1. Introduction

Nanophtonic devices, such as ultra-high density optical data storage, nano-sized optical telecommunication switches, quantum-confined lasers, nanolithography and nano optical sensor [1–3], have made significant progress in recent years. Nevertheless, light in subwavelength optical devices are not easily manipulated because of diffraction limit. If an object is much smaller than the wavelength and the spatial frequency spectrum of the scattered light from the object is broad, the high-spatial-frequency parts that miss the numerical aperture (N.A.) of the lens can not be detected. The limitation of detectable spatial frequencies in the far field determines the diffraction limits of optical system. In order to realize the object of nanophotonics, it is imperative to find practical approaches which overcome diffraction limits.

High spatial frequency fields (evanescent waves) are corresponding to subwavelength fine details of the object, and these components exponentially decay away from the object [4–6]. Those evanescent components are lost in the far field, and subwavelength details are indistinguishable in the far-field region. In 1928, Synge proposed to detect objects by a small hole on a metallic sheet in near-field region [7]. Using a small aperture close to sample surface can convolute high spatial frequency fields (evanescent waves) from small objects into detectable signals (propagating waves) [8,9]. This concept was first realized by a scanning microwave microscopy with λ/60 resolution in 1972 [10]. With fast progress of fabrication and manipulation techniques, this ideal was accomplished in visible regime by Pohl. et al. in 1984 [11]. In conventional near-field optical techniques, near-field optical probes can resolve subwavelength objects far beyond diffraction limits. However, this technology has difficulties of low scanning speed and near-field distance control which are major hurdles on commercial applications.

The idea of using a masking layer to overcome diffraction limit in optical disks appeared as early as in early 1990s [12,13]. In 1998, a multilayer near-field optical nanostructure was proposed to overcome the diffraction limit by internal disk structures [14]. Subsequently, the AgO_x-type [15], ZnO-type [16] and PtO_x-type [17] near-field optical disk were developed for higher carrier-to-noise ratio (CNR). A series of studies about the mechanism of AgO_x-type near-filed optical disk structure have been reported, and its super-resolution capability is due to the localized surface plasmon enhancements of Ag nanoparticles [18–28]. The random nanostructures of AgO_x layer play an important role in its superior spatial resolution [26], and work as near-field optical probes which couple evanescent waves into propagating signals. In the pursuit of high-density near-field optical disks, various super-resolution structures have been found to exhibit super-resolution capability, including dielectric, organic, and other complicated structures [29–32]. It is necessary to establish a general understanding about the physical mechanism of super-resolution for such a wide variety of near-field optical disks.

In this paper, Fourier optics approach and angular spectrum representation [4–6] are used to analyze the optical effect of a random nanostructure in near-field optical disks. To describe the general property of random nanostructure, its averaged characteristic is considered through the statistical properties. An analytical expression of readout contrast of near-field optical disks with diluted random-distributed apertures is derived. Readout contrasts with random apertures of various densities are simulated numerically.

2. Angular Spectrum Representation of Scattered Fields

If a monochromatic scalar wave propagate in (x,z) plane in a two dimensional model, giving the complex field U(x,0) at z = 0 axis, the Fourier transform of the incident complex field U(x,0),

is the angular spectrum of U(x,0). The angular spectrum can be regarded as a decomposition of plane waves with wave vector $\stackrel{\rightharpoonup }{k}$. The field U(x,z) can be written in terms of the angular spectrum of incident field

where $k_z=\sqrt{k^2-k_x^2},k=\frac{2\pi }{\lambda }$, and λ is the wavelength of incident light. The intensity of the component with wave vector k_x is

In commercial CD or DVD, the data is recorded by serial recording marks, and the realistic recording mark chains are very complicated. To simplify calculation, the periodic recording mark chain in the optical disk is represented by a grating. The near-field active layer consists of random nanostructures and its optical property can be characterized by a random optical function. Angular spectrum representation is used to calculate far fields of the near-field optical disks, and the model is shown in Fig. 1. The beam width of incident Gaussian wave is a. The far-field intensity of the near-field optical disk is

where G(x) is the optical function of the grating and F(x) is the optical function of the random nanostructure in the near-field optical disk. The refractive index of recording mark is different from the surrounding, and the optical reflectance changes when reading laser beam scan from surrounding to a recording mark. In experimental measurements, the reflection contrasts between recording marks and surrounding are represented by Carrier-to-Noise-Ratio (CNR). In order to compare with the experimental CNR of optical disks, far-field intensity differences between on-mark and off-mark situations are calculated to correspond to readout contrasts. The on-mark situation means that the incident light focuses on the recording mark, and off-mark situation means that the incident light focuses between adjacent marks. The grating optical function of on-mark case is

where g =π / L and L is the recording mark size. For the off-mark case, the grating shift a half period.

So the far-field intensity difference between on-mark and off-mark situations, which represents readout contrast of an optical disk, is

 

Fig. 1. Model of the near-field optical disk with a random nanostructure.

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It is noted that the far-field intensity differences of a near-field optical disk with a particular random nanostructure only relates to that specific random nanostructures. To avoid this problem, statistical approach is used to realize general properties of the near-field optical disk with random nanostructures. For the averaged random structure, <F(x)F(x’)> is decided by the statistical properties, autocorrelation function, of random structure. The autocorrelation function can be described as < F(x)F(x') > = σ ² C(x − x') + h ², where C(x−x’) is correlation function of two random functions F(x) and F(x’), σ is root-mean-square of random structure F(x), and h is mean value of F(x). The correlation function usually is assumed to be Gaussian C(x − x') = exp(−(x − x')² / l ²), where l is the correlation length. The averaged intensity difference is

The first term of average intensity difference is from mean value h ² of random structure and is the same as the case of uniform thin film with transmittance h ². The second term is from the correlation function of random structure. However, the exp(−g²a²/8) term dominates while mark size approaches to 0 (g→∞), and the average intensity differences are close to cases of uniform thin film with subwavelength record marks. But in Eq. (8) the average intensity difference is still subject to diffraction limit and disappears for subwavelength recording marks. The reason is that far-field intensity of on-mark case may be higher or lower than that of off-mark cases because of random structure, and the intensity difference (I_on-mark−I_off-mark) of recording marks much smaller than diffraction limit could be positive or negative for different random structures. We believe the contrast inversion in the intensity difference signal of is real and can be detected experimentally for small recording marks, although experimental CNR measurements of near-field optical disks concerned only the variations of readout signals along recording mark train. The enhancements of difference signal are lost in the average process, so the reasonable contrasts between on-mark and off-marks cases are absolute values of intensity differences.

It is difficult to average absolute values of intensity differences, so the root mean square (RMS) of far-field intensity differences would provide more reasonable average difference signals. From Eq. (7), the square of intensity difference is

The optical function of random nanostructures is the combination of fluctuations at random positions

where the x_i, x_j, x_l, x_m are independent random variables which are the center positions of fluctuations. Those random variables are assumed to be uniformly distributed. From Eq. (9) and (10), the square of far-field intensity difference is a function of x_i, x_j, x_l, x_m

The expectation value of square of intensity difference is

where the P(x_i), P(x_j), P(x_l), P(x_m) are probability density functions of those random variables and C is the (linear) density of random fluctuations. For a dilute case, the density is small and the first term of Eq. (12) dominates the behavior. It is straightforward to calculate the RMS of intensity differences from the first term of Eq. (12) for dilute cases. The RMS of the difference signals for both dilute and dense cases ae simulated by numerical method. The numerical RMS is calculated from Eq. (9) directly, and 12800 realizations of random nanostructures with the same statistical nature are averaged to obtain the far-field difference signals. Standard deviations of numerical results are under 1% of the readout contrast signals. To make the physical meaning clear, we use readout contrast signals instead of RMS of the far-field difference signals in the following sections.

3. Super-resolution of random nanostructures

In previous studies of near-field optical disks, the near-field intensity enhancements induced by nano-scatters are closer to Gaussian function than rectangle function [17–28]. Here Gaussian apertures, instead of rectangle apertures, are chosen as random fluctuations mentioned in previous section, since the well-behaved Gaussian apertures greatly simplify the integration process. The optical function of random nanostructures is assumed to be the sum of Gaussian apertures at random positions

where x_i are uniformly-distributed random positions and N is the amount of apertures. From the first term of Eq. (12) and (13), the readout contrast (RMS of far-field intensity difference) for dilute cases is

The readout contrast of random nanostructure is normalized to a transparent uniform thin film.

There are two main parts in Eq. (14). The first part is a combination of two Gaussian distributions in spatial frequency domain, and the interval between two Gaussian distributions is g (the spatial frequency of the grating). The detectable propagating waves are limited and decided by the N.A. of optical lens as illustrated in Fig. 2. While the aperture size is close to the beam width a, the width of Gaussian distribution is about 1/a. When the recording marks are smaller and its spatial frequency g is larger, the detectable components become fewer. On the other hand, when the aperture size D is much smaller than the beam width, the width of Gaussian distribution is about 1/D, which is much larger than 1/a. Between two Gaussian distributions, there is more overlapping, which means that detectable signals are enhanced even for a small recording mark size (Fig. 2(b)). The second part of Eq. (14) is a combination of two exponential decay functions which are related to the spatial frequency of grating. The first term is the same as the uniform thin film case which shows the averaged transmittance of random nanostructures. The second term is the same as the case of an opaque thin film with single Gaussian aperture and this term describe the characteristic of random nanostructures.

 

Fig. 2 Illustration of readout contrast in spatial frequency domain: (a) Aperture size D is close Gaussian beam width a. (b) Aperture size D is much smaller than the beam width a.

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Using the conventional DVD scheme, the wavelength λ is 650 nm and the Gaussian beam width a is 542 nm. Figure 3(a) shows readout contrast with different aperture size for a normal transmitting light, i.e., k_x = 0. The readout contrast signals of a uniform thin film are used as reference, and the signals drop around 0.9λ mark size due to diffraction limit. For the cases of D = λ, the size of random apertures is close to the wavelength, so the behavior of readout contrast signal is similar to the reference case. While the size of random apertures is smaller than the wavelength, the signals of small recording marks decrease very slowly. The signals of D = 0.1λ case apparently drop around 0.1λ mark size, because the aperture size decides the resolution limit of random nanostructures. For the cases of D = 0.01λ, the signals are almost flat for mark size larger than 0.02λ, but become weaker than the signals of the D = 0.1λ cases. Fig. 3(b) shows the results for N.A. = 0.6 cases, and these result are similar to the cases of normal transmitting limit. The numerical results of dilute apertures are also calculated to compare with the analytic results. The ratio between aperture area and total calculating regime is 5%. The numerical results are normalized to the analytic readout contrast signals at 0.8λ mark size. The behavior of numerical results agrees very well with the analytic results and it confirms the dilute approximation of Eq. (12).

 

Fig. 3 Readout contrast signals of analytical results (solid line) and numerical results (mark) with various aperture size for (a) for normal transmitting (k_x = 0) cases and (b) N.A. = 0.6 cases.

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Both analytical and numerical results demonstrate that readout contrast signals of recording marks smaller than diffraction limit decrease slowly for small random apertures. The readout contrast signals of small random apertures drop only when recording mark size approaches to aperture size. This behavior reveals the general properties of random aperture, i.e., broad spatial frequency of small random apertures convolute out evanescent signals and the size of random apertures determine the resolution capability. Readout contrast signals of large recording mark size smear because of random nanostructure, but the readout contrast signals of small recording mark size are highly enhanced. On the other hand, an equivalent process is to readout the random nanostructures using the diffraction grating with a specific spatial frequency and the matched evanescent wave from the random nanostructures is convoluted into propagating wave.

 

Fig. 4 Numerical readout contrast signals of dilute case for (a) k_x = 0, and (b) N.A. = 0.6 cases. The coverage rate is 65%.

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The numerical results of dense cases are simulated to compare with dilute cases. The ratio between aperture area and total calculating regime is 65% in Fig. 4. The readout contrast signals are normalized at 6λ mark size. The readout contrast signals of D = 0.1λ cases drop around 0.1λ mark size which are similar to the dilute situations in Fig. 3. For the cases of D = 0.01λ, the readout contrast signals are almost flat for mark size are larger than 0.01λ. Both the cases of k_x = 0 and N.A. = 0.6 exhibit similar behaviors which also appear in the results of dilute situation, though the readout contrast signals in Fig. 4 are about 5 db weaker than those of dilute situation. Those cases with even higher coverage ratio are also simulated and there is no characteristic difference between dilute cases and dense cases, even when the coverage ratio is as high as 99%. This surprising phenomenon can be explained by Babinet’s principle. The distribution of light diffracted by a screen with apertures is the same as the case with complementary screen. So signals of dense cases are similar to those of dilute cases.

4. Waveform of near-field optical disks with random nanostructures

From previous studies, readout signals of near-field optical disks with subwavelength recording marks have been observed in experiments and clearly indicate position and size of subwavelength recording marks of various lengths [33,34]. To compare more closely with previous experiments about readout waveform, the intensity variations (waveform) while the focusing Gaussian beam scans through periodic mark chain are examined. While the Gaussian beam shifts a distance l from on-mark situation, the optical function of grating is

Using Eq. (3), (13) and (15), the transmitting intensity is

A constant part in Eq. (16) is only related to wave vector k_x and g. In experimental measurements, the readout signals of optical disks are related only to the variations of received intensity while the Gaussian beam scans through periodic recording mark chain. So the background constant part can be neglected. There are phase differences for the apertures at different random positions and intensity variations caused by random apertures are diminished in statistical averaged process. As shown in section 2, RMS of I(l) provides more reasonable results. Following the averaged procedure in Eq. (12), averaged square of I(l) is

 

Fig. 5 (a) Waveform of single frequency case for k_x = 0 and D = 0.1λ ; (b) Numerical waveform of two-frequency case for k_x = 0 and D = 0.1λ .

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In Eq. (17), the exp(−a²g²/4) term decrease dramatically for small recording mark (g→∞), and the exp(−9a⁴g²/16(a²+D²)) term is much smaller than exp(−a⁴g²/16(a²+D²)) term which is the dominating term. The analytical RMS of I(l) (waveform) for different mark size (L) is shown in Fig. 5 (a), and the aperture size D is 0.1λ. The RMS of transmitting intensity displays a clear readout waveform while mark size L is 0.5λ(above diffraction limit). The readout waveform are well readable even for mark size smaller than diffraction limit (L = 0.4 λ and 0.35λ ). While mark size L is 0.3λ, which is close to the aperture size (0.1λ), the readout waveform becomes weaker but still distinguishable. The analytical result in Eq. (17) implies that readout contrast signals of near-field optical disks diminish much slower than those of conventional optical disks.

Above results present the readout waveform of a single-frequency pattern, however, realistic recording mark chain in optical disk consists of random patterns of recording marks. Here a two-frequency pattern is calculated to serve as an example. The optical function of two-frequency grating is assigned to be G(x) = 1 + cos(gx+θ ) + cos(1.7gx+θ ), and the phase θ indicates the shift of Gaussian beam, which is corresponding to l in Eq. (15). The simulated RMS of intensity I(θ ) is shown in Fig. 5 (b), and the coverage ratio of random apertures is 5%. Results show that transmitting variations are distinguishable for both cases of 0.77λ and 0.3λ mark size. The intensity minimums of I(θ ) are around 1.25π which is different from the single frequency case( minimum is at π ), and waveforms show asymmetric behaviors. The waveform differences between single frequency and two-frequency cases indicate that multi-frequency signals are possible to be coupled out by random apertures.

5. Near-field optical disks with random nanostructures

The concept of RMS signals in Eq. (12) can be easily applied to a more realistic model of near-field optical disks. The structure of a near-field optical disk is shown in Fig. 5. There is a 15 nm dielectric spacing layer between the random nanostructure and periodic recording layer. Incident Gaussian beam with beam width a illuminates from above through the random nanostructure, then it passes the spacing layer to the periodic recording layer. The optical waves are reflected from the periodic recording layer, and finally, the reflected wave passes through the spacing layer and the random nanostructure again. The far-field signals are obtained from angular spectrum representation of the reflected waves.

 

Fig. 5 A realistic model of near-field optical disk with random nanostructures.

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The optical function of random nanostructures is random Gaussian apertures as in Eq. (13). The wavelength λ is 650 nm and the beam width a is 542 nm. 15000 realizations of random nanostructures with the same statistical properties are used to obtain the readout contrast between on-mark and off-mark situations. The coverage ratio of apertures is 65%. The average RMS signals for N.A. = 0.6 cases are shown in Fig. 6 and all signals are normalized to the far-field signals of 6λ mark size. In the cases without random nanostructures, the far-field signals drop around 0.6λ mark size because of diffraction limit. Signals of smaller recording marks are enhanced for D = 0.1λ and D = 0.01λ cases. For example, in the case of D = 0.1λ, the readout contrast signals decrease slowly for recording marks larger than 0.1λ and drop around 0.1λ mark size. The behaviors of readout contrast signals are similar to results of Fig. 4, only the readout contrast signals decrease more for smaller recording marks. It is because the wave travels extra distance. Those results indicate that contrast between on-mark and off-mark situations is determined by the size of random nanostructures.

 

Fig. 6 Readout contrasts of numerical results for N.A. = 0.6 for a model of near-field optical disk.

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In this paper, both analytical and numerical results demonstrate that readout contrast signals are highly enhanced for cases with random nanostructures smaller than recording mark size. Those random nanostructures which can generate optical fields with local random fluctuations provide super-resolution ability. The random optical function can be realized by several ways, for example, from nano rough surface [14, 15], nano random scatters [16–28], or nano-composite [30]. Localized surface plasmons from metallic nanostructures are well known to be able to generate enhanced local fields which are excellent examples of the random optical function. These results give a reasonable explanation how evanescent signals convert to propagating signals through random nanostructure, which have been observed in previous studies [19–28]. However, the complex interactions between incident light and complicated optical disks have been simplified in our model. Multiple scattering and vector diffraction in near-field optical disks are not included in this study, and some behaviors shown in experiments may relate to those effects.

6. Conclusion

In this research, the super-resolution mechanism of near-field optical disks with random nanostructures is studied by Fourier optics and angular spectrum approach. We offer a general explanation about how the evanescent signals of recording marks are converted to detectable propagating signals through random nanostructure. The statistical properties of random nanostructures are used to obtain the general characteristic of near-field optical disks. Although the intensity differences are lost in the average process when the size of the recording mark approaches zero, the RMS of both intensity differences and readout waveforms persist for subwavelength record marks, which can be picked up via the differential signal process in the optical recording devices. Analytical results with dilute approximation show that the readout contrast signal is the combination of the averaged transmittance of random structure and light through an opaque thin film with an aperture. The resolution capability is determined by the size of random nanostructures. The numerical results agree with the analytic results in dilute approximation. A realistic model of near-field optical disk is also simulated and the numerical results are consistent with previous experimental measurements of near-field optical disks. The readout waveforms of a single-frequency pattern and a two-frequency pattern have also been simulated and they further demonstrate the super-resolution capability of near-field optical disks.

Though the complex phenomena of realistic optical disks are not fully included in this model, this theory describes a clear picture about how the random nanostructures in high-density near-field optical disks enhance the optical resolution. It provides an approach to explore optics in subwavelength regime with random nanostructures and lays a foundation for high-density near-field optical storage.