## Abstract

We propose a new method of optically reconstructing binary data formed by nanostructures with an elemental size several tens of nanometers smaller than the diffraction limit, implemented with an interference microscope and a complex-amplitude image pattern matching method. We examine the size dependency of the data reconstruction capacity using a light propagation simulation based on the finite-difference time-domain (FDTD) method and the Fourier spatial frequency filtering method. We demonstrated that the readable size of the binary nanostructure depends on the magnitude of noise.

©2013 Optical Society of America

## 1. Introduction

The amount of processed data has been continuing to increase day by day. The demands for high-speed, high-capacity data storage are also increasing. The demands on optical recording media are the same as those on other kinds of media, but it is well-known that, in principle, the recording density is limited by the optical diffraction limit. Therefore, new technologies have been developed to improve the density, including volume optical recording and reading methods using holography [1] and two-photon absorption [2], as well as two-dimensional optical recording and reading methods using super-resolution near field structures (super-RENS) [3] and solid immersion lenses (SILs) [4].

In this paper, we investigate a method for identifying nanostructures having an elemental size several tens of nanometers smaller than the diffraction limit by using an interference microscope with a pattern matching method.

It is technically challenging to determine the minimum size in identifying nanostructures using propagating light. It is well-known that propagating light fails to reveal most of the information about nanostructures, and the amount of missing information increases as the elemental size of the nanostructures decreases. However, small remaining differences between the propagating light components reflected by nanostructures are detectable, depending on the signal-to-noise ratio of the whole imaging system, and if we know the one-to-one relation between these differences and the original pattern, that is to say, under a special constrain condition and given prior information, we will be able to identify the nanostructures.

The complex amplitude obtained by interference microscopy gives more useful information for identifying the nanostructures. This identification is an important issue for reconstructing the data in optical data storage. We propose a new method of optically reconstructing binary data using complex amplitude images as new information for identifying the binary data formed by nanostructures smaller than the diffraction limit by using an interference microscope. The nanostructure we treat here is a binary structure formed by a pit or land having a size of several tens of nanometers. We examined the size dependency of 4-bit data reconstruction using the finite difference time domain method (FDTD) and light propagation based on the spatial frequency filtering technique representing imaging optics.

## 2. Complex amplitude reconstruction of binary nanostructures

#### 2.1. Binary nanostructures

A pit (convex structure) expresses a binary digital value of 1 (high) and the absence of a pit, i.e., a land, expresses 0 (low). A linear arrangement of pits and lands expresses multiple bits. Figure 1 shows a binary nanostructure representing the 4-bit binary digital data 1101. The pit width and height are denoted as *w*_{p} and *h*_{p}, respectively. In the initial stage of our research, in order to demonstrate the potential of our concept, in this paper we consider a simple one-dimensional arrangement of the pits because of the low computational costs involved. In future research, we will investigate two-dimensional and three-dimensional arrangements of pits.

#### 2.2. Reconstruction flow

Figure 2 shows a flow chart of the data reconstruction. A target binary nanostructure is observed with an interference microscope, and a complex-amplitude image is obtained. The method of obtaining the complex-amplitude image from the interference images is not limited to a specific method, and in this paper the phase-shifting method with four buckets is used. The target binary nanostructure is reconstructed with the 1:*N* matching method, in which *N* complex-amplitude templates corresponding to *N* binary data patterns are prepared in advance using an interference microscope simulated in a computer. The template that most resembles the observed complex amplitude image is selected from the *N* templates. Finally, the binary data is reconstructed.

The estimation in the template matching is performed using the degree of difference (*DoD*) between two complex amplitude images, defined as

*u*and

_{i}*u*are complex-amplitude images, [･] is the operator that obtains the real part of a complex number, and

_{j}*Cor*(

*u*,

_{i}*u*) is the complex correlation between

_{j}*u*and

_{i}*u*, defined as

_{j}*DoD*is [0, 1]; and

*DoD*= 0 if

*u*(

_{i}*x*) =

*u*(

_{j}*x*), and

*DoD*= 1 if

*u*(

_{i}*x*) = -

*u*(

_{j}*x*).

#### 2.3. Computer simulation of interference microscope

The interference microscope shown in Fig. 3 was simulated by using the FDTD method [5, 6] for calculating the light propagation near the nanostructure and by using spatial filtering based on the fast Fourier transform for simulating the microscope imaging system. The light source was a laser with a wavelength *λ* of 400 nm. The numerical aperture of the objective lens was *NA* = 0.85, and the magnification of the microscope was 250 × .

First, a complex-amplitude image of the reflected light from the nanostructure was obtained using the FDTD method. The transverse electric mode FDTD (TE-FDTD) was used. The *E*_{x} and *E*_{z} components of the electric field ** E =** (

*E*

_{x,}

*E*

_{y,}

*E*

_{z}) were located along the sides of a square called a cell, the

*H*component of the magnetic field

_{y}**(**

*H =**H*

_{x,}

*H*

_{y,}

*H*

_{z}) was located at the center of the cell, and the

*E*,

_{y}*H*, and

_{x}*H*components were 0. The binary nanostructure was a perfect conductor (

_{z}**=**

*E***0**in the structure), the other region was a vacuum, and they were surrounded by a perfectly matched layer (PML) [7,8] absorption boundary, allowing the analysis space to be treated as an open region, as shown in Fig. 4. The incident light was a TE-polarized focused beam, and the near-field reflected electric field distribution in the focal plane of the objective lens

*u*(

^{near}*x*) =

*E*(

_{x}*x*) was obtained. The number of cells, each having a size of 2 nm × 2 nm, was 512 × 200; that is, the calculation region was 1.024 µm × 0.4 µm.

The magnified complex-amplitude image, *u*(*x*), was obtained by spatial frequency filtering, described by the following equation:

*F*[･] and

*F*

^{−1}[･] are the Fourier transform and its inverse, and

*H*is a transfer function depending on

*NA*and

*λ*, given by

In this calculation, the number of pixels in a one-dimensional image was 512. The filtered complex amplitude was imaged on an image sensor with a pixel size of 5.0 µm, and the detected image was transferred to a computer.

Figure 5 shows an example of the light propagation calculation simulating an interference microscope. Figure 5(a) shows the reflected near-field complex amplitude distribution. The upper and lower images are the amplitude and phase distributions, respectively. Figure 5(b) shows the complex amplitude distribution on the focal plane, *u*^{near}(*x*). Figure 5(c) shows the complex amplitude distribution on the image sensor, *u*(*x*), after passing through the microscope, which lost the high-spatial-frequency components contained in *u*^{near}(*x*).

#### 2.4. Templates

When the binary nanostructure has four bits, the number of complex amplitude templates is 2^{4} = 16. There are two methods for generating templates: one is to generate them by a computer simulation of an interference microscope, and the other is to generate them with a real optical microscope. If the number of templates is small, the use of a real optical system identical to the reconstruction optical system used for obtaining the templates will be effective for achieving higher performance. However, when the number of templates is large, it is more efficient to generate the templates by computer simulation. In this paper, the templates were generated by computer simulation of the interference microscope. Figure 6 shows templates with *h _{p}* =

*w*

_{p}= 100 nm. The 16 templates are denoted as

*u*

_{0000}(

*x*) to

*u*

_{1111}(

*x*).

## 3. Analysis and estimation of reconstruction performance

#### 3.1. Degree of difference between two templates

The degree of difference, *DoD* calculated by Eq. (1), is a useful index for comparison of two complex-amplitude vectors. Figure 7 shows the *DoD*s between every pair of templates in the 16 templates *u*_{0000}(*x*) to *u*_{1111}(*x*) versus the elemental size (pit size) of the nanostructure. The number of pairs is _{16}C_{2} = 120. The pit size was varied while keeping *p*_{w} = *p*_{h}. *DoD*_{max}, *DoD*_{min} and *DoD*_{ave} are the maximum, minimum, and average *DoD* values for each pit size. As the pit size decreased, the *DoD* values decreased due to the decreasing difference between templates.

#### 3.2. Multi-dimensional scaling analysis of templates and the relative position mapping

In order to visualize the relative relation of the templates, we used multi-dimensional scaling (MDS), which is a method for locating patterns from their distances in *N*-dimensional space [9]. When a position vector of an *i*-th pattern in *N* patterns is *x** _{i}*, and the

*j*-th element is

*x*, an

_{ij}*N*×

*N*position matrix is defined as

**= {**

*X**x*}. When the inner product of

_{ij}

*x**and*

_{i}

*x**is*

_{j}*b*, the inner product matrix is

_{ij}**= {**

*B**b*} =

_{ij}

*XX**. If the position is in Euclidean space, the relation of the distance*

^{t}*d*and the inner product

_{ij}*b*between two positions

_{ij}

*x**and*

_{i}

*x**is given by*

_{j}*b*is the distance between the position

_{ii}

*x**and the origin O of the Euclidean space. When a position*

_{i}

*x**is the origin O, the inner product*

_{n}*b*is given byThus, the distance

_{ij}*d*is an element of a distance matrix

_{ij}**= {**

*D**d*} that transforms to the matrix

_{ij}**, which is a symmetric and diagonalizable matrix. Therefore, introducing a matrix**

*B***= diag(**

*Λ**λ*

_{0},…,

*λ*

_{N-1}) using the eigenvalues

*λ*

_{0},…,

*λ*

_{N}_{-1}and a matrix

**= (**

*P*

*p*_{0},…,

*p*_{N-1}) using the eigenvectors

*p*_{0},…,

*p*

_{N}_{-1},

**is diagonalized by**

*B***=**

*B***. Because**

*PΛP*^{t}**=**

*B*

*XX**, the position matrix*

^{t}**is given byThus, the distance matrix**

*X***constructs the position matrix**

*D***.**

*X*In this paper, the position of a template *u _{i}*(

*x*) is

*x*, and the difference between two templates is

_{ij}*d*

_{ij}

*=**DoD*(

*u*,

_{i}*u*). However, templates are located in N-dimensional space. Therefore, the eigenvalues exclude the largest value

_{j}*λ*

_{i}, and the next-largest value

*λ*

_{j}was altered to 0. Then, a relation position map, to which the templates contributed, was obtained in a two-dimensional space.

Figures 8, 9, and 10 show relative position maps constructed by MDS for templates with *h*_{p} = *w*_{p} = 100 nm, *h*_{p} = *w*_{p} = 70 nm, and *h*_{p} = *w*_{p} = 40 nm, respectively. The difference between the templates was quantitatively measured using the degree of difference *DoD*, and the relative positions of the templates were constructed. A larger distance between templates indicates the ease of discriminating these templates. The binary structure 0000 is equivalent to a flat mirror; therefore, *u*_{0000}(*x*) was a sinc function defocused with a distance of 2*p*_{h}. The pattern remained constant with varying pit size; therefore, the position of this point on the relation position map was fixed at (0.5, 0.0).

In Fig. 8, the templates are widely spread over the whole space. Therefore, it is expected that the nanostructures with an elemental size of 100 nm (*p*_{w} = *p*_{h} = 100 nm) are readily discriminated. The templates *u*_{0000} and *u*_{1111} are discretely located, and *DoD*(*u*_{0000}, *u*_{1111}) was close to 1, indicating that they were located at the farthest positions from each other. The *DoD* between two templates tended to be large when their Hamming distance was large. The value on the vertical axis increased with an increasing number of pits (binary value 1). Moreover, two nanostructures with a symmetrical structure were located at symmetrical positions on the horizontal axis. Even though the number of pits was the same, the template was farther away from *u*_{0000}(*x*) when the nanostructure had a pit near the center. This is because a nanostructure with a pit near the center more strongly affects the reflected light of the focused incident beam. To summarize what this relative position map reveals, the vertical axis represents the number of pits (binary value 1), the horizontal axis represents the pit symmetry, and the distribution of irradiation light distorts the relative position distribution of the templates on this map.

As shown in Figs. 9 and 10, the positions of the templates converged toward the origin *u*_{0000} with the smaller pit size. When *p*_{w} = *p*_{h} = 40 nm shown in Fig. 10, the values on the vertical axis corresponded to the number of pits, because the light distribution became more uniform. The distribution along the horizontal axis was small because the variance of the nanostructures decreased with decreasing pit size. However, a difference between the complex amplitudes (the binary patterns) still remained, and it should be possible to distinguish them using an interference microscope with a very high signal-to-noise ratio (*SNR*).

#### 3.3. Evaluation of reconstruction performance using bit error rate (BER) in the presence of electronic noise

The characteristics of the binary data reconstruction in the presence of random noise were evaluated. A complex-amplitude image was obtained by analyzing interference images captured by an image sensor using the phase shifting method with four steps [10]. The *n*-th interference image, *I _{n}*(

*x*), is given by

*u*is reference light. A complex-amplitude image obtained from four fringe images is calculated by

_{r}*σ*

^{2},

*N*(0,

_{n}*σ*

^{2}), is present in the fringe images, a complex amplitude containing noise,

*u’*(

*x*), is given by

*N*

_{0},

*N*

_{1},

*N*

_{2}, and

*N*

_{3}are independent of each other, and

*N*

_{02}=

*N*

_{0}–

*N*

_{2}and

*N*

_{31}=

*N*

_{3}–

*N*

_{1}.

When binary data *i* was reconstructed by performing matching between *u _{i}*'(

*x*) and the templates

*u*

_{0000}(

*x*) to

*u*

_{1111}(

*x*), the number of false reconstructions was counted, and the bit error rate (

*BER*) was computed. This process was repeated while varying the noise level. The signal-to-noise ratio (

*SNR*) is defined as

*I*is the maximum value in all fringe images, which is calculated as

_{max}*σ*

^{2}was determined for

*I*, and noise was added to all pixels of the complex-amplitude image.

_{max}Figure 11 shows the complex amplitudes with noise. Figures 11(a) and 11(b) show the complex-amplitude images with *SNR* = 30 dB and *SNR* = 50 dB, respectively. The reference light *u*_{r} had the same distribution as *u*_{0}.

Figure 12 shows *BER* versus *SNR*, for different elemental sizes. The reconstructions were repeated 10^{5} times while changing the noise for each pattern. The number of patterns was 16, and each pattern had 4 bits; thus, the total number of trial bits was 6.4 × 10^{6}. Increasing the *SNR* of the imaging sensor decreased the *BER*. To correctly reconstruct the binary nanostructure with a small element size requires a high *SNR*. If *BER* = 10^{−4} is assumed to be low enough for error-free reconstruction by introducing an error-correcting coding method, this *BER* can be achieved with *SNR* = 55 dB when *w*_{p} = *h*_{p} = 30 nm, *SNR* = 75 dB when *w*_{p} = *h*_{p} = 20 nm, and *SNR* = 95 dB when *w*_{p} = *h*_{p} = 10 nm. Because high-performance image sensors have a dynamic range of more than 50 dB, the data can be well-reconstructed even if *p*_{w} = *p*_{h} = 30 nm.

## 4. Conclusions

We demonstrated a new method of optically reconstructing binary data formed by nanostructures smaller than the diffraction limit using an interference microscope. When the pit size of the nanostructure was small, the variation of the reconstructed complex-amplitude images was small. We demonstrated a *BER* < 10^{−4} when *w*_{p} = *h*_{p} = 30 nm and *SNR* = 55 dB, and when *w*_{p} = *h*_{p} = 20 nm and *SNR* = 75 dB. This computer simulation was performed on an ideal material and ideal devices, except for the noise in the image sensor, and therefore, the results presented here give the upper limit of the system performance. Microscopy based on complex amplitude observation and the pattern matching method described here is a highly promising technique for super-resolution imaging of nanostructures such as an imaging for biological samples and an optical inspection for semiconductor devices [11].

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