1. Introduction

As the demand for data storage capacity continually grows, data storage technologies are being driven to higher area densities. The maximum achievable area density with conventional optical recording techniques is determined by the diffraction limit. Solid immersion microscopy, a technique similar to oil immersion microscopy, extends the diffraction limit by filling the object space with a high-refractive-index material. Since Mansfield and Kino introduced a solid immersion lens (SIL) in optical microscopes [1], this technology has been developed for high-density data storage [2–5], photolithography [6], microscopy [7–9], and other applications [10–12].

A SIL microscopy system generally consists of a high-NA (numerical aperture) lens and a hemisphere or supersphere SIL [1–9]. In addition, some non-sphere solid immersion mirrors are also designed and demonstrated [13–16]. The vector diffraction theory established by Richards and Wolf [17] can be used to describe the focal field distribution in a high-NA system [18]. The focal spot is asymmetrical when a linearly polarized light is incident on a high-NA SIL system. By using amplitude and phase masks, this asymmetry can be moderately rectified and the so-called super-resolution effect is generated [19,20]. In addition, some amplitude and phase masks are also used to overcome the disadvantage of the short focal depth in near-field SIL recording [21–23].

On the other hand, a focused radially polarized beam can overcome the disadvantage of the asymmetry of a linearly polarized beam. The intensity distributions of cylindrical vector beams that have radial or azimuthal polarization in the focal region of a high-NA lens were calculated by Youngworth and Brown [24] by using the vector diffraction theory [17,18]. When a radially polarized beam illuminates a high-NA lens, a tighter and symmetrical spot was obtained [25,26]. Helseth studied the optical field distribution for the radially polarized beam focused to a solid immersion medium [27]. Recently, Kozawa and Sato calculated the intensity distribution near the focal point of a high-NA lens illuminated by higher-order radially polarized laser beams [28,29]. In this paper, we use high-order radially polarized beams to illuminate a high-NA SIL system and calculate the intensity distributions in a recording sample by using the vector diffraction theory. Numerical results show that some high-order radially polarized beams can improve the recording density and the focal depth of the near-field SIL recording system compared with conventional radially polarized beam focusing.

2. Theory

A radially polarized beam can be described as a superposition of a TEM₀₁ Gauss-Laguerre mode with a polarization direction parallel to the x axis and a TEM₁₀ mode with a polarization direction parallel to the y axis [25]. The radially polarized beam frequently referred to as a single-ring or doughnut-mode beam, R-TEM₀₁ ^* has completely cylindrical symmetry in both intensity and polarization distributions. Similarly, higher-order modes with multi-ring beam patterns that are expressed as R-TEM_p1 ^* are also rotationally symmetric and have p+1 rings (where p is the radial mode number). The lowest-order R-TEM₀₁ ^* mode can be generated from a laser cavity [30], and higher-order radially polarized R-TEM_p1 ^* modes can be generally produced by using a polarization-selective optical element or mechanism [31,32]. When a radially polarized beam is tightly focused by an aplanatic lens, the electric field in the focal region is divided into two components, namely the radial (transverse) field E _ρ and the axial (longitudinal) field E _z. No azimuthal (tangential) component occurs through the focusing as long as an incident beam is completely radially polarized [24]. For the SIL optical recording system shown in Fig. 1, these components are written, in the cylindrical coordinate centered at the geometrical focus O, as

where θ _1max = arcsin(NA) is the convergence angle related to the numerical aperture of the lens. θ ₂ = arcsin(n ₁ sinθ ₁/n ₂) and θ ₃ = arcsin(n ₂ sinθ ₂/n ₃) are the refractive angles in the gap and in the recording sample, respectively. h is the thickness of the gap between the sample and the SIL, which represents the flight height of SIL optical head for optical storage. k_i = 2π n_i/λ (i = 1, 2, 3) is the wave number in the dielectric I, and n_i is the refractive index of dielectric material for three regions, as shown in Fig. 1. J _n is the nth Bessel function of the first kind. η is the energy factor defined as the ratio of the energy of light passing through the pupil and the total energy in the pupil plane. l ₀(θ ₁) is the relative amplitude of the electric field in front of a pupil. t^p _sys is the effective transmittance complex amplitude coefficient of the system, which can be written as

In Eq. (2), the multiple reflection effect within the gap has been considered but the multiple reflection within the SIL has been neglected [5]. r^p_lm and t^m_lp are the Fresnel reflection and transmission coefficients for a single interface separating regions l and m, and δ _gap = 4πn ₂ hcosθ ₂/λ.

We assume that the amplitude distribution of the incident R-TEM_p1 ^* beams takes the form

where L ¹ _p is the generalized Laguerre polynomial and β ₀ is the size parameter of the incident beam determined by the ratio of the pupil radius to the incident beam waist in front of the objective. β ₀ should be larger than 1 because the outer ring of the R-TEM₁₁ ^* beam will be completely blocked by the pupil if β ₀ < 1.

 

Fig. 1. Schematic plot of an optical recording system with a solid immersion lens.

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3. Numerical aperture dependence on the focal spot pattern for several R-TEM_p1^* modes

Figure 2 shows the intensity profiles at the inner surface of the recording layer for various radial mode numbers p. The calculation parameters are chosen as n ₃=1.656-i0.004 (which represents a mildly absorbing photoresist), n ₂ =1 (air), n ₃=1.845 (LaSF9 glass), and the NA of the lens is assumed to be 1. The value of the incident beam width parameter β ₀ for each mode beam is determined so that 90% of the incoming energy in front of the objective is focused. Each intensity profile is normalized to the maximum value of the total intensity (solid curves) for the R-TEM₀₁ ^* beam in Fig. 2(a). It is seen from the figure that in any mode beam focusing, only the longitudinal component (dashed curves) emerges on the beam axis, whereas no radial component (dotted curves) is seen on the axis. When p is small, for example p = 0, 1, 2, the intensity (|E _z|²) of the longitudinal component is much larger than the intensity (|E _ρ|²) of the radial component, and thus the total intensity profile is basically determined by the longitudinal component. As the radial mode number p increases, the peak value of the total intensity (|E _total|²) decreases rapidly and the radial component broadens outwards. In addition, the intensity of the radial intensity relative to the intensity of the longitudinal component increases with the increase of p, even larger than the intensity of the longitudinal component, as shown in Figs. 2(e) and 2(f).

 

Fig. 2. Intensity profiles in the inner surface of the sample for the focusing of (a) R-TEM₀₁ ^*, (b) R-TEM₁₁ ^*, (c) R-TEM₂₁ ^*, (d) R-TEM₃₁ ^*, (e) R-TEM₄₁ ^*, and (f) R-TEM₅₁ ^* modes when h=50 nm and NA=1. The total intensity (solid curves), the radial component (dotted curves), and the longitudinal component (dashed curves) are drawn in each figure. Every profile is normalized to the peak intensity in (a).

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To characterize the pattern of the recording spot in optical storage, we first define four parameters. The first parameter is the FWHM (full width at half-maximum) of the focal spot, which gives a measure of the resolution. The second parameter is the peak intensity S of the spot, which gives a measure of the brightness of the spot. The third parameter is the maximum side-lobe intensity M, measured with respect to the focal spot. The final parameter is the focal depth D, which is defined as the distance between the interface and axial position where the value of intensity is half of the intensity at the interface. For optical storage, a certain level of the peak intensity is required for proper recording. Additional power can be used to increase the energy density of the optical stylus and thus compensate for the low level of the peak intensity. Special attention must be paid to the side lobes to prevent unwanted pits from being recorded and to the focal depth when trying to obtain small FWHM. The acceptable level of side-lobe intensity is dependent on the recording medium, but in general it is preferable to keep it as low as possible. If we assume that the acceptable side-lobe intensity is M ≤ 0.3 in optical storage, a usable field is presented only by the R-TEM₀₁ ^*, R-TEM₁₁ ^*, and R-TEM₂₁ ^* beams in Fig. 2. The field distribution presented by the R-TEM₃₁ ^*, R-TEM₄₁ ^*, and R-TEM₅₁ ^* beams may be useful for confocal microscopy, where contribution from strong side lobes can be eliminated by using point-by-point detection [33].

Figure 3 plots the longitudinal and total intensity distribution for each mode. The peak intensities are normalized to 1. For the longitudinal component shown in Fig. 3(a), it is clearly seen that the larger radial mode number p is, the sharper the center peak of the longitudinal component is. For the total intensity, however, the total intensity distribution is different from the longitudinal component. When p increases from 0 to 2, the center peak of the total intensity becomes sharper, and then as p increases, the center peak of the total intensity becomes wider, even wider than that of R-TEM_01^* beam focusing, as shown in Fig. 3(b). For intensity distributions of the longitudinal component, the FWHM values of the center spot are obtained to be 0.2767λ, 0.2336λ, 0.2211λ, 0.2133λ, 0.2074λ, and 0.2048λ, for radial mode number p from 0 to 5, respectively. For the total intensity distributions shown in Fig. 3(b), the corresponding FWHMs are 0.3399λ, 0.2460λ, 0.2456λ, 0.2706λ, 0.3563λ, and 0.5174λ. It is noted that the FWHM 0.2456λ of the total intensity for the incident R-TEM₂₁ ^* beam is a little smaller than 0.2460λ for the incident R-TEM₁₁ ^* beam. However, it is found that the peak intensity of the focal spot for the incident R-TEM₁₁ ^* beam is about 2.5 times larger and the side-lobe intensity 0.8 times smaller than that for the incident R-TEM₂₁ ^* beam. Based on comprehensive consideration of the side-lobe intensity, peak intensity, and FWHM of the focal spot, a conclusion can be drawn that an R-TEM11* mode beam may provide a sharper, comparatively strong peak-intensity spot size with a comparatively lower side-lobe intensity in high-NA SIL recording. Note that the recording spot size (FWHM) of an incident R-TEM₁₁ ^* beam is reduced by 27.6% in total intensity compared with an incident R-TEM₀₁ ^* beam.

 

Fig. 3. Normalized intensity profiles of (a) longitudinal component and (b) total intensity in the inner surface of the sample for different radial modes when h=50nm and NA=1.

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Figure 4 shows the total intensity distributions in the ρ-z meridional plane for the incident R-TEM₀₁ ^* (a) and R-TEM₁₁ ^* (b) beams. From Fig. 4 it can be clearly seen that as the depth going into the sample increases the intensity of the recording spot decreases rapidly for the incident R-TEM₀₁ ^* beam, but for the incident R-TEM₁₁ ^* beam the intensity of the recording spot and its FWHM are kept almost unchanged in quite a large range. The focal depth of the system for the incident R-TEM₁₁ ^* beam is 2.64 times larger than that for the incident R-TEM₀₁ ^* beam, which is listed in Table 1. These long focal-depth and unaltered FWHM characteristics of R-TEM₁₁ ^* beam focusing will be very useful for subsurface high-density optical storage and photolithography.

 

Fig. 4. Normalized total intensity distributions in the ρ-z meridional plane for the incident R-TEM₀₁ ^* (a) and R-TEM₁₁ ^* (b) beams when h=50nm. and NA=1.

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In previous calculations the NA of the lens is assumed to be 1 as an extreme case. Table 1 gives the spot pattern (described by the parameters of the FWHM, focal depth D, peak intensity S, and side-lobe intensity M) for two incident beams of R-TEM₀₁ ^* and R-TEM₁₁ ^* when the NA varies from 0.6 to 1.0. All of the peak intensities are normalized to the peak intensity for R-TEM₀₁ ^* beam focusing when NA=1. Note that the parameters of the spot pattern for R-TEM₁₁ ^* are not shown for the two lower NAs and are indicated by “-” because the side-lobe intensity M of the focal spot becomes larger than 0.5; therefore, the effective recording pits are no longer formed in the real optical data storage. Table 1 shows clearly that the FWHM value of the focal spot decreases and the peak intensity increases as the NA increases. The FWHM for R-TEM₁₁ ^* beam focusing is smaller than that for R-TEM01* beam focusing, but the peak intensity is smaller and the side-lobe intensity is larger. It is of interest to note that the side-lobe intensity for R-TEM₁₁ ^* beam focusing decreases greatly as NA increases.

Table 1. Pattern of the Focal Spot for High-NA SIL Focusing at h=50nm

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4. Beam width dependence for R-TEM₁₁^* mode beam focusing

In previous calculations the incident beam width parameter β ₀ for each mode was determined to focus 90% of the whole energy in front of the objective. However, the intensity distribution in the recording sample will vary with β ₀. Figures 5 (a)–(d) show the intensity profiles in the inner surface of the sample for different beam width parameters of β ₀= 2.5, 2.0, 1.3, and 0.5, respectively, when an R-TEM₁₁ ^* mode is focused. Every intensity profile is normalized to the maximum value of the total intensity for β ₀=2.0. It is shown that for any beam width parameter β ₀, the longitudinal component (dashed curves) maintains the maximum value on the beam axis, especially when β ₀= 1.3, which is not same as that of a high-NA lens focusing in an homogeneous medium [29], whereas the radial component (dotted curves) maintains the null on the axis and an annular intensity distribution. The peak intensity and FWHM of the recording spot varies drastically with β ₀. When β ₀=2.0 [Fig. 5(b)] the peak intensity is largest and the FWHM is smallest while when β ₀=1.3 [Fig. 5(c)] the peak intensity is smallest and the FWHM is largest. Although the FWHMs in Figs. 5(a) and 5(d) are also quite small, the peak intensity is smaller than in Fig. 5(b). Thus, Fig. 5 suggests that a strong peak intensity and a sharper recording spot are satisfactorily formed at β ₀=2.0 for R-TEM₁₁ ^* mode.

Figure 6 shows the variation of the peak intensity and FWHM of recording spot with the beam width parameter β ₀ where the solid and dashed curves are the cases of the R-TEM₁₁ ^* and R-TEM₀₁ ^* beams, respectively. From Fig. 6 it can be seen that the peak intensity and FWHM of the focused spot for the R-TEM₁₁ ^* beam vary drastically with the beam width parameter β ₀, in contrast to that of the R-TEM₀₁ ^* beam. Our explanation for these complex behaviors is as follows. It is well known that the R-TEM₁₁ ^* mode has two rings and the phase difference between the rings is π. When the beam width is very large (β ₀ is very small), the outer ring is completely obstructed by the pupil, and a part of light in the inner ring is also obstructed. Consequently it is easy to understand that the intensity of the focused spot is very small. Small FWHM is due to the evanescent wave. In this case of a very small β ₀, the radius of the annular light beam is quite large, and thus the incident angle of all lights is larger than the critical angle at the SIL-air interface; therefore, the transmitted light from the SIL has only the evanescent component. The focused evanescent wave can generate a small spot size. As β ₀ increases in the range of _{β0 < 0.75}, the more lights are incident on the lens, but the radius of the annular beam decreases so that the peak intensity and FWHM of the spot size increases. In the range of _{075 < β0 < 1}, the lights in the inner ring of the R-TEM₁₁ ^* mode almost all shift into the pupil. They are divided into two parts: one with large angle of incidence and the other with small angle of incidence. Thus the transmitted wave consists of two components, the evanescent wave that originates from the lights with a large angle of incidence and the propagating wave that originates from the lights with a small angle of incidence. It is well known that the SIL changes not only the magnitude of the incident light but also the phase of the incident light when the incident angle is larger than the critical angle. The destructive interference between evanescent and propagating waves results in the decrease of the peak intensity. The increase of the FWHM is due to the reduction of the radius of the annular beam. When _{β0 > 1}, the light in the outer part of the R-TEM₁₁ ^* mode starts to enter into the pupil. The intrinsic π phase difference between the inner and outer rings of the R-TEM₁₁ ^* mode aggravates the destructive interference between the evanescent wave and the propagating wave, and consequently the peak intensity decreases rapidly in the range of _{1 < β 0 < 1.3} until the minimum peak intensity emerges at _{β0 = 1.3}. Because the small quantity of lights in the outer ring passes through the pupil, the FWHM rapidly enlarges and the spot size is mainly determined by the propagating wave close to the optical axis. In the range of _{1.3 < β0 < 1.55}, the peak intensity starts to increase and the FWHM starts to decrease, which both result from the increasing evanescent wave. When _{1.55 < β0 < 2.5}, the lights in the outer ring of the R-TEM₁₁ ^* mode almost completely shift into the pupil and the evanescent wave dominates; thus, the peak intensity is very large but the FWHM is very small. When β ₀ > 2.5, all light shifts to the vicinity of the optical axis and there is no more evanescent wave. Consequently the peak intensity decreases rapidly and the FWHM increases rapidly as β ₀ increases.

 

Fig. 5. Calculated intensity profiles in the inner surface of the sample for different beam width parameters of β ₀= (a) 2.5, (b) 2.0, (c) 1.3, and (d) 0.5, where h=50nm and NA=1, and the incident beam is the R-TEM₁₁ ^* mode. The total intensity (solid curve), the longitudinal component (dashed curve), and the radial component (dotted curve) are represented in each figure. Every intensity profile is normalized to the maximum of the total intensity in (b).

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Fig. 6. (a) Peak intensity (a) and FWHM (b) of the focused spot in the inner surface of the sample plotted against the beam width parameter β ₀ for R-TEM₁₁ ^* (solid curve) and R-TEM₀₁ ^* (dashed curve) mode focusing. The FWHM value is in units of wavelength.

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5. Conclusion

In this research the intensity distributions of the recording optical field in an optical recording system with a SIL have been calculated for different incident radially polarized beams based on the vector diffraction theory. We find that two higher-order radially polarized modes, R-TEM₁₁ ^* and R-TEM₂₁ ^*, are useful to the near-field optical data storage in a high-NA SIL system. However, more higher-order modes, such as R-TEM₃₁ ^*, R-TEM₄₁ ^*, and R-TEM₅₁ ^*, are not useful since the side-lobe intensity rapidly increases as the radial mode number increases in the SIL system. Our calculation results show that the FWHM of the recording spot for R-TEM₁₁ ^* beam focusing can be reduced 27.6% and its focal depth can be lengthened 2.64 times compared with R-TEM₀₁ ^* beam focusing. At the same time, the FWHM value is kept almost unchanged in quite a large distance from the air-sample interface. The effects of the beam width parameter on the FWHM and peak intensity are also presented and explained simply by using the interference theory.