## Abstract

We demonstrate that a two-layer shape-engineered nanostructure exhibits asymmetric polarization conversion efficiency thanks to near-field interactions. We present a rigorous theoretical foundation based on an angular-spectrum representation of optical near-fields that takes account of the geometrical features of the proposed device architecture and gives results that agree well with electromagnetic numerical simulations. The principle used here exploits the unique intrinsic optical near-field processes associated with nanostructured matter, while eliminating the need for conventional scanning optical fiber probing tips, paving the way to novel nanophotonic devices and systems.

© 2013 Optical Society of America

## 1. Introduction

Nanophotonics has been extensively studied with the aim of unveiling and exploiting optical near-field interactions associated with nanostructured matter [1]. The technologies used for characterizing optical near-field processes [2,3] and related materials technologies [4] have been rapidly advancing. In addition, investigations based on information physics have been revealing some unique and emergent attributes of nanophotonics that are useful for various applications [5]; examples include, but are not limited to, basic logic circuits [6,7], computing paradigms that go beyond the von Neumann architecture [8,9], and applications related to information security [10].

What we particularly address in this paper stems from advancements in shape-engineered nanostructures and some technological concerns in conventional near-field optics that necessitate precision mechanical control, for example, in controlling optical fiber probing tips. Advances in electron-beam lithography, nano-imprinting, and other areas allow the fabrication of well-controlled reliable nanostructures [11–14], and interesting information applications, such as information security, have been made possible. For instance, we have demonstrated a “*hierarchical hologram*” that works in both optical far-fields and near-fields, the former being associated with conventional holographic images, and the latter being associated with the optical intensity distribution originating from a nanometric structure embedded in the hologram, which is accessible only via optical near-fields [12,15]. In other words, information hiding can be realized by using optical near-fields and nanofabrication technologies. Also, authentication functions can be implemented by using two shape-engineered nanostructures and their associated optical near-fields [11]. In this system, the two nanostructures respectively work as a *lock* and *key*, where authenticity is guaranteed by the nanoscale-precision shapes of the structures.

The physical principles of nano-optics can contribute to the development of novel functionalities. The common feature across these demonstrations is that they are based on high-precision mutual relations between nanostructured matter, mediated by optical near-fields. Technologically, in turn, they require high-precision alignment between nanostructured matter, such as an optical near-field fiber probe tip and the device under study. Although this attribute is one fundamentally superior aspect in terms of increased security, at the same time it is a severe technological difficulty in terms of stability and practical use.

Therefore, in this paper, we demonstrate a two-layer united nanostructure in which the layers interact via optical near-fields and which exhibits unique optical properties that are observable in the optical far-field. More concretely, the proposed device architecture exhibits “asymmetry” in its associated polarization properties; specifically, the polarization conversion efficiency from *x*-polarized input light to *y*-polarized output light (X$\to $Y) and that from *y*-polarized input light to *x*-polarized output light (Y$\to $X) results in different values. We demonstrate its rigorous theoretical foundation based on an angular spectrum representation of optical near-fields that takes account of the geometries of the two-layer nanostructure, and in which representative features are characterized by electromagnetic numerical calculations. With such an architecture, we are able to exploit optical near-field processes occurring at the subwavelength scale, while at the same time allowing them to be retrieved by a macro-scale optical measurement, thus considerably relaxing the stringent requirements of precision alignment in conventional optical near-field setups.

The asymmetric polarization property discussed in this paper, defined as described above, is *not* related to the magneto-optical chiral effects [16], chiral plasmonic structures [13,17–21]. Although it may be possible to achieve equivalent asymmetric optical responses by combinations of conventional optical elements or anisotropic materials, the focus of this paper is to accomplish the asymmetric polarization properties defined above by using shape-engineered two-layer nanostructures formed of isotropic matter via their associated optical near-fields. We consider that such an approach will pave the way to new functional nanophotonic devices and optical security applications based on near-field processes, without the need for technologically difficult demands, such as those required in scanning probe-based measurements. That is to say, the asymmetric polarization properties, which can be associated with *information*, are implemented with the built-in shape of the nanostructures themselves. The asymmetric transmission of linearly polarized light in optical metamaterials was demonstrated by Menzel et al. [22]. The findings of our study, emphasizing the role of optical near-fields associated with nanostructures, will give greater physical insights regarding asymmetric polarization, provide a systematic approach for designing designated functions, and offer fundamentals for novel applications, such as information security.

This paper is organized as follows. First, Section 2 characterizes one fundamental feature of precision alignment requirements involving optical near-field processes via a rigorous theory based on an angular-spectrum representation of electromagnetic fields on the nanoscale. Section 3 introduces asymmetry of polarization conversion efficiency in a two-layer nanostructure. Section 4 describes the theoretical background based on the angular spectrum framework introduced in Section 2. Section 5 demonstrates some representative features based on electromagnetic calculations and presents methods of characterizing devices via some figures-of-merit that correspond to the theoretical framework discussed in Section 4. Section 6 concludes the paper.

## 2. Theoretical foundations for describing precision mutual relations via optical near-fields

First, we characterize the fundamental properties of precision mutual relations required in order to retrieve a proper signal via optical near-fields with a simple but rigorous theoretical approach. These theoretical elements will be used in discussing the asymmetry of polarization conversion to be discussed in Section 3 and later sections.

Optical near-fields are the localized, non-propagating components of electromagnetic fields in the vicinity of materials [23]. We need to locate certain kinds of *reader* to induce interactions with the device under study. In order to characterize the structure of the system, we denote the entities of the system as follows. Let the device under study be denoted by *D*, and the reader by *R*. One of the characteristic consequences of nano-optical systems is that the output signal depends precisely on both *D* and *R*, a relationship which is represented by *v* = *g*(*D*,*R*). In order to theoretically represent the fundamental characters of the output signal we take the following approach.

The device under study is regarded as an oscillating electric dipole ** D** with frequency

*K*placed in a vacuum at the origin of a Cartesian space in which the velocity of light is taken as unity (

*c*= 1). To deal with scattering problems based on assumed planar boundary conditions, as well as offering a physically intuitive understanding, it is convenient to represent the complex amplitude

**of the electric dipole radiation as a superposition of plane waves having complex wave vectors—an approach called the angular spectrum representation [24–26]. In this representation,**

*E***is defined by**

*E**xy*plane.

Suppose that the dipole ** D** is oriented at an angle

*θ*with respect to the

*z*axis and at an angle

*ϕ*in the

*xy*plane, that is, $D=d(\mathrm{sin}\theta \mathrm{cos}\varphi ,\mathrm{sin}\theta \mathrm{sin}\varphi ,\mathrm{cos}\theta )$, as schematically shown in Fig. 1(a). Suppose also that we observe the radiation from the electric dipole at a position displaced from the dipole by $R=({r}_{\left|\right|}^{}\mathrm{cos}\phi ,{r}_{\left|\right|}^{}\mathrm{sin}\phi ,Z)$; that is to say, we assume that the reader is placed at

**. The angular spectrum representation of the**

*R**z*component of the electric field for evanescent waves (namely, $1\le {s}_{\left|\right|}<+\infty $) from the dipole

**is given by**

*D**J*(

_{n}*x*) represents a Bessel function of the first kind, where

*n*is an integer, and the term ${f}_{z}({s}_{\left|\right|},D,R)$ is called the angular spectrum of the electric field.

Now, we consider that ${f}_{z}({s}_{\left|\right|},D,R)$ is equivalent to the signal *v* = *g*(*D*,*R*) characterized in the system model. Assuming that the dipole is oriented parallel to the *x*-axis, we have *ϕ* = 0 and *θ* = π/2. Also, assuming that the reader *R* is located on the *xz*-plane, we have *φ* = 0. In such a model, letting ${r}_{\left|\right|}$ be denoted by *X*, we characterize the horizontal (*X*) and vertical (*Z*) dependences. Note that *X* and *Z* are given in units of wavelength.

The solid curve in Fig. 1(b) shows the angular spectrum when *X* = 1/20 and *Z* = 1/20. We consider that this corresponds to a *genuine* device ** D** and a

*genuine*reader

**. Differences of the reader**

*R***are equivalent to differences of**

*R**Z*; for instance, when

*Z*is shifted by distance Δ

*Ζ*= −1/100, the angular spectrum is given by the dotted curve in Fig. 1(b). Similarly, when Δ

*Ζ*= 1/100, the angular spectrum is given by the dashed curve in Fig. 1(b). (As described below, Fig. 1(c) is for the horizontal (

*X*) direction.) As shown by the changes of the curve in Fig. 1(b), a slight difference with respect to

*Z*results in a different output signal from the system. In order to quantitatively evaluate the

*Z*-dependence, the correlation coefficient of the angular spectrum is calculated as a function of Δ

*Ζ*, as summarized in Fig. 1(d). More specifically, let the angular spectrum of the genuine device

*D*and genuine reader

*R*be given by ${f}_{a}({s}_{\left|\right|})$

_{,}and let the angular spectrum of the genuine device

*D*and a reader displaced from the genuine reader by an amount of Δ

*Ζ*be given by ${f}_{\Delta Z}({s}_{\left|\right|})$; hence, the correlation coefficient is defined by $\int ({f}_{\Delta Z}({s}_{\left|\right|})-\overline{{f}_{\Delta Z}({s}_{\left|\right|})}})({f}_{a}({s}_{\left|\right|})-\overline{{f}_{a}({s}_{\left|\right|})})d{s}_{\left|\right|}/\sqrt{{\displaystyle \int {\left({f}_{\Delta Z}({s}_{\left|\right|})-\overline{{f}_{\Delta Z}({s}_{\left|\right|})}\right)}^{2}}d{s}_{\left|\right|}{\displaystyle \int {\left({f}_{a}({s}_{\left|\right|})-\overline{{f}_{a}({s}_{\left|\right|})}\right)}^{2}}d{s}_{\left|\right|}$

_{,}where $\overline{f({s}_{\left|\right|})}$ indicates the average value of $f({s}_{\left|\right|})$. If we determine that a genuine signal should yield a correlation coefficient larger than 0.9, Δ

*Ζ*should be between −1/37 and 1/34, which would be an extremely small absolute value in real dimensions. This indicates that nano-optics exhibits a strong reader-dependence, in agreement with reports on near-field scanning optical microscopy [27,28]. Similarly, by considering the horizontal position of the dipole as the identity of the device, a different

*X*provides a different angular spectrum. The solid, dotted, and dashed curves in Fig. 1(c) indicate the angular spectra when Δ

*X*is given by 0, −1/100, and + 1/100, respectively. The correlation coefficient is evaluated as shown in Fig. 1(e); it is larger than 0.9 when Δ

*X*is between −1/77 and 1/91, indicating that the output signal is sensitive to subtle differences of the device

**.**

*D*## 3. Polarization asymmetry induced by a two-layer nanostructure

As demonstrated in the simple model shown Section 2, tiny mutual differences between ** D** and

**result in large differences in the output signal via optical near-fields. The fundamental idea of this paper is to combine, or unite,**

*R***and**

*D***in the first place; that is to say, we consider a device architecture where two nanostructures are located in close proximity.**

*R*The basic architecture of the proposed structure is schematically shown in Fig. 2(a), where the first layer is composed of square shapes, whereas the second layer is composed of rectangular shapes located at the lower right corners of the square shapes in the first layer. In other words, the first layer has a symmetric shape, and the second layer has an asymmetric shape and is placed at an asymmetric position with respect to the first layer.

The square and circular marks in Fig. 2(b) respectively denote the polarization conversion efficiency from *x*-polarized input light to *y*-polarized output light (denoted by ${E}_{X\to Y}$) and from *y*-polarized input light to *x*-polarized output light (${E}_{Y\to X}$), calculated by a finite-difference time-domain (FDTD) simulation. The absolute value of the difference between the polarization conversion efficiencies is denoted by triangular marks in Fig. 2(c). The horizontal axes in Figs. 2(b) and 2(c) are accompanied by schematic illustrations of the elemental nanostructures to be evaluated.

The operating wavelength used was 688 nm. As the material, we assumed gold, which has a refractive index of 0.16 and an extinction ratio of 3.8 at a wavelength of 688 nm [29]. The dimensions of each square in the first layer were 300 nm $\times $ 300 nm, and the dimensions of each rectangle in the second layer were 75 nm (*x*) $\times $ 150 nm (*y*), which are respectively 1/4 and 1/2 the sizes of the squares in the first layer. These shapes are periodically arranged at regular intervals. (In the calculation, the interval between the neighboring units was 200 nm.) The thicknesses of the first and second layers were 100 nm. The gap between the two layers was 10 nm.

The nanostructures shown in Figs. 2(b,i) and 2(b,ii) were used to evaluate the polarization conversion efficiency of the first layer only and the second layer only, respectively, resulting in nearly no asymmetry. The nanostructures shown in Figs. 2(b,iii) and 2(b,iv) have a single-layer shape that mimics the letter “Z”. With such a shape, although polarization conversion efficiencies (${E}_{X\to Y}$ and ${E}_{Y\to X}$) appeared [30], these efficiencies had almost the same value, and so the difference of the polarization conversion efficiency, which is the definition of polarization asymmetry in this paper, as summarized in Fig. 2(c), was negligible.

The nanostructure shown in Fig. 2(a), corresponding to Fig. 2(b,v), exhibits asymmetry in the polarization conversion efficiencies. The nanostructure shown in Fig. 2(b,vi) consisted of the same-sized elements and the same inter-layer gap as those in Fig. 2(b,v), but the second-layer rectangles were placed at the *centers* with respect to the first-layer squares. This led to significantly reduced polarization conversion efficiencies, and the corresponding difference was calculated to be nearly zero; that is, there was no asymmetry.

Before discussing the theoretical background behind such asymmetry in Section 4 based on optical near-fields, here we make a few remarks on related work in the literature. Regarding the Jones matrix of a polarization rotation *θ*, the polarization conversion efficiencies ${E}_{X\to Y}$ and ${E}_{Y\to X}$ are respectively given by the (2,1) and (1,2) elements of the Jones matrix, which are sin *θ* and -sin *θ*, respectively. The difference of their absolute values results in zero, meaning that the polarization asymmetry discussed in this paper, i.e., ${E}_{X\to Y}\ne {E}_{Y\to X}$, does not appear. A polarizer extracting polarized light oriented at angle *θ* with respect to the *x*-axis exhibits polarization conversion efficiencies ${E}_{X\to Y}$ and ${E}_{Y\to X}$ having the same value, given by sin *θ* cos *θ*; that is to say, there is no asymmetry, as defined in this paper.

In the field of metamaterials, classification of periodic metamaterials regarding their polarization properties has been studied [31]. The asymmetric polarization conversion efficiencies discussed in this paper correspond to the “fifth” group in the context discussed in [31]. In a more general context, the relevance to general optical activity [32,33] could be considered. Optical activity is described in its most general form by a Jones matrix, given by

*γ*is called the gyromagnetic coefficient [33]. The first term is related to a mirror symmetry, and the second is related to pure optical rotation. A difference in the absolute value of the non-diagonal elements of the matrix

*J*, corresponding to the “asymmetry” discussed in this paper, could give a non-zero result if

_{o.a.}*B*and

*γ*are given by non-zero, complex numbers. Further insights regarding the relation between these generalized schemes [31,33] and the effects offered by near-field interactions may be interesting topics of future work.

## 4. Theory of asymmetry by two-layer nanostructures based on angular spectrum representation

Here we give a theoretical description of the asymmetry induced by a two-layer nanostructure based on the angular spectrum representation of optical near-fields introduced in Section 2. With *x*-polarized input light irradiating the square-shaped first-layer nanostructure, we consider that electron charges are concentrated in the corners of the nanostructure due to the so-called plasmon resonance effect [34,35]. Also, the phases of the oscillating electron charges differ by π between the right-hand and left-hand sides. In Fig. 3(a), this is represented by four dipoles, two on the right and two on the left, with opposite orientations. The side length of the first-layer shape is assumed to be λ/4, where λ is the operating wavelength.

Now we turn our attention to the induced electric fields and their associated induced electron charges in the second layer. Since the *y*-polarized component is of primary concern, we focus our attention on points A and B in Fig. 3(a), which correspond to the upper and lower center edges of the second layer, respectively. Note that the side length of the second-layer structure is λ/16 along the *x*-axis and λ/8 along the *y*-axis, and the second layer is located at the bottom right corner with respect to the first layer. Also, the inter-layer distance between the first and the second layer is assumed to be λ/20.

Similarly, with *y*-polarized input light, we assume that four oscillating dipoles are induced in the corners of the first layer, with the upper two and lower two being reversed in phase. In order to characterize the *x*-to-*y* polarization conversion, this time we focus on points C and D in Fig. 3(b), which are the centers of the right and left edges of the second layer, respectively.

Concerning the relative phase arrangements of the dipoles, we derive the angular spectrum corresponding to points A, B, C, and D, as shown in Fig. 4(a,i), based on Eq. (5). We observe that the difference of the angular spectrum at points A and B is more significant compared with that at points C and D.

Second, we locate the second layer at the center with respect to the first layer. In this case, as demonstrated in Fig. 4(a,ii), the angular spectra at points A and B exhibit exactly the same profile, as do those at points C and D. That is, via Eq. (5), the geometrical features of an array of dipoles and the relative evaluation positions give rise to identical angular spectra. Therefore, the difference between points A and B (and also points C and D) is zero, which indicates that such a configuration exhibits no polarization conversion from *x* to *y* or *y* to *x*. This agrees with the numerical calculations shown in Fig. 2(b,vi).

Third, we assume a square shape, that is, a symmetric structure, in the second layer, located at the lower right corner with respect to the first layer, as schematically shown in Fig. 4(a,iii). The side length of the second-layer squares is half that of the first-layer squares, namely, λ/8. The angular spectra at points A, B, C, and D behave differently; however, the “difference” of the curves between points A and B and between points C and D is not as significant as in the case of Fig. 4(a,i), as will be evaluated numerically below. Finally, we assume the same rectangular second-layer structure as in Fig. 4(a,i) but with an increased inter-layer distance of λ/2 (Fig. 4(a,iv)). The angular spectra at points A and B, as well as those at points C and D, exhibit nearly the same profiles, and the high-frequency components disappear. This is purely due to the increased inter-layer distance, causing the spectra to exponentially decay from the first layer, which is a manifestation of Eq. (5).

Here, we define the following metric for the asymmetry of polarization conversion predicted by the theory based on the angular spectrum representation:

*x*- and

*y*-polarized input light evaluated at point

*P*, respectively. The first term of Eq. (7) represents the X to Y polarization conversion, and the second describes the Y to X conversion; we consider that the difference between the two represents the asymmetry of the polarization conversion. As summarized in Fig. 4(b), which contains the numerical values based on Eq. (7), the two-layer nanostructure in which the asymmetric second layer is placed at an asymmetric position with respect to the first layer and in close proximity to it yields a significantly larger value. This is a clear manifestation that the asymmetric polarization conversion stems from the two-layer shape-engineered nanostructure in which the layers interact via optical near-fields.

## 5. Electromagnetic calculation and multi-polar analysis

In order to characterize some basic features, we first evaluate the inter-layer-distance dependences. Figures 5(a) and 5(b) show cross-sectional profiles of the electric field intensity $|{E}_{y}{|}^{2}$ when two-layer nanostructures with inter-layer distances of 10 nm and 50 nm, respectively, are irradiated with *x*-polarized input light. As is clearly observed from the images, the increased inter-layer distance decreases the inter-layer interactions. The square and circular marks in Fig. 5(c) represent the calculated polarization conversion efficiencies ${E}_{X\to Y}$ and ${E}_{Y\to X}$, respectively, whose difference diminishes as the inter-layer distance increases.

Based on the theory discussed in Section 4, for instance with *x*-polarized input light, we focus on the differences of the angular spectrum of the second layer in the vertical direction through an analytical scheme concerning the geometries. Numerically, we can discuss the induced charge distribution by calculating the divergence of the electric fields in the numerical simulations. As schematically shown in Fig. 5(d), we can decompose the induced charge distribution in the second layer into representative components corresponding to a constant value, a vertically different component, a horizontally different component, and higher-order components. In other words, we can factorize the charge distribution on an orthogonal basis, like a multi-polar expansion. We consider that the vertically different component corresponds to *y*-polarized output light (denoted by “$X\to Y$” in Fig. 5(d)), and the horizontally different component corresponds to *x*-polarized output light (denoted by “$Y\to X$”). More specifically, let the difference between the sum of the induced charge in the upper half and the sum of the induced charge in the lower half of the surface of the second layer facing the first layer with the *x*-polarized input light be ${p}_{X\to Y}$. Also, with the *y*-polarized input light, the difference between the sum of the induced charge in the left-hand half and the sum of the induced charge in the right-hand half of the surface of the second layer facing the first layer is ${p}_{Y\to X}$. Here we consider that the figure-of-merit (FoM) representing the asymmetric polarization conversion originating from the charge induced in individual elements is given by

*for the charge distribution in the second layer, which show good agreement with the calculated polarization conversion efficiencies.*

_{intrinsic}Additionally, we discuss some related concerns regarding structural attributes and asymmetry in polarization conversion. The first is related to the dependencies on inter-element-distances or the arrangement of elemental two-layer structures. (This is also referred to as the coupling between meta-atoms or the meta-atom arrangement [36,37].) While keeping the dimensions of the elemental two-layer structure the same as the one shown in Fig. 2(b,v), the circular marks in Fig. 6(a) represent the difference of the polarization conversion efficiencies as a function of the inter-element distance. The inter-element distances are the same for both vertical and horizontal directions. The asymmetry is maximized with an inter-element distance of 200 nm, which is actually the case shown in Fig. 2(b,v), indicating that inter-element distances that are too small or too large eliminate the asymmetry.

A presumable physical reason for this is as follows. With *x*-polarized input light, a vertically uniform charge distribution is induced in each of the second-layer rectangular elements, which is introduced above as ${p}_{X\to Y}$. The effect originating from vertically adjacent elements can be represented by $-{p}_{X\to Y}\times {L}_{y}/({L}_{y}+{L}_{y}^{(G)})$, where ${L}_{y}$ is the vertical length of the second-layer rectangular elements (150 nm), and ${L}_{y}^{(G)}$ is the distance between the vertically adjacent second-layer elements (Fig. 6(a)). The coefficient $-{L}_{y}/({L}_{y}+{L}_{y}^{(G)})$ indicates that the greater the inter-element distance, manifested by ${L}_{y}^{(G)}$, the lower the impact on the polarization, and the minus sign means that the electron charge density in the upper half of an element and that in the lower half of an element located above and adjacent to the former are in an opposite relation with respect to the polarity of the element-intrinsic attribute (${p}_{X\to Y}$), as schematically shown in Fig. 6(a). Similarly, the inter-element-dependent *x*-polarization with *y*-polarized input light is represented by $-{p}_{Y\to X}\times {L}_{x}/({L}_{x}+{L}_{x}^{(G)})$, where ${L}_{x}$ is the horizontal length of the second-layer rectangular element (75 nm), and ${L}_{x}^{(G)}$ is the distance between the horizontally adjacent second-layer elements. The net figure-of-merit concerning both the element-intrinsic attribute and the inter-element-distance-dependent component is given by

Another concern is the spectral dependency and the thickness dependency. The solid and dashed blue curves in Fig. 6(b) show the spectrum of the *y*-polarized output light with *x*-polarized input light and that of the *x*-polarized output light with *y*-polarized input light, respectively. Here, we assume an input optical pulse with a differential Gaussian form whose width is 0.9 fs, corresponding to a bandwidth of around 200–1300 THz. The conversion efficiency is given by calculating the Fourier transform of the output electric field evaluated at a point 2 μm away from the output surface of the second layer. Similarly, the solid and dashed red curves show the spectra for the nanostructures whose thicknesses and inter-layer gap are reduced by half (50%). The solid and dashed green curves, on the other hand, are the spectra for the nanostructure whose thicknesses and inter-layer gap are doubled (200%). The particular nanostructures investigated in earlier sections (blue curves) exhibit larger polarization conversion efficiencies (and strong asymmetric properties) around 680 nm. In any case, the asymmetric property exhibits wavelength and thickness dependencies. Detailed studies regarding these inter-element couplings (or meta-atom couplings) and spectra will be interesting topics of future work.

## 6. Conclusion

In summary, we demonstrate an asymmetry in polarization conversion induced by a two-layer shape-engineered nanostructure formed of isotropic materials that interact via optical near-fields. By using an angular spectrum representation of electric fields in the subwavelength regime, we first characterize the stringent alignment requirements in the conventional setup in which the device and reader are separated. With a view to achieving responses that can be detected in the far-field while at the same time exploiting the intrinsic optical near-fields associated with nanostructured matter, we propose a two-layer unified device architecture consisting of a square-shaped (or symmetric) first layer and a rectangular-shaped (or asymmetric) second layer located in an asymmetric position with respect to the first layer. The geometrical features of the nanostructure are systematically taken into account in a theoretical framework based on the angular-spectrum representation of optical near-fields, giving an asymmetric polarization conversion which agrees well with numerical calculations. The dependence on the inter-layer distance also clearly indicates the involvement of near-field interactions between the two layers.

Finally, we make a few additional remarks about future work regarding applications. In Sections 1 and 2, we addressed the precision alignment requirements required between the “device” and the “reader”. With the two-layer unified nanostructures, the inter-layer precisions are still important, as indicated for instance in Figs. 4(a,i) and 4(a,ii). We have to emphasize that the stringent alignment requirements addressed in the introduction are for the case where optical near-field probing tips are used. On the other hand, while alignment is indeed crucial in the manufacturing process in the case of the two-layer nanostructures proposed here, the device can be implemented in a fixed configuration, without any further alignment needed, in the form of a module together with some other optical elements, such as emitters and detectors. We should note that the intrinsic optical near-field processes associated with nanostructured matter are still utilized. The precision shape- and layout-dependencies of the polarization properties can then be exploited, for example, as the identity of the devices, as in the application known as “artifact-metrics” for anti-counterfeit technologies proposed by Matsumoto et al. [38]. Indeed, Matsumoto et al. succeeded in demonstrating “nano artifact-metrics” by utilizing randomly formed silicon nanostructures by leveraging resist collapse in electron beam lithography [39]. A discussion of such applications, as well as physical insights into the optical properties made possible by nanostructures [40], is an interesting topic for future work. Also, the theoretical approach based on optical near-field processes demonstrated in Section 4 will be applicable to various metamaterial structures aimed at, for instance, environmental applications [41] and telecommunications [42], among others.

## Acknowledgments

The authors would like to thank many collaborators for illuminating discussions over several years, in particular T. Kawazoe, T. Yatsui, and W. Nomura. This work was supported in part by Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS) and the Strategic Information and Communications R&D Promotion Programme (SCOPE) of the Ministry of Internal Affairs and Communications.

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