systems, and suitable architectures may be built to exploit this. This chapter discusses system architectures for nanophotonics, taking into consideration the unique physical principles of optical near-field interactions, and also describes their experimental verification based on enabling technologies, such as quantum dots and engineered metal nanostructures. In particular, two unique physical processes in light–matter interactions on the nanometer scale are examined. One is optical excitation transfer via optical near-field interactions, and the other is the hierarchical property of optical near-fields. Also, shape-engineered nanostructures and their associated polarization properties are characterized from a system perspective, and some applications are shown. The architectural and physical insights gained enable realization of nanophotonic information systems that overcome the limitations of conventional light and provide unique functionalities that are only achievable using optical near-field processes.

Keywords

Elemental Shape Content Addressable Memory Layout Factor Polarization Conversion Efficiency Original Hologram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

23.1 Introduction

To accommodate the continuously growing amount of digital data and ubiquitous devices, as well as qualitatively new requirements demanded by industry and society, such as safety and security, optical technologies are expected to become more highly integrated and to play a wider role in enhancing system performance. However, many technological difficulties remain to be in overcome in adopting optical technologies in critical information and communication systems; one problem is the poor integrability of optical hardware due to the diffraction limit of light [1, 2].

Nanophotonics, on the other hand, which is based on local interactions between nanometer-scale matter via optical near-fields, offers ultrahigh-density integration since it is not constrained by the diffraction limit. Fundamental nanophotonic processes, such as optical excitation transfer via optical near-fields between nanometer-scale matter, have been studied extensively [3, 4, 5]. This higher integration density is not the only benefit of optical near-fields over conventional optics and electronics. From a system architectural perspective, nanophotonics drastically changes the fundamental design rules of functional optical systems, and suitable architectures may be built to exploit this. As a result, nanophotonics will have a strong impact in terms of qualitative improvements to information and communication systems.

This chapter discusses system architectures for nanophotonics, taking into consideration the unique physical principles of optical near-field processes, and describes their experimental verification based on technological vehicles such as quantum dots and engineered metal nanostructures. In particular, two unique physical processes in light-matter interactions on the nanometer scale are exploited. One is optical excitation transfer via optical near-field interactions, and the other is the hierarchical property of optical near-field interactions.

The overall concept of this chapter is outlined in Fig. 23.1. Section 23.2 discusses system architectures based on optical excitation transfer. Section 23.3 investigates networks of optical near-field interactions. Section 23.4 discusses hierarchical architectures based on optical near-field processes, followed by discussions on shape-engineered nanostructures in Sect. 23.5. The architectural and physical insights gained enable the realization of nanophotonic information and communications systems that can overcome the integration-density limit imposed by the diffraction of light while providing ultra low-power operation and unique functionalities that are only achievable using optical near-field interactions.
Fig. 23.1

Overview of the chapter: Nanophotonic systems based on optical excitation transfer mediated by optical near-field interactions and hierarchical properties in optical near-fields are examined

23.2 System Architectures Based on Optical Excitation Transfer

23.2.1 Optical Excitation Transfer via Optical Near-Field Interactions and Its Functional Features

In this section, optical excitation transfer processes involving optical near-field interactions are reviewed from a system perspective. First, their fundamental principles are briefly reviewed, and then their functional features are introduced for later discussion.

The interaction Hamiltonian between an electron and an electric field is given by
$$\hat{H}_{int} = -\displaystyle\int \hat{{\psi }}^{\dag }({r}){\mu }\hat{\psi }({r}) \bullet \hat{{D}}({r})d{r},$$
(23.1)
where \({\mu }\) is the dipole moment, \(\hat{{\psi }}^{\dag }({r})\) and \(\hat{\psi }({r})\) are respectively creation and annihilation operators of an electron at \({r}\), and \(\hat{{D}}({r})\) is the operator of electric flux density. In usual light–matter interactions, the operator \(\hat{{D}}({r})\) is a constant since the electric field of propagating light is considered to be constant on the nanometer scale. Therefore, as is well known, one can derive optical selection rules by calculating a transfer matrix of an electric dipole. As a consequence, in the case of cubic quantum dots, for instance, transitions to states described by quantum numbers containing an even number are prohibited. In the case of optical near-field interactions, on the other hand, due to the steep electric field of optical near-fields in the vicinity of nanoscale matter, an optical transition that violates conventional optical selection rules is allowed.
Optical excitations in nanostructures, such as quantum dots, can be transferred to neighboring ones via optical near-field interactions [3, 4]. For instance, assume two cubic quantum dots whose side lengths L are a and \(\sqrt{ 2}a\), which are called QD S and QD L , respectively (see Fig. 23.2a). Suppose that the energy eigenvalues for the quantized exciton energy level specified by quantum numbers \((n_{x},n_{y},n_{z})\) in a QD with side length L are given by
$$E_{(n_{x},\,n_{y},\,n_{z})} = E_{B} + \frac{{\hslash {}^{2}\pi }^{2}} {2M{L}^{2}}\left (n_{x}^{2} + n_{ y}^{2} + n_{ z}^{2}\right ),$$
(23.2)
where E B is the energy of the bulk exciton and M is the effective mass of the exciton. According to Eq. (23.2), there exists a resonance between the level of quantum number (1,1,1) for QD S and that of quantum number (2,1,1) for QD L . There is an optical near-field interaction, which is denoted by U​, due to the steep electric field in the vicinity of QD S . Therefore, excitons in QD S can move to the (2,1,1)-level in QD L . Note that such a transfer is prohibited for propagating light since the (2,1,1)-level in QD L contains an even number [6]. In QD L , the exciton sees a sublevel energy relaxation, denoted by Γ, which is faster than the near-field interaction, and so the exciton goes to the (1,1,1)-level in QD L . It should be emphasized that the sublevel relaxation determines the unidirectional exciton transfer from QD S to QD L .
Fig. 23.2

(a) Optical excitation transfer from a smaller quantum dot to a larger one, mediated by optical near-field interactions. (b) State filling induced at the lower energy level in the larger dot results in different flows of optical excitation

Now, several unique functional aspects should be noted in the above excitation transfer processes. First, as already mentioned, the transition from the (1,1,1)-level in QD S to the (2,1,1)-level in QD L is usually a dipole-forbidden transition. In contrast, such a transition is allowed when intermediated by the optical near-field. Second, in the resonant energy levels of those quantum dots, optical excitations can go back and forth between QD S and QD L , which is called optical nutation. The direction of the excitations is determined by the energy dissipation processes. When another excitation sits in the (1,1,1)-level in QD L , the excitation in the (1,1,1)-level in QD S cannot be transferred to the (1,1,1)-level in QD L , as schematically shown in Fig. 23.2b. Therefore, based on the above mechanisms, the flow of optical excitations can be controlled in quantum dot systems via optical near-field interactions.

From an architectural standpoint, such a flow of excitations directly leads to digital processing systems and computational architectures. First of all, two different physical states can be made to appear by controlling the dissipation processes in the larger dot; this is the principle of the nanophotonic switch [7]. Also, such flow control itself allows an architecture known as a binary decision diagram, where an arbitrary combinatorial logic operation is determined by the destination of a signal flowing from a root [8].

Such optical excitation transfer processes also lead to unique system architectures. In this regard, Sect. 23.2.2 discusses a massively parallel architecture and its nanophotonic implementations. Also, Sect. 23.2.3 demonstrates that optical excitation transfer provides higher tamper resistance against attacks than conventional electrically wired devices by focusing on environmental factors for signal transfer. It should also be noted that optical excitation transfer has been the subject of a wide range of research. For example, Pistol et al. demonstrated resonant energy transfer-based logic [9] using DNA-assisted self-assembly technologies [10] for sensing and other applications. Other system-related investigations using optical excitation transfer include interconnections [11], pulsation mechanisms [12], and skew dependence [13].

23.2.2 Parallel Architecture Using Optical Excitation Transfer

Memory-Based Architecture This section discusses a memory-based architecture where computations are regarded as a lookup table or database search problem, which is also called a content addressable memory (CAM) [14]. The inherent parallelism of this architecture is well matched with the physics of optical excitation transfer and provides performance benefits such as high-density, low-power operation [15].

In this architecture, an input signal (content) serves as a query to a lookup table, and the output is the address of data matching the input. This architecture plays a critical role in various systems, for example, in data routers where the output port for an incoming packet is determined based on lookup tables.

All-optical means for implementing such functions have been proposed, for instance, by using planar lightwave circuits [16]. However, since separate optical hardware for each table entry would be needed in implementations based on today’s known methods, if the number of entries in the routing table is on the order of 10,000 or more, the overall physical size of the system would become impractically large. On the other hand, by using diffraction-limit-free nanophotonic principles, huge lookup tables can be realized with compact configurations.

It is important to note that the lookup table problem is equivalent to an inner product operation. Assume an N-bit input signal \(\mathrm{\mathbf{S}} = (s_{1},\cdots \,,s_{N})\) and reference data \(\mathrm{\mathbf{D}} = (d_{1},\cdots \,,d_{N})\). Here, the inner product \(\mathrm{\mathbf{S}} \bullet \mathrm{\mathbf{D}} =\sum \nolimits _{ i=1}^{N}s_{i} \bullet d_{i}\) will provide a maximum value when the input perfectly matches the reference data with an appropriate modulation format [17]. Then, the function of a CAM is to derive j that maximizes \(\mathrm{\mathbf{S}} \bullet \mathrm{\mathbf{D}}_{j}\).

Global Summation Using Near-Field Interactions As discussed above, the inner product operations are the key functionality of the memory-based architecture. The multiplication of 2 bits, namely, \(x_{i} = s_{i} \bullet d_{i}\), has already been demonstrated by using a combination of three quantum dots [7]. Therefore, one of the key operations remaining is the summation, or data gathering scheme, denoted by \(\sum x_{i}\), where all data bits should be taken into account, as schematically shown in Fig. 23.3.
Fig. 23.3

Memory-based architecture of content addressable memory based on optical excitation transfer. Global summation and broadcast interconnects, which are both important subfunctions for the memory-based architecture, were demonstrated based on optical excitation transfer using different-sized QDs

In known optical methods, wave propagation in free space or in waveguides, using focusing lenses or fiber couplers, for example, is well-matched with such a data gathering scheme because the physical nature of propagating light is inherently suitable for collection or distribution of information, such as in global summation. However, the achievable level of integration of these methods is restricted due to the diffraction limit of light. In nanophotonics, on the other hand, the near-field interaction is inherently physically local, although functionally global behavior is required.

The global data gathering mechanism, or summation, is realized based on the unidirectional energy flow via an optical near-field, as schematically shown in Fig. 23.3, where surrounding excitations are transferred toward a quantum dot QD C located at the center [18, 19]. This is based on the excitation transfer processes presented in Sect. 23.2.1 and in Fig. 23.2a, where an optical excitation is transferred from a smaller dot (QD S ) to a larger one (QD L ) through a resonant energy sublevel and a sublevel relaxation process occurring at the larger dot. In the system shown in Fig. 23.3, similar energy transfers may take place among the resonant energy levels in the dots surrounding QD C so that excitation transfer can occur. The lowest energy level in each quantum dot is coupled to a free photon bath to sweep out the excitation radiatively. The output signal is proportional to the (1,1,1)-level population in QD L .

A proof-of-principle experiment was performed to verify the nanoscale summation using CuCl quantum dots in an NaCl matrix, which has also been employed for demonstrating nanophotonic switches [7] and optical nano-fountains [19]. A quantum dot arrangement in which three small QDs (QD1 to QD3) surrounded a large QD at the center (QD C ) was chosen. Here, at most three light beams with different wavelengths, 325, 376, and 381.3 nm, are radiated to excite the respective quantum dots QD1 to QD3, having sizes of 1, 3.1, and 4.1 nm. The excited excitons are transferred to QD C , and their radiation is observed by using a near-field fiber probe tip. Notice the output signal intensity at a photon energy level of 3.225 eV in Fig. 23.3, which corresponds to a wavelength of 384 nm, or a QD C size of 5.9 nm. The intensity varies approximately as 1:2:3, depending on the number of excited QDs in the vicinity, as observed in Fig. 23.3. The spatial intensity distribution was measured by scanning the fiber probe, as shown in the bottom-right corner of Fig. 23.3, where the energy is converged at the center. Hence, this architecture works as a summation mechanism, counting the number of input channels, based on exciton energy transfer via optical near-field interactions.

Such a quantum-dot-based data gathering mechanism is also extremely energy efficient compared with other optical methods, such as focusing lenses or optical couplers. For example, the transmittance between two materials with refractive indexes n 1 and n 2 is given by 4n 1 n 2/\({(n_{1} + n_{2})}^{2}\); this gives a 4 % loss if n 1 and n 2 are 1 and 1.5, respectively. The transmittance of an N-channel guided wave coupler is 1/N from the input to the output. In nanophotonic summation, the loss is attributed to the dissipation between energy sublevels, which is significantly smaller. Incidentally, it is energy and space efficient compared with electrical CAM VLSI chips [15, 20, 21].

Broadcast Interconnects For the parallel architecture shown above, it should also be noted that the input data should be commonly applied to all lookup table entries. In other words, a broadcast interconnect is another important requirement for parallel architectures. Broadcast is also important in applications such as matrix–vector products [22, 23] and switching operations, for example, broadcast-and-select architectures [24]. Optics is in fact well suited to such broadcast operations in the form of simple imaging optics [22, 23] or in optical waveguide couplers, thanks to the nature of wave propagation. However, the integration density of this approach is physically limited by the diffraction limit, which leads to bulky system configurations.

The overall physical operation principle of a broadcast using optical near-fields is as follows [25]. Suppose that arrays of nanophotonic circuit blocks are distributed within an area whose size is comparable to the wavelength. 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 dipole-allowed energy sublevels.

The far- and near-field coupling mentioned above is explained based on a model assuming cubic quantum dots, which was introduced in Sect. 23.2.1. According to Eq. (23.2), 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 \(\sqrt{ 2}a\). Energy transfer from the smaller QD to the larger one occurs via optical near fields, which is forbidden for far-field light [7].

The input energy level for the QDs, that is, the (1,1,1)-level, can also couple to the far-field excitation. This fact can be utilized 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 \(\Omega _{i} = \left \{\omega _{i,1},\omega _{i,2},\cdots \,,\omega _{i,A}\right \}\), should be distinct values and should not overlap with the output channel frequencies \(\Omega _{o} = \left \{\omega _{o,1},\omega _{o,2},\cdots \,,\omega _{o,B}\right \}\). 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, which are not used for either input or output, denoted by \(\Omega _{n} = \left \{\omega _{n,1},\omega _{n,2},\cdots \,,\omega _{n,C}\right \}\), where C is the number of those frequencies. Therefore, the design criteria for global data broadcasting are to exclusively assign input, output, and internal frequencies, Ω i , Ω o , and Ω n , respectively.

In a frequency multiplexing sense, this interconnection method is similar to multi-wavelength chip-scale interconnects [26]. Known methods, however, require a physical space comparable to the number of diffraction-limited input channels due to wavelength demultiplexing, whereas in the nanophotonic scheme, the device arrays are integrated on the subwavelength scale, and multiple frequencies are multiplexed in the far-field light supplied to the device.

To verify the broadcasting method, the following experiments were performed using CuCl QDs inhomogeneously distributed in an NaCl matrix at a temperature of 22 K [25]. To operate a three-dot nanophotonic switch (two-input AND gate) in the device, at most two input light beams (IN1 and IN2) are radiated. When both inputs exist, an output signal is obtained from the positions where the switches exist, as described above. In the experiment, IN1 and IN2 were assigned to 325 and 384.7 nm, respectively. They were radiated over the entire sample (global irradiation) via far-field light. The spatial intensity distribution of the output, at 382.6 nm, was measured by scanning a near-field fiber probe within an area of approximately 1 ×1 μm. When only IN1 was applied to the sample, the output of the AND gate was ZERO (OFF state). When both inputs were radiated, the output was ONE (ON state). Note the regions marked by \(\blacksquare, {\bullet} \), and \(\blacklozenge\) in Fig. 23.3. In those regions, the output signal levels were respectively low and high, 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 are accomplished by diffraction-limited far-field light.

Combining this broadcasting mechanism with the summation mechanism will allow the development of nanoscale integration of massively parallel architectures, which have conventionally resulted in bulky configurations.

23.2.3 Secure Signal Transfer in Nanophotonics

In addition to breaking through the diffraction limit of light, such local interactions of optical near-fields also have important functional aspects, such as in security applications, particularly tamper resistance against attacks [27]. One of the most critical security issues in present electronic devices is so-called side-channel attacks, by which information is tampered with either invasively or noninvasively. This may be achieved, for instance, merely by monitoring their power consumption [28].

In this subsection, it is shown that devices based on optical excitation transfer via near-field interactions are physically more tamper resistant than their conventional electronic counterparts. The key is that the flow of information in nanoscale devices cannot be completed unless they are appropriately coupled with their environment [29], which could possibly be the weakest link in terms of their tamper resistance. A theoretical approach is presented to investigate the tamper resistance of optical excitation transfer, including a comparison with electrical devices.

Here, tampering of information is defined as involving simple signal transfer processes, since the primary focus is on their fundamental physical properties.

In order to compare the tamper resistance, an electronic system based on single charge tunneling is introduced here, in which a tunnel junction with capacitance C and tunneling resistance R T is coupled to a voltage source V via an external impedance Z(ω), as shown in Fig. 23.4a. In order to achieve single charge tunneling, besides the condition that the electrostatic energy E C = e 2 ∕ 2C of a single excess electron be greater than the thermal energy k B T, the environment must have appropriate conditions, as discussed in detail in Ref. [30]. For instance, with an inductance L in the external impedance, the fluctuation of the charge is given by
$$<\delta {Q}^{2} >= \frac{{e}^{2}} {4\rho } \coth \left (\frac{\beta \hslash \omega _{s}} {2} \right ),$$
(23.3)
where \(\rho = E_{C}/\hslash \omega _{S}\), \(\omega _{S} = {(LC)}^{-1/2}\), and \(\beta = 1/k_{B}T\). Therefore, charge fluctuations cannot be small even at zero temperature unless \(\rho \gg 1\). This means that a high-impedance environment is necessary, which makes tampering technically easy, for instance by adding another impedance circuit.
Fig. 23.4

Model of tamper resistance in devices based on (a) single charge tunneling and (b) optical excitation transfer. Dotted curves show the scale of a key device, and dashed curves show the scale of the environment required for the system to work

Here, let us define two scales to illustrate tamper resistance: (I) the scale associated with the key device size and (II) the scale associated with the environment required for operating the system, which are respectively indicated by the dotted and dashed lines in Fig. 23.4. In the case of Fig. 23.4a, scale I is the scale of a tunneling device, whereas scale II covers all of the components. It turns out that the low tamper resistance of such wired devices is due to the fact that scale II is typically the macroscale, even though scale I is the nanometer scale.

In contrast, in the case of the optical excitation transfer shown in Fig. 23.3b, the two quantum dots and their surrounding environment are governed by scale I. It is also important to note that scale II is the same as scale I. More specifically, the transfer of an exciton from QD S to QD L is completed due to the non-radiative relaxation process occurring at QD L , which is usually difficult to tamper with. Theoretically, the sublevel relaxation constant is given by

$$\Gamma = 2\pi \vert g{(\omega )\vert }^{2}D(\omega ),$$
(23.4)
where \(\hslash g(\omega )\) is the exciton–phonon coupling energy at frequency ω, is Planck’s constant divided by 2π, and D(ω) is the phonon density of states [31]. Therefore, tampering with the relaxation process requires somehow “stealing” the exciton–phonon coupling, which would be extremely difficult technically.

It should also be noted that the energy dissipation occurring in the optical excitation transfer, derived theoretically as E (2, 1, 1)E (1, 1, 1) in QD L based on Eq. (23.2), should be larger than the exciton–phonon coupling energy of \(\hslash \Gamma \), otherwise the two levels in QD L cannot be resolved. This is similar to the fact that the condition ρ ≫ 1 is necessary in the electron tunneling example, which means that the mode energy \(\hslash \omega _{S}\) is smaller than the required charging energy E C . By regarding \(\hslash \Gamma \) as a kind of mode energy in the optical excitation transfer, the difference between the optical excitation transfer and a conventional wired device is the physical scale at which this mode energy is realized: nanoscale for the optical excitation transfer and macroscale for electric circuits.

23.3 Networks of Optical Near-Field Interactions

23.3.1 Optimal Network of Near-Field Interactions

Consider the quantum dot system in Fig. 23.5a, where multiple smaller dots (denoted by S i ) can be coupled with one larger dot, denoted by L. We assume inter-dot interactions between adjacent smaller quantum dots; that is, (i) S i interacts with S i + 1 (\(i = 1,\cdots \,,N - 1\)), and (ii) S N interacts with S 1, where N is the number of smaller quantum dots. For instance, the system shown in Fig. 23.5b(i) consists of two smaller quantum dots and one larger quantum dot, denoted by S2–L1. Similarly, S3–L1, S4–L1, and S5–L1 systems, respectively, shown in Figs. 23.5b(ii–iv) are composed of three, four, and five smaller quantum dots in addition to one larger quantum dot.
Fig. 23.5

(a) Network of optical near-field interactions in a system composed of multiple smaller quantum dots and one larger quantum dot networked via optical near-field interactions. (b) Example systems composed of multiple smaller quantum dots and one larger quantum dot. (c) Optimal ratio of the number of smaller quantum dots to larger quantum dots so that the optical excitation transfer is most efficiently induced

Now, what is of interest is to calculate the flow of excitations from the smaller dots to the larger one. The theoretical and experimental details can be found in Ref. [32]; we introduce the information necessary for discussing the topology dependency and autonomy in optical excitation transfer in Sect. 23.3.2.

We deal with the problem theoretically based on a density matrix formalism. In the case of the S2–L1 system, which is composed of two smaller quantum dots and one larger quantum dot, the inter-dot interactions between the smaller dots and the larger one are denoted by \(U_{S_{i} L}\), and the interaction between the smaller dots is denoted by \(U_{S_{1}S_{2}}\), as schematically shown in Fig. 23.5a. The radiations from S 1, S 2, and L are, respectively, represented by the relaxation constants \(\gamma_{S_{1} }\), \(\gamma _{S_{2}}\), and γ L . We suppose that the system initially has two excitations in S 1 and S 2. With such an initial state, we can prepare a total of eleven bases where zero, one, or two excitation(s) occupy the energy levels. In the numerical calculation, we assume \(U_{S_{i}L}^{-1} = 200\) ps, \(U_{S_{1}S_{2}}^{-1} = 100\) ps, \(\gamma _{L}^{-1} = 1\) ns, \(\gamma _{S_{1}}^{-1} = 2.92\) ns, and Γ− 1 = 10 ps as parameter values. Following the same procedure, we also derive quantum master equations for the S3–L1, S4–L1, and S5–L1 systems that have initial states in which all smaller quantum dots are excited. Finally, we calculate the population of the lower level of the larger quantum dot, whose time integral we regard as the output signal.

We compare the output signal as a function of the ratio of the number of smaller dots to the number of larger dots assuming that the total number of quantum dots in a given unit area is the same, regardless of their sizes (smaller or larger). As shown by the circles in Fig. 23.5c, the most efficient transfer is obtained when the ratio of the number of smaller dots to the number of larger dots is 4. In other words, increasing the number of smaller quantum dots beyond a certain level does not necessarily contribute to increased output signals. Because of the limited radiation lifetime of large quantum dots, not all of the initial excitations can be successfully transferred to the large quantum dots due to the states occupying the lower excitation levels of the large quantum dots. Therefore, part of the input populations of smaller quantum dots must decay, which results in a loss in the transfer from the smaller quantum dots to the large quantum dots when there are too many excitations in the smaller quantum dots surrounding one large quantum dot.

An optimal mixture of smaller and larger quantum dots was experimentally demonstrated by using two kinds of CdSe/ZnS core/shell quantum dots whose diameters were 2.0 and 2.8 nm [32, 33]. In the experiment detailed in Ref. [32], an increase of the photocurrent was measured for the output signal. As shown by the squares in Fig. 23.5c, the maximum increase was obtained when the ratio of the number of smaller quantum dots to larger dots was 3:1, which agrees well with the theoretical optimal ratio discussed above.

Efficient optical excitation transfer in layered quantum dot structures has also been experimentally demonstrated. The radiation from layered graded-size CdTe quantum dots exhibits a signal nearly four times larger than that from structures composed of uniform-size quantum dots, a phenomenon which has been called exciton recycling [34] or superefficient exciton funneling [35]. Adopting the theory of networks of optical excitation transfer mediated by optical near-field interactions, the theoretical approach allows systematic analysis of layered quantum dot systems, revealing dominant factors contributing to the efficient optical excitation transfer and demonstrating good agreement with previous experimental observations [36].

23.3.2 Autonomy in Optical Excitation Transfer

In the previous section, we observe that the amount of optical excitation transferred from smaller quantum dots to larger quantum dots depends on the ratio of their numbers. This suggests that we could increase the output by engineering the network structure of the quantum dots. This section takes the S5–L1 system in Fig. 23.5b(iv) as an example and demonstrates that it is possible to increase the output signal by appropriately configuring the network of quantum dots. We set all of the inter-dot interaction times to 100 ps while keeping all other parameter values the same as those in Sect. 23.3.1.

In Fig. 23.6a, the original S5–L1 system, denoted by E0, is the same as the system shown in Fig. 23.5b(iv). Assume that some of the interactions between the smaller quantum dots (denoted by S 1 to S 5) and the large quantum dot surrounded by them are degraded, or lost, due to, for instance, material disorders, such as a violation of the condition represented by Eq. (23.2). In total, there are eight such configurations when symmetries are taken into account. For instance, when one of the five links between the smaller quantum dots and the large quantum dot is degraded, we obtain the system E1 in Fig. 23.6a. The mark “X” indicates a degraded interaction between S 1 and L. Similarly, when there are two degraded links, the system should be represented either by the system E2 or the system E2 shown in Fig. 23.6a.
Fig. 23.6

(a) Eight different network topologies in the S5–L1 system, where some of the interactions between the smaller quantum dots QD S and the large quantum dot QD L are degraded, or lost. (Degraded interactions are indicated by “X.”) The notation EN indicates that the system contains N-degraded interactions. (b) Time-integrated populations for the systems in (a). (c) The evolution of the populations associated with the number of excitations (ranging from 1 to 5) in systems E0 and E2. (d) Time evolutions of the populations associated with the smaller quantum dots (S 1 to S 5) in system E2 in (a) while assuming that all smaller quantum dots contain excitations in the initial setup

Figure 23.6b summarizes the integrated populations as a function of the network configurations in Fig. 23.6a. Interestingly, except for the system E5, which has no valid links between the smaller quantum dots and the large quantum dot, systems with degraded interactions exhibit a higher output signal than the system E0 without the link defects. System E2 exhibits an output signal that is about 1.64 times higher than system E0. This corresponds to the results described in Sect. 23.3.1, where the output is maximized when the ratio of the number of smaller dots to large dots is 4, meaning that the excessively high number of excitations in the smaller dots cannot be transferred to the large dot they surround. Due to the “limited” interactions between the smaller dots and the large dot, such as in the case of systems E2 and E2 , the excitations located in the smaller dots have a higher probability to be transferred to the larger dot. Figure 23.6c demonstrates the evolution of populations associated with the total number of excitations contained in the system, ranging from 1 to 5. The solid and dashed curves in Fig. 23.6c respectively refer to systems E0 and E2. The populations containing one excitation increase dramatically in E2 as compared with E0, which is another indication that the excitations can be kept in the system until they are successfully transferred to the destination, exhibiting a topology-dependent efficiency increase [37].

The autonomous behavior traceable memory of optical excitation transfer is emphasized in Fig. 23.6d, which summarizes the evolutions of populations associated with S 1 to S 5 in system E2, where both the interaction between S 2 and L and the interaction between S 3 and L are negligible. Initially, all of the smaller dots contain excitations. Note that the populations associated with S 2 and S 3 remain at a higher level for a short initial time, indicating that the excitations in S 2 and S 3 are effectively “waiting” in the smaller dots until they have the opportunity to be transferred to a large dot.

23.4 Hierarchical Architectures in Nanophotonics

23.4.1 Physical Hierarchy in Nanophotonics and Functional Hierarchy

In this section, we describe another feature of nanophotonics that can be exploited, the inherent hierarchy in optical near-field interactions [38, 39]. As schematically shown in Fig. 23.7a, there are multiple layers associated with the physical scale between the macroscale world and the atomic-scale world, which are primarily governed by propagating light and electron interactions, respectively. Between those two extremes, typically in scales ranging from a few nanometers to wavelength size, optical near-field interactions play a crucial role. In this section, we describe how such hierarchical properties in this mesoscopic or subwavelength regime can be exploited.
Fig. 23.7

(a) Hierarchical system architecture based on the inherent hierarchical properties in optical near-field processes. (b) Signal contrast as a function of the ratio of the radii of the sample and the probe based on dipole–dipole interactions

This physical hierarchy in optical near-field interactions will be analyzed by using a simple dipole–dipole interaction model as demonstrated in Sect. 23.4.2. Before going into details of the physical processes, functionalities required for system applications are first briefly reviewed in terms of hierarchy.

One of the problems for ultrahigh-density nanophotonic systems is interconnection bottlenecks, which were addressed in Sect. 23.2.2 above with regard to broadcast interconnects [25]. In fact, a hierarchical structure can be found in these broadcast interconnects by relating far-field effects at a coarser scale and near-field effects at a finer scale.

In this regard, it should also be mentioned that such physical differences in optical near-field and far-field effects can be used for a wide range of applications. The behavior of usual optical elements, such as diffractive optical elements, holograms, or glass components, is associated with their optical responses in optical far-fields. In other words, nanostructures can exist in such optical elements as long as they do not affect the optical responses in the far-field. Designing nanostructures that are accessible only via optical near-fields allows additional, or hidden, information to be recorded in those optical elements while maintaining the original optical responses in the far-fields. In fact, a hierarchical hologram and hierarchical diffraction grating have been experimentally demonstrated, as described in Sect. 23.4.3.

Since there is more hierarchy in the optical near-field regime, further applications should be possible; for example, it should be possible for nanometer-scale high-density systems to be hierarchically connected to coarse layer systems.

Hierarchical functionalities are also important for several aspects of memory systems. One is related to recent high-density, huge-capacity memory systems, in which data retrieval or searching from the entire memory archive has become even more difficult. One approach for solving such a problem is to employ a hierarchical system design, that is, by recording abstract data, metadata, or tag data in addition to the original raw data.

Hierarchy in nanophotonics provides a physical way of achieving such functional hierarchy. As will be introduced below, low-density, rough information is readout at a coarser scale, whereas high-density, detailed information is readout at a finer scale. Section 23.4.2 will show physical mechanisms for such hierarchical information retrieval.

Another issue in hierarchical functionalities will be security. High-security information is recorded at a finer scale, whereas less-critical security information is associated with a coarse layer. Also, in addition to associating different types of information with different physical scales, another kind of information could also be associated with one or more layers of the physical hierarchy. For instance, a traceable memory has been demonstrated based on such hierarchical properties in optical near-fields [40, 41].

23.4.2 Hierarchical Memory Retrieval

This section describes a physical model of optical near-field interactions based on dipole–dipole interactions [42] and their application to hierarchical memory retrieval [38]. 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 . The inset in Fig. 23.7b 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 sample; 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 Δ p P = Δα P E 0. Similarly, p P changes the electric dipole moment in the sample by Δ p 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
$$\displaystyle\begin{array}{rcl} I& =&{ \left \vert \mbox{ $p$}_{P} + \Delta \mbox{ $p$}_{P} + \mbox{ $p$}_{S} + \Delta \mbox{ $p$}_{S}\right \vert }^{2} \\ & \approx & {(\alpha _{P} +\alpha _{S})}^{2}\vert E{_{ 0}\vert }^{2} + 4\Delta \alpha (\alpha _{ P} +\alpha _{S})\vert E{_{0}\vert }^{2}\end{array}$$
(23.5)
where Δα = Δα S = Δα P [42]. The second term in Eq. (23.5) 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. (23.5) is the background signal for the measurement. Therefore, the ratio of the second term to the first term in Eq. (23.5) 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. 23.7b. Thus, one 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 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 associated with its scale.
Based on the above simple hierarchical mechanism, a hierarchical memory system has been constructed. Consider, for example, a maximum of N nanoparticles distributed in a region of subwavelength scale. Those nanoparticles can be nicely resolved by a scanning near-field microscope if the size of its fiber probe tip is comparable to the size of individual nanoparticles; in this way, the first-layer information associated with each distribution of nanoparticles is retrievable, corresponding to 2 N -different codes. By using a larger-diameter fiber probe tip instead, the distribution of the particles cannot be resolved, but a mean-field feature with a resolution comparable to the size of the probe can be extracted, namely, the number of particles within an area comparable to the size of the fiber probe tip. Thus, the second-layer information associated with the number of particles, corresponding to (N + 1) different signal levels, is retrievable. Therefore, one can access different sets of signals, 2 N or N + 1, depending on the scale of observation. This leads to hierarchical memory retrieval by associating this information hierarchy with the distribution and the number of nanoparticles using an appropriate coding strategy, as schematically shown in Fig. 23.8d.
Fig. 23.8

Hierarchical information retrieval. (a) Each section consists of small particles. (b) Experimental setup. (c) Calculated scattering cross sections depending on the number of particles in each section (squares) and peak intensity of each section in the intensity pattern shown in (d). (d) SEM image of an Au particle array and intensity pattern captured by a NSOM with a fiber probe having a 500-nm-diameter aperture

For example, in encoding N-bit information, (N − 1)-bit signals can be encoded by distributions of nanoparticles while associating the remaining 1-bit with the number of nanoparticles. Details of encoding/decoding strategies will be found in [38].

Simulations were performed assuming ideal isotropic metal particles to see how the second-layer signal varies depending on the number of particles by using a finite-difference time-domain simulator. Here, 80-nm-diameter particles are distributed over a 200-nm-radius circular grid at constant intervals. The solid circles in Fig. 23.8c show calculated scattering cross sections as a function of the number of particles. A linear correspondence to the number of particles was observed. This result supports the simple physical model described above.

In order to experimentally demonstrate such principles, an array of Au particles, each with a diameter around 80 nm, was distributed over a SiO2 substrate in a 200-nm-radius circle. These particles were fabricated by a liftoff technique using electron beam (EB) lithography with a Cr buffer layer. Each group of Au particles was spaced by 2 μm. A scanning electron microscope (SEM) image is shown in Fig. 23.8d in which 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, a near-field scanning optical microscope (NSOM) with a large-diameter-aperture (500-nm)-metalized fiber probe was used in an illumination–collection setup. 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 23.8d shows an intensity profile captured by the probe, from which the second-layer information is retrieved. The solid squares in Fig. 23.8c indicate the peak intensity of each section, which increased linearly. Those results show the validity of hierarchical memory retrieval from nanostructures.

23.4.3 Hierarchical Optical Elements

Holography, which generates natural three-dimensional images, is one of the most common anticounterfeiting techniques [43]. In the case of a volume hologram, the surface is ingeniously formed into microscopic periodic structures which diffract incident light in specific directions. A number of diffracted light beams can form an arbitrary three-dimensional image. Generally, these microscopic structures are recognized as being difficult to duplicate, and therefore, holograms have been widely used in the anticounterfeiting of bank notes, credit cards, etc. However, conventional anticounterfeiting methods based on the physical appearance of holograms are nowadays not completely secure [44]. Nanophotonic solutions, utilizing light–matter interactions on the nanoscale, would provide higher anticounterfeiting capability and would potentially enable other novel applications, such as artifact–metric systems [45].

The hierarchical hologram works in both optical far-fields and near-fields, the former being associated with conventional holographic images (Fig. 23.9a) and the latter being associated with the optical intensity distribution based on a nanometric structure (Fig. 23.9b) that is accessible only via optical near-fields (Fig. 23.9c). In principle, a structural change occurring at the subwavelength scale does not affect the optical response function, which is dominated by propagating light. Therefore, the visual aspect of the hologram is not affected by such a small structural change on the surface. Additional data can thus be written by engineering structural changes in the subwavelength regime so that they are only accessible via optical near-field interactions without having any influence on the optical response obtained via the conventional far-field light. By applying this hierarchy, new functions can be added to conventional holograms.
Fig. 23.9

Hierarchical optical elements, such as hierarchical holograms, based on the different optical responses obtained in (a) the optical far-field and (c) the optical near-field. (b) A nanostructure is embedded in the original grating structure of the hologram. (d) Evident polarization dependency in optical near-fields results based on the surrounding structures

There are at least two strategies for realizing a hierarchical hologram.

One strategy is to apply nanometric structural changes to the surface structure of a conventional hologram. In Ref. [46], a thin metal layer is coated on a conventional hologram and diffraction grating, followed by nanostructure patterning by a focused ion beam machine. Additional information corresponding to the fabricated nanostructures is successfully retrieved while preserving the macroscopic view of the original hologram or the diffraction efficiency of the diffraction grating.

The other strategy, employed in the case of embossed holograms composed of diffraction gratings, is to locally engineer the original hologram pattern from the beginning, that is, to embed nanostructures within the original pattern of the hologram [47]. In this case, since the original hologram is basically composed of one-dimensional grating structures, evident polarization dependence is obtained in retrieving the nanostructures via optical near-fields, as detailed below. There are some additional benefits with this approach: one is that we can fully utilize the existing industrial facilities and fabrication technologies that have been developed for conventional holograms, yet providing additional information in the hologram. Another is that the polarization dependence facilitates the readout of nanostructures via optical near-fields, as mentioned below.

As shown in Fig. 23.9, we created a sample device to experimentally demonstrate retrieval of the nanostructures embedded within an embossed hologram. The entire device structure, whose size was 15 ×20 mm, was fabricated by electron beam lithography on a Si substrate, followed by sputtering a 50-nm-thick Au layer. The cross-sectional profile is shown in Fig. 23.9d.

As indicated in Fig. 23.9a, we can observe a three-dimensional image of the earth reconstructed from the device. More specifically, the device was based on the design of Virtuagram®;, developed by Dai Nippon Printing Co., Ltd., Japan, which is a high-definition computer-generated hologram composed of binary-level one-dimensional modulated gratings, as shown in the SEM image in Fig. 23.9b. Within the device, we slightly modified the shape of the original structure of the hologram so that the nanostructural change was accessible only via optical near-field interactions. As shown in Fig. 23.9b, square- or rectangle-shaped structures, whose associated optical near-fields correspond to the additional or hidden information, were embedded in the original hologram structures. The unit size of the nanostructures ranged from 40 to 160 nm. Note that the original hologram was composed of arrays of one-dimensional grid structures, spanning along the vertical direction in Fig. 23.9b. To embed the nanophotonic codes, the grid structures were partially modified in order to implement the nanophotonic codes. Nevertheless, the grid structures remained topologically continuously connected along the vertical direction. On the other hand, the nanostructures were always isolated from the original grid structures. These geometrical characteristics provide interesting polarization dependence.

The input light induces oscillating surface charge distributions due to the coupling between the light and electrons in the metal. Note that the original one-dimensional grid structures span along the vertical direction. The y-polarized input light induces surface charges along the vertical grids. Since the grid structure continuously exists along the y-direction, there is no chance for the charges to be concentrated. However, in the area of the embedded nanophotonic code, we can find structural discontinuity in the grid; this results in higher charge concentrations at the edges of the embedded nanostructure. On the other hand, the x-polarized input light sees structural discontinuity along the horizontal direction due to the vertical grid structures, as well as in the areas of the embedded nanostructures. It turns out that charge concentration occurs not only in the edges of the embedded nanostructures but also at other horizontal edges of the environmental grid structures. When square-shaped nanophotonic codes are isolated in a uniform plane, both x- and y-polarized input light have equal effects on the nanostructures. These mechanisms indicate that the nanostructures embedded in holograms could exploit these polarization dependences.

In the experimental demonstration, optical responses in near-mode observation were detected using a NSOM operated in an illumination–collection mode with an optical fiber tip having a radius of curvature of 5 nm. The observation distance between the tip of the probe and the sample device was set at less than 50 nm. The light source used was a laser diode (LD) with an operating wavelength of 785 nm, and scattered light was detected by a photomultiplier tube (PMT).

We examined NSOM images in the vicinity of nanostructures that were embedded in the hologram and nanostructures that were not embedded in the hologram using light from a linearly polarized radiation source, with polarizations rotated by 0–180 at 20 intervals. In the case of nanostructures embedded in the hologram, clear polarization dependence was observed. To quantitatively evaluate the polarization dependency of the embedded nanophotonic code, we adopted a figure of merit that we call recognizability for the observed NSOM images [47], indicating the relative intensity compared with the surroundings.

The square and circular marks in Fig. 23.9e respectively show the recognizability of isolated nanostructures and those embedded in the hologram. Clear polarization dependency is observed in the case of the nanostructures embedded in the holograms, facilitating near-field information retrieval.

23.5 Shape-Engineered Nanostructures for Polarization Control for Nanophotonic Systems

Shape engineering of nanostructures is one of the most useful and important means to implement nanometer-scale photonic systems, an example of which has already been demonstrated in the hierarchical hologram in described in Sect. 23.4.3. Electric field enhancement based on the resonance between light and free electron plasma in a metal nanostructure is one well-known feature [48] that has been used in many other applications, such as optical data storage [49], biosensors [50], and integrated optical circuits [51, 52, 53]. Such resonance effects are, however, only one of the possible light–matter interactions on the nanometer scale that can be exploited for practical applications. For example, it is possible to engineer the polarization of light in the optical near-field and far-field by controlling the geometries of metal nanostructures. Since there are a vast number of design parameters potentially available on the nanometer scale, an intuitive physical picture of the polarization associated with geometries of nanostructures can be useful in restricting the parameters to obtain the intended optical responses.

In this section, we consider polarization control in the optical near-field and far-field by designing the shape of a metal nanostructure, based on the concepts of elemental shape and layout, to analyze and synthesize optical responses brought about by the nanostructure [54]. In particular, we focus on the problem of rotating the plane of polarization. Polarization in the optical near-field is an important factor in the operation of nanophotonic devices [55]. Polarization in the far-field is, of course, also important for various applications. Devices including nanostructures have already been employed, for instance, in so-called wire-grid polarizers [56, 57].

The concepts of elemental shape and layout are physically related, respectively, to the electric current induced in the metal nanostructure and the electric fields, that is, the optical near-fields, induced between individual elements of the metal nanostructures, which helps in understanding the induced optical responses. For example, it will help to determine if a particular optical response originates from the shape of the nanostructure itself, that is to say, the elemental shape factor, or from the positional relations between individual elements, that is to say, the layout factor. Such analysis will also help in the design of more complex structures, such as multilayer systems. What should be noted, in particular in the case of multilayer systems, is that the optical near-fields appearing between individual elemental shapes, including their hierarchical properties, strongly affect the resultant optical response. This indicates that the properties of the system are not obtained by a superposition of the properties of individual elements, in contrast to optical antennas, whose behavior is explained by focusing on factors associated with individual elements [58].

23.5.1 Polarization and Geometry on the Nanometer Scale

The nanostructure we consider is located on an xy-plane and is irradiated with linearly polarized light from the direction of the normal. We first assume that the nanostructure has a regular structure on the xy-plane; in other words, it has no fine structure along the z-axis. Here we consider the concepts of elemental shape and layout, introduced above, to represent the whole structure. Elemental shape refers to the shape of an individual structural unit, and the whole structure is composed of a number of such units having the same elemental shape. Layout refers to the relative positions of such structural units. Therefore, the whole structure is described as a kind of convolution of elemental shape and layout. This is schematically shown in Fig. 23.10a.
Fig. 23.10

(a) Elemental shape and layout factors used to describe the configuration of the entire structure. This chapter deals with two representative structures, what we call I- and Z-shapes. (b) Electric field intensity in near- and far-fields produced by I-shape and Z-shape structures

We begin with the following two example cases, which, as will shortly be presented in Fig. 23.10b, exhibit contrasting properties in their optical near-field and far-field responses. One is what we call an I-shape, which exhibits a strong electric field only in the optical near-field regime, while showing an extremely small far-field electric field. The other is what we call a Z-shape, which exhibits a weak near-field electric field, while showing a strong far-field electric field. They are schematically shown in the top row in Fig. 23.10a. In the case of the I-shape, the elemental shape is a rectangle. Such rectangular units are arranged as specified by the layout (second row in Fig. 23.10a); they are arranged with the same interval horizontally (along the x-axis) and vertically (along the y-axis), but every other row is horizontally displaced by half of the interval. In the case of the Z-shape, the element shape is like the letter “Z,” and they are arranged regularly in the xy-plane as specified by the layout shown in the second row in Fig. 23.10a.

We calculate the optical responses in both the near-field and far-field based on a finite-difference time-domain (FDTD) method [59, 60]. As the material, we assume gold, which has a refractive index of 0.16 and an extinction ratio of 3.8 at a wavelength of 688 nm [61]. Representative geometries of the I-shape and Z-shape structures in the xy-plane are shown in Fig. 23.11a, b, respectively. The width (line width) of the structures is 60 nm, and the thickness is 200 nm. The light source is placed 500 nm away from one of the surfaces of the structures. We assume periodic boundary conditions at the edges in the x- and y-directions and perfectly matched layers in the z-direction.
Fig. 23.11

Charge distributions induced in (a) I-shape and (b) Z-shape structures with (c) x-polarized and (d) y-polarized input light. The arrows in (c-1) and (d-1) are associated with the induced electric currents within the elemental shapes and those in (c-2) and (d-2) are associated with near-fields among elemental shapes

The near-field intensity is calculated at a plane 5 nm away from the surface that is opposite to the light source, which we call the near-field output plane. With continuous-wave, linearly polarized 688-nm light parallel to the x-axis as the input light, we analyze the y-component of the electric field at the near-field output plane. From this, we evaluate the polarization conversion efficiency in the near-field regime, defined by
$$T_{x\rightarrow y}^{NEAR} = \frac{{\left \vert E_{y}(\mbox{ $p$}_{n})\right \vert }^{2}} {{\left \vert E_{x}(\mbox{ $p$}_{i})\right \vert }^{2}},$$
(23.6)
where p n is the position on the near-field output plane, p i is the position of the light source, and E x (p) and E y (p), respectively, represent the x- and y-components of the electric field at position p. Since T xy NEAR varies depending on the position, we focus on the maximum value in the near-field output plane. The metric defined by Eq. (23.6) can be larger than 1 due to electric field enhancement. The energy conversion efficiency can be obtained by calculating Poynting vectors existing in the near-field output plane; however, the region of interest in the near-field regime is where the charge distributions give their local maximums and minimums, as discussed shortly in this section. Therefore, we adopted the metric in the near-field regime given by Eq. (23.6).
The far-field optical response is calculated at a plane 2 μm away from the surface of the structure opposite to the light source, which we call the far-field output plane. We assume an input optical pulse with a differential Gaussian form whose width is 0.9 fs, corresponding to a bandwidth of around 200–1,300 THz. The transmission efficiency is given by calculating the Fourier transform of the electric field at the far-field output plane divided by the Fourier transform of the electric field at the light source. Since we are interested in the conversion from x-polarized input light to y-polarized output light, the transmission is given by
$$T_{x\rightarrow y}^{FAR}(\lambda ) ={ \left \vert \frac{F[E_{y}(t,\mbox{ $p$}_{f})]} {F[E_{x}(t,\mbox{ $p$}_{i})]} \right \vert }^{2},$$
(23.7)
where p f denotes the position on the far-field output plane and F[E(t, p)] denotes the Fourier transform of E(t, p). Here, T xy FAR is also dependent on position, as well as wavelength, but it is not strongly dependent on p f . In this chapter, we represent \(T_{x\rightarrow y}^{FAR}\) by a value given at a position on the far-field output plane with a wavelength λ = 688 nm.

Figure 23.10b summarizes the electric field intensity of y-polarized output light from x-polarized input light and that of x-polarized output light from y-polarized input light, in both the near- and far-fields at the wavelength of 688 nm. The horizontal scale in Fig. 23.10b is physically related to the polarization conversion efficiency.

We first note the following two features. First, the near-field electric field intensity, represented by dark gray bars in Fig. 23.10b, is nearly 2,000 times higher with the I-shape than with the Z-shape. The far-field electric field intensity, shown by light gray bars, on the other hand, is around 200 times higher with the Z-shape than with the I-shape. One of the primary goals of this chapter is to explain the physical mechanism of these contrasting optical responses in the near- and far-fields in an intuitive framework, which will be useful for analyzing and designing more complex systems.

Here we derive the distribution of induced electron charge density (simply referred to as charge hereafter) by calculating the divergence of the electric fields to analyze the relation between the shapes of the structures and their resultant optical responses. Figure 23.11 shows such charge distributions for I-shape and Z-shape structures.

First, we describe Fig. 23.11c, which relates to x-polarized input light. The images shown in Fig. 23.11(c-1), denoted by “Shape,” represent the distributions of charges at each unit, namely, charges associated with the elemental shape. The images in Fig. 23.11(c-2), denoted by “Layout,” show the distributions of charges at elemental shapes and their surroundings.

We can extract positions at which induced electron charge densities exhibit a local maximum and a local minimum. Then, we can derive two kinds of vectors connecting the local maximum and local minimum, which we call flow vectors. One is a vector existing inside an elemental shape, denoted by dashed arrows in Fig. 23.11(c-1), which is physically associated with an electric current induced in the metal. The other vector appears between individual elemental shapes, denoted by solid arrows in Fig. 23.11(c-2), which is physically associated with near-fields between elemental shapes. We call the latter ones inter-elemental-shape flow vectors.

From those flow vectors, first, in the case of the I-shape structure shown in Fig. 23.11a, we note that:
  1. 1.

    Within an elemental shape in Fig. 23.11(c-1), the flow vectors are parallel to the x-axis (There is no y-component in the vectors).

     
  2. 2.

    At the layout level in Fig. 23.11(c-2), flow vectors that have y-components appear. Also, flow vectors that have y-components are in opposite directions between neighboring elemental shapes.

     

From these facts, the y-components of the flow vectors are arranged in a quadrupole manner, which agrees with the very small radiation in the far-field demonstrated in Fig. 23.10b. Also, these suggest that the appearance of y-components in the flow vectors originates from the layout factor, not from the elemental shape factor. This indicates that the polarization conversion capability of the I-shape structure is layout sensitive, which will be explored in more detail in Sect. 23.5.2.

Second, in the case of the Z-shape structure, we note that:
  1. 1.

    In the elemental shape, y-components of the flow vectors appear.

     
  2. 2.

    In the layout, we can also find y-components in the flow vectors. Also, at the layout level, the y-components of all vectors are in the same direction.

     

In complete contrast to the I-shape structure, the Z-shape structure has y-components in the flow vectors arranged in a dipole-like manner, leading to strong y-polarized light in the far-field, as demonstrated in Fig. 23.10b. Also, the ability to convert x-polarized input light to y-polarized output light in the far-field, as quantified by \(T_{x\rightarrow y}^{FAR}\), primarily originates from the elemental shape factor, not from the layout factor.

23.5.2 Layout-Dependent Polarization Control

As indicated in Sect. 23.5.1, the polarization conversion from x-polarized input light to y-polarized output light with the I-shape structure originates from the layout factor. Here, we modify the layout while keeping the same elemental shape, and we evaluate the resulting conversion efficiencies.

In Fig. 23.12, we examine such layout dependencies by changing the horizontal displacement of elemental shapes between two consecutive rows, indicated by the parameter D in the inset of Fig. 23.12c. The polarization conversion efficiency, \(T_{x\rightarrow y}^{FAR}\), at the wavelength of 688 nm as a function of D is indicated by the circles in Fig. 23.12a. Although it exhibits very small values for the I-shape structure, it has a large variance depending on the layout: a maximum value of around 10− 9 when D is 200 nm and a minimum value of around 10− 12 when D is 80 nm, a difference of three orders of magnitude. On the other hand, the Z-shape structure exhibits an almost constant \(T_{x\rightarrow y}^{FAR}\) with different horizontal positional differences, as indicated by the squares in Fig. 23.12a, meaning that the Z-shape structure is weakly dependent on the layout factor.
Fig. 23.12

(a) Conversion efficiency dependence on the layout factor. The I-shape structure exhibits stronger dependence on layout than the Z-shape structure. (b) and (c) Current distributions and inter-elemental-shape flow vectors for I-shape structure when (b) D = 80 nm and (c) D = 200 nm

To account for such a tendency, we represent the I-shape structure by two inter-elemental-shape flow vectors denoted by p 1 and p 2 in Fig. 23.12b. Here, R i and θ i , respectively, denote the length of p i and its angle relative to the y-axis. All of the inter-elemental-shape flow vectors are identical to those two vectors and their mirror symmetry. Physically, a flow vector with a large length and a large inclination to the y-axis contributes weakly to y-components of the radiation. Therefore, the index \(\cos \theta _{i}/R_{i}^{2}\) will affect the radiation. Together with the quadrupole-like layout, we define the following metric:
$$\left \vert \cos \theta _{1}\left /\right . R_{1}^{2} -\cos \theta _{ 2}\left /\right . R_{2}^{2}\right \vert ,$$
(23.8)
which is denoted by the triangles in Fig. 23.12a; it agrees well with the conversion efficiency \(T_{x\rightarrow y}^{FAR}\) of the I-shape structure.

23.5.3 Application to Authentication Function

The structures that have been discussed so far contain a regular structure on a single plane, namely, they are single-layer structures. Now, we consider stacking another layer on top of the original layer. As an example, we consider adding another layer on top of the I-shape structure so that the y-component in the far-field radiation increases (recall that the original I-shape-only structure exhibits very small far-field radiation). We call the two nanostructures Shape A and Shape B hereafter.

Shape A and Shape B were designed as aligned rectangular units on an xy-plane at constant intervals horizontally (along the x-axis) and vertically (along the y-axis), as, respectively, shown in Fig. 23.13a, b. When we irradiate Shape A with x-polarized light, surface charges are concentrated at the horizontal edges of each of the rectangular units. The relative phase difference of the oscillating charges between the horizontal edges is π, which is schematically represented by + and − marks in Fig. 23.13a. Now, consider the y-component of the far-field radiation from Shape A, which is associated with the charge distributions induced in the rectangle. When we draw arrows from the + marks to the − marks along the y-axis, we find that adjacent arrows are always directed oppositely, indicating that the y-component of the far-field radiation is externally small. In other words, Shape A behaves as a quadrupole regarding the y-component of the far-field radiation. It should also be noted that near-field components exist in the vicinity of the units in Shape A. With this fact in mind, we put the other metal nanostructure, Shape B, on top of Shape A. Through the optical near-fields in the vicinity of Shape A, surface charges are induced on Shape B. What should be noted here is that the arrows connecting the + and − marks along the y-axis are now aligned in the same direction, and so the y-component of the far-field radiation appears; that is, the stacked structure of Shape A and Shape B behaves as a dipole (Fig. 23.13c). Also, Shape A and Shape B need to be closely located to invoke such effects since the optical near-field interactions between Shape A and Shape B are critical. In other words, far-field radiation appears only when Shape A and Shape B are correctly stacked; that is to say, a quadrupole–dipole transform is achieved through shape-engineered nanostructures and their associated optical near-field interactions.
Fig. 23.13

(a)–(c) Shapes and the associated distributions of induced charge density. (a) Shape A, (b) Shape B, and (c) stacked structure of Shapes A and B. (d) Comparison of calculated far-field intensity for various combinations of shapes. Far-field intensity appears strongly only when Shape A matches the appropriate Shape B. (e) Measured polarization conversion efficiency for three areas of the fabricated device. Conversion efficiency exhibited a larger value specifically in the areas where the stacked structure of Shapes A and B was located. SEM images of each area are also shown

In Fig. 23.13d, we also consider the output signals when we place differently shaped structures on top of Shape A, instead of Shape B. With Shape B , Shape B\(^{{\prime\prime}}\), and Shape B\(^{{\prime\prime\prime}}\), which are respectively represented in the insets of Fig. 23.13d, the output signals do not appear, as shown from the fourth to the sixth rows in Fig. 23.13d, since the condition necessary for far-field radiation is not satisfied with those shapes, namely, the correct key is necessary to unlock the lock [62].

We fabricated structures consisting of (i) Shape A only, (ii) Shape B only, and (iii) Shapes A and B stacked. Although the stacked structure should ideally be provided by combining the individual single-layer structures, in the following experiment, the stacked structure was integrated in a single sample as a solid two-layer structure to avoid the experimental difficulty in precisely aligning the individual structures mechanically. The fabrication process was detailed in Ref. [63]. The lower side in Fig. 23.13e also shows SEM images of fabricated samples of (i)–(iii). Because the stacked structure was fabricated as a single sample, the gap between Shapes A and B was fixed at 200 nm.

The performance was evaluated in terms of the polarization conversion efficiency by radiating x-polarized light on each of the areas (i), (ii), and (iii) and measuring the intensity of the y-component in the transmitted light. The light source was a laser diode with an operating wavelength of 690 nm. Two sets of polarizers (extinction ratio 10− 6) were used to extract the x-component for the input light and to extract the y-component in the transmitted light. The intensity was measured by a lock-in controlled photodiode. The position of the sample was controlled by a stepping motor with a step size of 20 μm. Figure 23.13e shows the polarization conversion efficiency as a function of the position on the sample, where it exhibited a larger value specifically in the areas where the stacked structure of Shapes A and B was located, which agrees well with the theoretically predicted and calculated results shown in Fig. 23.13d.

23.6 Summary

This chapter reviews nanophotonic systems based on localized and hierarchical optical near-field processes. In particular, optical excitation transfer involving optical near-field interactions and its hierarchical properties are highlighted while demonstrating their enabling functionalities. Networks of optical excitation transfers mediated by near-field interactions and shape-engineered nanostructures and their associated polarization properties have also been presented. It should be emphasized that those basic features enable versatile applications and functionalities besides the example demonstrations shown here. Also, there are many other degrees of freedom in the nanometer scale that need to be deeply understood in terms of their implications for systems. Further exploration and attempts to exploit nanophotonics for future devices and systems will certainly be exciting.

Notes

Acknowledgements

Part of this work was supported by the Strategic Information and Communications R&D Promotion Programme (SCOPE) of the Ministry of Internal Affairs and Communications and Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science. The author acknowledges Dai Nippon Printing Co., Ltd., for fabrication of the hierarchical hologram.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Makoto Naruse
    • 1
  1. 1.Photonic Network Research InstituteNational Institute of Information and Communications TechnologyTokyoJapan

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