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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 wp and hp, 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
where ui and uj are complex-amplitude images, [･] is the operator that obtains the real part of a complex number, and Cor(ui, uj) is the complex correlation between ui and uj, defined asThe range of DoD is [0, 1]; and DoD = 0 if ui(x) = uj(x), and DoD = 1 if ui (x) = -uj(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 Ex and Ez components of the electric field E = (Ex, Ey, Ez) were located along the sides of a square called a cell, the Hy component of the magnetic field H = (Hx, Hy, Hz) was located at the center of the cell, and the Ey, Hx, and Hz components were 0. The binary nanostructure was a perfect conductor (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 unear(x) = Ex (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:
where F[･] and F−1[･] are the Fourier transform and its inverse, and H is a transfer function depending on NA and λ, given byThis process was implemented with the fast Fourier transform (FFT) algorithm.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, unear(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 unear(x).
Fig. 5 Example calculation results, showing (a) reflected near-field distribution, Ex(x, z), (b) near-field complex amplitude distribution on focal plane, unear(x), and (c) complex amplitude distribution on image sensor, u(x), after passing through microscope.
2.4. Templates
When the binary nanostructure has four bits, the number of complex amplitude templates is 24 = 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 hp = wp = 100 nm. The 16 templates are denoted as u0000(x) to u1111(x).
Fig. 6 Complex amplitude templates (four bits). Solid line indicates the amplitude distribution and dashed lines indicates the phase distribution.
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 DoDs between every pair of templates in the 16 templates u0000(x) to u1111(x) versus the elemental size (pit size) of the nanostructure. The number of pairs is 16C2 = 120. The pit size was varied while keeping pw = ph. DoDmax, DoDmin and DoDave 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.
Fig. 7 Degree of difference (DoD) between all pairs of templates versus the pit size. The curves labeled DoDmax, DoDmin, and DoDave are the maximum, minimum, and average DoD.
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 xi, and the j-th element is xij, an N × N position matrix is defined as X = {xij}. When the inner product of xi and xj is bij, the inner product matrix is B = {bij} = XXt. If the position is in Euclidean space, the relation of the distance dij and the inner product bij between two positions xi and xj is given by
where bii is the distance between the position xi and the origin O of the Euclidean space. When a position xn is the origin O, the inner product bij is given byThus, the distance dij is an element of a distance matrix D = {dij} that transforms to the matrix B, which is a symmetric and diagonalizable matrix. Therefore, introducing a matrix Λ = diag(λ0,…, λN-1) using the eigenvalues λ0,…, λN-1 and a matrix P = (p0,…, pN-1) using the eigenvectors p0,…, pN-1, B is diagonalized by B = PΛPt. Because B = XXt, the position matrix X is given byThus, the distance matrix D constructs the position matrix X.In this paper, the position of a template ui(x) is xij , and the difference between two templates is dij = DoD(ui, uj). However, templates are located in N-dimensional space. Therefore, the eigenvalues exclude the largest value λi, and the next-largest value λj 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 hp = wp = 100 nm, hp = wp = 70 nm, and hp = wp = 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, u0000(x) was a sinc function defocused with a distance of 2ph. 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).
Fig. 10 Relative positions of complex amplitude templates when wp = hp = 40 nm. Inset shows a magnified view of the region where the templates were clustered.
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 (pw = ph = 100 nm) are readily discriminated. The templates u0000 and u1111 are discretely located, and DoD(u0000, u1111) 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 u0000(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 u0000 with the smaller pit size. When pw = ph = 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, In(x), is given by
where ur is reference light. A complex-amplitude image obtained from four fringe images is calculated byIf white Gaussian noise with mean 0 and variance σ2, Nn(0, σ2), is present in the fringe images, a complex amplitude containing noise, u’(x), is given bywhere the random variables N0, N1, N2, and N3 are independent of each other, and N02 = N0 – N2 and N31 = N3 – N1.When binary data i was reconstructed by performing matching between ui'(x) and the templates u0000(x) to u1111(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
where Imax is the maximum value in all fringe images, which is calculated asThe noise variance σ2 was determined for Imax, and noise was added to all pixels of the complex-amplitude image.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 ur had the same distribution as u0.
Figure 12 shows BER versus SNR, for different elemental sizes. The reconstructions were repeated 105 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 × 106. 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 wp = hp = 30 nm, SNR = 75 dB when wp = hp = 20 nm, and SNR = 95 dB when wp = hp = 10 nm. Because high-performance image sensors have a dynamic range of more than 50 dB, the data can be well-reconstructed even if pw = ph = 30 nm.
Fig. 12 BER versus SNR of image sensor in binary data reconstruction with different elemental sizes.
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 wp = hp = 30 nm and SNR = 55 dB, and when wp = hp = 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].
References and links
1. P. J. van Heerden, “Theory of optical information storage in solids,” Appl. Opt. 2(4), 393–400 (1963). [CrossRef]
2. J. H. Strickler and W. W. Webb, “Three-dimensional optical data storage in refractive media by two-photon point excitation,” Opt. Lett. 16(22), 1780–1782 (1991). [CrossRef] [PubMed]
3. J. Tominaga, T. Nakano, and N. Atoda, “An approach for recording and readout beyond the diffraction limit with an Sb thin film,” Appl. Phys. Lett. 73(15), 2078–2080 (1998). [CrossRef]
4. I. Ichimura, S. Hayashi, and G. S. Kino, “High-density optical recording using a solid immersion lens,” Appl. Opt. 36(19), 4339–4348 (1997). [CrossRef] [PubMed]
5. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]
6. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, 1993).
7. J.-P. Berenger, “A perfectly matched layer for the absorption of electro-magnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]
8. J. Fang and Z. Wu, “Closed-form expression of closed numerical reflection coefficient at PML Interfaces and optimization of PML Performance,” IEEE Microw. Guided Wave Lett. 6(9), 332–334 (1996). [CrossRef]
9. W. S. Torgerson, “Multidimensional scaling: I. Theory and method,” Psychometrika 17(4), 401–419 (1952). [CrossRef]
10. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef] [PubMed]
11. K. P. Bishop, S. M. Gasper, L. M. Milner, S. S. H. Naqvi, and J. R. McNeil, “Grating line shape characterization using scatterometry,” Proc. SPIE 1545, 64–73 (1991). [CrossRef]
