We analyze the effects of optical variables, such as illumination state, focal position variation, near-field air-gap height, and refractive index mismatch, in immersion lens-based near-field optics on the resultant field propagation characteristics, including spot size, focal depth, and aberrations. First, to investigate the general behaviors of various incident polarization states, focused fields near the focal planes in simple two- or three-layered media structures are calculated under considerations of refractive index mismatch, geometric focal position variations, and air-gap height in a multi-layered medium. Notably, for solid immersion near-field optics, although purely TM polarized illumination generates a stronger and 15% smaller beam spot size in the focal region than in the case of circularly polarized incident light, the intensity of the focused field decreases sharply from the interface between air and the third medium. For the same optical configurations, we show that changes in geometric focal position to the recording or detecting medium increases focal depth. Finally, through focused field analysis on a ROM (read-only memory) and a RW (rewritable) medium, compound effects of considered variables are discussed. The resultant field propagation behaviors described in this study may be applicable to the design of either highly efficient reflection or transmission near-field optics for immersion lens based information storage, microscopy and lithographic devices.
© 2008 Optical Society of America
Since the real-time optical microscope with a solid immersion lens (SIL) was invented by Mansfield and Kino, studies in near-field (NF) optics applications based on immersion optics in near-field microscopy, spectroscopy, optical lithography, and optical and magnetic information devices have been actively carried out . The high resolution, optical and mechanical stability, measurement speed, and compactness (by virtue of the actuated near-field air-gap servo) of immersion lens-based optics renders it practically useful in the fields of optical data storage devices and microscopy [2–3]. Chen et al. proposed compact high-resolution microscopy that can be used to measure binary grating structures with pitches below 200 nm with a solid immersion lens (SIL)-based numerical aperture (NA) of 1.5 . Shinoda et al. demonstrated the feasibility of 150 Gbyte data storage capacity on a single recording layer of a CD-sized disk . For research on high-resolution optical lithographic systems, liquid immersion and SIL-based NF optics with ArF lasers have been developed [6, 7]. Recently, Smith et al. reported fabrication results of 25-nm half track pitch with solid immersion lithography at a wavelength of 193 nm . In the above-mentioned research areas, common technical issues are achieving high resolution, deep focal depth, and either high transmission or reflection efficiencies. Therefore, analyzing the individual or compound effects of many optical variables on the resultant highly focused field in a multi-layered medium structure becomes critical.
Vector field theory based on the work of Richards and Wolf can calculate the beam spot distribution near the focal plane in high NA optical systems . Ichimura et al. calculated a point spread function for the electromagnetic field on the top surface of the medium given multiple beam interference caused by SIL, air-gap, and a medium interface . Using this formalism, extraordinary spherical aberrations and astigmatism were predicted due to the existence and size of the air-gap, which would lead to phase differences between the transverse electric and magnetic polarized components [10, 11]. Török et al. focused on the phase difference and derived an electromagnetic diffraction formula for the planar interface between two materials of mismatched refractive indices [12–14]. Several studies calculated the electromagnetic field distribution in the case of multiple beam interference between multiple thin layers [15–17]. In the regime of higher resolution optics, studies on the focusing characteristics of cylindrical vector beams in immersion optics-based near-field microscopy indicated that the incidence of radially polarized light on aplanatic optics generates stronger and more confined longitudinal electric fields than transverse fields from circularly or linearly polarized light [18, 19]. Previous studies examined the vector field calculation formalism for diffracted electromagnetic fields in thin film stacks, but did not include the individual effects of the optical variables, such as mismatch in refractive indices, geometric focal position shift, and various types of illumination, and their compound effects.
In this study, we focus primarily on analyzing the effects of optical variables on the resultant optical characteristics, such as transverse and longitudinal intensity distribution, aberration performance and focal depth variation. Optical variables considered include various states of incident polarizations, mismatches in refractive indices in a thin film stack, changes in geometric focal position, and near-field air-gap variation, which is indispensable for SIL-based near-field optics. Zhang et al. recently completed a similar study, but they considered only the SIL and the single consecutive material directly attached to the bottom surface of the SIL . To calculate the electric field distribution within multi-layered media, we use the efficient calculation scheme described by van de Nes et al. . Finally, in this paper, compound effects of the optical variables on the focused field by a SIL-based near-field recording optics in a Si-ROM (Silicon Read Only Memory) and a RW (Rewritable) medium will be presented and discussed.
2. Fundamental theory
In this paper, we consider the aplanatic imaging optics system shown in Fig. 1 . The Richards and Wolf diffraction integral derived from the Debye integral for the field structure of E Img (E x, E y, E z) at the Cartesian coordinates (x, y, z) near the Gaussian focus is given by:
where a(k x, k y) is the amplitude of a plane-wave on the exit pupil described by the propagation vector k=(k x, k y, k z), r p is the position vector on the focal plane, and numerical integration is carried out over the exit pupil Ω for the designed NA of the optics . For an aplanatic system that satisfies the sine condition, the geometric focal length is defined as f and a(k x, k y)=f(k z/k 1)1/2·E(k x,k y). E(k x, k y) is the electric field distribution on the exit pupil. For a multilayered medium consisting of N media, the vector rotation of the rays at each boundary of N media and the multiple beam interference caused by the forward and backward traveling waves between each consecutive set of boundaries must be considered. The diffraction integral on a plane inside the stratified media structure can be rewritten in cylindrical coordinates:
where r p, ϕ p, and z p are cylindrical coordinates on the image plane inside the stratified media, and k zi and k z 1 are the longitudinal components of propagation vector in i th medium and the first medium, respectively, k 0 and k 1 are propagation vectors in free space and the first medium, respectively, and k r is the radial component of propagation vector. In Eq. (2), the matrices A i ± are the solved forms of the integral over the azimuthal entrance pupil angle k ϕ with respect to the propagation matrix, which describes the vector rotation and transmittance and reflection coefficients in each medium at entrance pupil. A i ± for incident linear, circular, radial, and azimuthal polarizations are defined, respectively, as:
where J N=i (N) J N(rk r)cos(N ϕ p), J̄N=i (N) J N(rk r)sin(N ϕ p), and J N(rk r) is the N th Bessel function of the first kind. The derivation of the effective transmission and reflection coefficients g i n ± is given in .
To investigate spot quality in the N th layer of the medium, we measure beam spot size and analyze aberration performance using the Zernike polynomial. Each diffracted and focusing wave on the image plane can be expressed as:
where ρ is the normalized exit pupil coordinate, defined as ρ=k r/(NA·k 0), and E Img(ρ, k ϕ) and W(ρ, k ϕ) are amplitude and phase of each focused and diffracted wave, respectively. Therefore, Zernike coefficients representing each aberration can be defined as:
where Z n,m(ρ, k ϕ) is the Zernike circle polynomial and the Kronecker δ m 0 is 1 or 0 when m=0 or m≠0, respectively.
3.1 Effects of sequential differences of refractive indices in media
In this section, to understand the fundamental effects of mismatch in refractive indices between consecutive media on the resultant electric field, a two-layered medium is considered. We consider two configurations to compare the propagation behaviors of each transverse magnetic (TM) and transverse electric (TE) wave in far- and near-field optics with NA enhancement by a solid immersion lens (SIL). The first configuration is composed of a system with NA of 0.911, SIL with refractive index 2.086, and a consecutive second medium with a lower refractive index (n 1>n 2) placed behind the geometrical focus. In the first configuration, the NA of the system is enhanced to 1.9 by a factor of the refractive index of the SIL. The second configuration consists of a system focused by a lens with an NA=0.911 in air and a glass substrate with a higher refractive index (n 1<n 2) placed behind the geometrical focus. For each simulation model, we vary the refractive index of the second layer and the state of the incident polarization. To observe the propagation characteristics of tightly focused fields, we assume the interface is located at the geometric focal position. With this assumption, the focal position shift and its related auxiliary aberrations caused by interface between the first and second layers can be disregarded. Generally, by applying radial and azimuthal polarizations simultaneously, the propagation characteristics of the TM and TE polarized waves can be investigated individually. The mixed natures of TM and TE polarized waves can also be studied in cases of incident linear and circular polarization.
As reported in previous studies and as shown in Figs. 3 and 4, radial (TM) polarization generates radial and longitudinal field components, E r, and E z, and azimuthal (TE) polarization generates only an azimuthal field component, E ϕ, in the focal region because TE polarization is associated only with azimuthal pupil angle, while TM polarization depends on radial and longitudinal directions [14, 17, 18]. Differences in refractive indices cause noticeable propagation behavior differences in purely TE and TM polarized waves. As the radial and azimuthal components of the propagation vector provoke destructive interference along the optical axis, the peak amplitudes of both fields are smaller than the amplitude of the longitudinal component, which constructively interferes along the optical axis, as the longitudinal components of the propagation vectors lie in the same direction. Also, differences in the field strength in the second medium between E z and E r, both generated by TM polarized incidence, can be deduced from the boundary conditions of Maxwell’s equations, which state that the transverse electric field is continuous and the longitudinal field is discontinuous at the medium transition. The longitudinal field component in the second medium is multiplied by a factor ε 1/ε 2, where ε 1 and ε 2 are dielectric constants in the first and the second medium, respectively. Therefore, in the first configuration, the field strength of the longitudinal field component is even higher than that of the radial field just inside the second medium, while in the second configuration, the field strength of the longitudinal field is smaller than that of the radial field inside the second medium. In the first configuration, the tightly focused spot of the longitudinal field, E z, penetrates into the second medium as a mixture of propagating and evanescent fields. Therefore, the transmitted peak intensities of each focused field just inside the second medium are even higher than those of the second configuration shown in Fig. 4. However, along the positive direction of the propagation axis, as the transmitted amplitude of the tightly focused E z field decreases due to exponential decay of the evanescent field as shown in Fig. 3(c), the transversal beam spot size grows significantly larger along the z-axis from the medium transition, and consequently, diverges abruptly from the end of the evanescent field penetration depth.
Figure 5 shows the electric field distribution given incident circular polarization. In contrast to the case of purely TM-polarized illumination, the longitudinal field component no longer generates an axially confined beam spot profile because the oscillating directions of the longitudinal components of the propagation vectors on the entrance pupil are oriented to provoke destructive interference along the optical axis. Therefore, in the case of illumination by circularly polarized light, the transverse components, E x and E y, of the focused field generate an axially focused beam spot near the focal region. Figure 6 shows the axial intensity distributions along the optical axis for two configurations. In both configurations, the focused field generated by radial polarization provokes a discontinuous intensity profile at the medium transition, unlike the linearly and circularly polarized cases. This occurs because the longitudinal component of the axially focused field, E z, in the radially polarized case scales with the ratio of the dielectric constant, ε 1/ε 2, over the transition. Otherwise, as the transverse components, E x and E y for linear and circular polarizations, are continuous, these field components can generate an axially focused field with higher transmittance in the second configuration than in the radial polarization case. In addition, as shown in Fig. 6(a), it can be confirmed that evanescent energy penetrates into the second medium by half the wavelength.
Figure 7 shows a comparison of transverse intensity profiles near the geometric focus for the first and second configurations, and Table 1 compares the changes in beam spot size along the optical axis for the four different models. In the case of linear polarization, as the transverse spot does not yield circular symmetry, the spot profile is represented separately for the 0° and 90° planes at several focal positions. In addition, azimuthal polarization is not considered, as it does not generate a focused beam spot along the optical axis. In both configurations, although the incidence of linearly polarized light induces the smallest spot size on the 90° plane, the overall spot shape does not have rotational symmetry. Therefore, it becomes much broader on the 0° plane. Specifically, spot size differs greatly on both planes in cases using the first configuration, models 1 and 2. In the first configuration, all the beam spot sizes of model 2 on the plane z=+λ/5 are smaller than those of model 1. This is natural because a smaller mismatch in refractive indices lengthens the penetration depth of the evanescent energy to the second layer. As noted previously, radial polarization generates the smallest beam spot size in a single medium. This tendency is shown in the first configuration, models 1 and 2, as the longitudinal field component dominates the radial field component over medium transitions. However, in the second configuration, as the longitudinal field diminishes abruptly from the interface, the transverse field, E r, dominates in the second medium. As a result, peak intensity is no longer generated along the optical axis.
3.2 Air-gap-dependent sensitivity of various polarizations
With SIL-based optics, proper selection of SIL material with high refractive index can achieve extremely high resolution. In high NA solid immersion optics, as it is impossible to maintain optical contact between the SIL and a consecutive medium or sample under test, there is inevitably an air-gap between the SIL and the next medium. In this section, we discuss the effects of air-gap on transmitted and reflected electric fields in immersion optics. The simulated system is composed of far-field optics with NA of 0.911 and a near-field structure consisting of a spherical SIL, λ/8-thick air-gap, and a consecutive medium with refractive indices of 2.086, 1.0, and 2.086, respectively.
Incident pure TM-polarized light generates extremely small spots in the region of the first, and inside the second, medium near the first interface. Specifically, radial polarization forms the smallest beam spot with circular symmetry inside the air-gap. However, as the intensity of the dominant E z field component is discontinuous at the medium transition from air to the third medium, the extremely high optical resolution achieved inside the air-gap by high NA optics and the radial polarization focusing characteristics cannot be obtained in the third medium. In contrast, circular and linear polarizations generate highly focused beam spots on the bottom surface of the SIL and inside the third medium. As shown in Table 2, the FWHM beam spot sizes on the plane z=+λ/5 are almost the same as those on the top surface of the third medium. Therefore, the mixed polarizations are suitable for SIL-based near-field optics applications in microscopy, recording optics, and lithographic optics. For liquid immersion optics applications, radial polarization can yield better transmission and higher resolution characteristics when the specimen has a lower refractive index than the liquid material.
To investigate the effect of air-gap thickness variation on beam spot quality, non-zero primary aberrations were calculated as shown in Tables 3 and 4. As a highly focused spot is generated inside the third medium with circular and linear polarization, only those incidence conditions were considered in these calculations. We also studied the effects of refractive index mismatch between the SIL and the third medium by considering the SIL material and the Si-disk as the third medium. The results shown in Tables 3 and 4 indicate that incident linear polarization and a mismatch in refractive index provoke higher air-gap-induced aberrations than configurations using circular polarization and an index-matched third medium. Especially in information readout optics, in which a Si-disk is used for the third medium and the system is illuminated by linearly polarized light, astigmatism becomes so severe that it is difficult to obtain diffraction-limited spot quality on the image plane with an air-gap thickness larger than λ/8. Aberration induced by the near-field air-gap is a critical problem in the application of solid immersion-based near-field optics in several fields, such as microscopy, information storage and lithographic optics, as it limits the resolution of the optics. However, simulation results also indicate that such astigmatisms disappear when using circularly polarized light. The rotationally symmetric aberrations that exist independent of the state of polarization can be compensated for by properly considering the amount of air-gap-induced aberration in pre-focusing lens design.
3.3 Focal position variation
In this section, propagation characteristics of highly focused fields inside the last medium given varying geometric focal position will be investigated. For the simple 4-layered recording medium, a stack of SIL (2.086), air-gap (1.0), and index-matched layer (2.086), focal depth variations, spot size, and longitudinal intensity are primarily studied. Separation of each layer transition, including air-gap distance, is fixed to λ/16 for sufficient coupling efficiency of the evanescent field. When the aberration due to mismatch in refractive index between the third and fourth layers is disregarded, the geometric focal plane can be assumed to be the plane z=0.
In the case of illumination by radially polarized light, the axially confined field, trapped almost inside the air-gap, as shown in Fig. 8(a) in the previous section, shows insufficient transmission to the third and fourth layers and broadens the spot size. Shifting the interface location in the negative z-direction with constant air-gap thickness, radially polarized illumination generates an axially confined spot inside the third medium due to the longitudinal field’s gradually increasing transmittance. Figure 10 shows the overall electric field distribution and beam spot profile on the geometric focal plane for the d 1=-λ and d 2=-λ (1-1/16) configuration. Although longitudinal field transmittance to the third layer is greatly enhanced by interface location adjustment, resolving higher spatial frequencies on the focal plane is impossible because the radial field component, which generates the “doughnut-like” spot, has a strong effect on the resultant field inside the third medium. In contrast, given illumination by circularly polarized light, shifting the interface in the negative z-direction lengthens the focal depth. Overall electric field distributions for the two cases of shifting transition layers are compared in Fig. 11.
As the first interface moves in the negative z-direction, focal depth (which is mainly induced by transverse fields) inside the third layer gradually increases up to the specific focal length for the case in which the entire focal region is filled with a single SIL material. As the auxiliary aberrations in SIL-based NFR optics are purely induced by air-gap distance, transverse and longitudinal intensity distributions for both cases are exactly identical along the positive longitudinal path from the plane z=0. On the z=-λ/4 plane, although the beam spot size becomes broader than that on the z=0 plane due to the effect of the longitudinal field component, the FWHM spot size is still equal to that on the plane z=0. In addition, the RMS aberration coefficient is lower than 70 m λ rms in the focal region from -λ/3 to λ/8. Therefore, in this region, diffraction-limited spot quality is satisfied. Notice that expansion of the focal depth can be achieved by simply shifting layer transitions. Practically, this shifting interface denotes the truncation of the SIL from the exact geometries of hemisphere and super-hemisphere used in SIL-based optics applications, such as microscopy and optical storage systems. For example, given an optical storage device, if a thin layer the refractive index of which is the same as the SIL is coated on the information layer, the focal depth of the optics can be increased directly and data protection improved. This increase in focal depth reduces readout signal error, the so-called jitter of the readout signal, when the thin layered stack of the recording medium is coated with small inhomogeneities. In addition, measurement speeds of longitudinal scans in three-dimensional near-field confocal microscopy can be improved.
4. Applications of near-field optics
4.1 Imaging characteristics of near-field information storage optics
In this section, we first investigate the imaging characteristics of near-field optics for data storage applications inside read-only memory (ROM) and a rewritable (RW) medium. In Fig. 13, simulation models for Si-ROM and RW media are presented. Media models for the ROM and RW media shown in Fig. 12 have been used in practical applications as readout and recording media structures by Zijp et al. and Shinoda et al. [5, 10], respectively. For the RW media configuration, we compared field propagation characteristics of circularly polarized illumination in conventional RW media with field propagation characteristics in a proposed media stack with a thick SiN layer of 200 nm, equal to λ/2. The purpose of this comparison is to examine the effects of focal position variation. In the first model, with ROM, the reflected electric field from the top surface of the Si disk is used as the readout signal. Therefore, electrical energy transferred to the first surface of the Si disk is more important than the field focused on the bottom surface of the SIL. As shown in Figs. 13 and 14, for the high reflectance angular spectrum region with NA>1, the interference phenomenon between the forward and backward propagating waves brightens noticeably. In these plots, values are clipped at 40% of the maximum value to increase the visibility of the field structure in the low amplitude regions.
As shown in Figs. 13 and 14(a), for the best focal position just in front of the first medium transition, incident radial polarization generates a strongly focused spot as much as 100 nm of FWHM beam spot, with side lobes below 10%. However, at the interface between the air layer and the Si disk, as the amplitude of its longitudinal field is discontinuous and yields almost zero transmittance, the high resolution of the optics cannot be preserved even on the top surface Si-disk. This means that given radially polarized illumination, resolving bit data mastered on the top surface of the ROM with a high spatial frequency corresponding to the NA of the optics is impossible. In contrast to radial polarization behavior, even though linearly and circularly polarized illuminations generate “doughnut-like” spots inside the air-gap due to the strong effect of the longitudinal field E z, transverse fields E x and E y induce axially confined electric fields on and inside the Si-disk. Therefore, optics with circularly or linearly polarized illuminations can read-out bit data on the ROM at its highest resolution. Moreover, as can be seen in Fig. 13(a)–13(c), the intensities of the circularly and linearly polarized light inside the Si disk are much higher than those of radially polarized light along the optical axis.
In the second model, the RW medium, as the GST layer is used to induce phase change (PC) effects for a recording process, the recording density is determined by the spot size in this layer. Thus, the geometric focal plane is assumed to be positioned at precisely the middle of the GST layer. Although the geometric focal plane must be placed accurately to minimize rotationally symmetric aberrations caused by the multi-layered stack, following the study of Zijp et al., this effect is disregarded in this analysis, as we consider relatively small mismatches of refractive indices among SiN, ZnS-SiO2, and GeSbTe layers . Figure 15 shows electric field distributions near the focal regions of two different models with top SiN layers of different thicknesses. For both cases, highly focused spots are generated inside the GST layer and abruptly dissipated in the Ag alloy reflective layer. Compared to the case with the 15-nm thick SiN layer, in which the brightest field is formed inside the air-gap, the RW medium with a thicker SiN layer (200 nm) forms the brightest field inside the RW medium with uniform longitudinal beam spot quality, as shown in Fig. 15(d). Even on the z=-λ/4 (-100 nm in the model) plane, the FWHM size of the spot is almost the same as that on the geometric focal plane, z=0, as shown in Fig. 15(c). Therefore, although the thick SiN layer is spin-coated with roughness ranging to several tens of nanometers, phase change effects on the GST layer can be ensured. From this result, we conclude that the effect of broadening focal depth inside the RW medium with a thicker SiN layer can alleviate difficulties in manufacturing high-density RW media as well as improve data protection.
Recently, due to successful near-field air-gap servo results, there has been increasing interest in applying near-field optics to the high resolutions characteristic of information storage, microscopy, and lithographic devices. In this study, we focused on the effects of various optical variables, such as mismatches in refractive indices and variations in focal position and air-gap distance, to the resultant field propagation characteristics. To understand qualitatively the propagation characteristics of pure TM and TE polarized lights and mixed illuminations, various illumination states were considered for simple models composed of the SIL and a consecutive medium with refractive indexes higher and lower than that of the SIL.
First, we found that the longitudinal component of the focused field generated by purely TM polarized illumination, which generates the smallest beam spot among the various illumination conditions in the focal region, yields the lowest transmittance to the next consecutive medium with higher refractive index than the first medium. This phenomenon results in poor imaging characteristics in the recording medium consecutively following the air layer. Therefore, we conclude that radially polarized illumination of SIL-based near-field optics is not suitable for near-field information recording or detection systems with extremely high imaging resolution, as there must be an air-gap for the flying optical head.
Second, from analysis of the beam spot characteristics on the top surface of the third medium, we found that spot size increases abruptly from an air-gap distance of λ/8, and that air-gap distance should be maintained below λ/8 to ensure diffraction-limited spot quality for near-field optics with the extremely high NA of 1.9. In addition, rotationally symmetric aberrations, such as spherical aberration and defocus, are not much affected by the refractive index of the third medium for cases of circular and linearly polarized illumination. In contrast to the rotationally symmetric aberrations, the refractive index of the third medium greatly affects astigmatism, which is only caused by linearly polarized illumination.
Third, as we examined the changes in geometric focus position in sections 3.3 and 4.1, we found that movement toward the inside of the third medium improves beam propagation characteristics inside the third medium. Practically, this configuration can be achieved with a truncated SIL and a top coat on the recording medium. Therefore, if the recording or detection layer is covered by a dielectric layer as thick as λ/2, we expect that recording and detection stability, as well as data protection from collision of the SIL with the medium, can be greatly improved.
Finally, we used the conventional Si-ROM disk for analysis of a specific storage medium structure. As predicted, although radially polarized illumination generates the smallest beam spot on the best focal plane, its dominant E z intensity decreases sharply at the medium transition between the second and third media. Consequently, an axially focused field is not achieved even on the top surface of the Si-disk. For the RW medium, which uses a phase change effect inside the GST layer, we investigated changes in geometric focal position. As the focal depth and beam spot size improve markedly near the plane of the GST, we expect to achieve better data protection and stable information recording or detection using a truncated SIL and top coat on information layers as thick as λ/2.
This work was supported by the Korea Science and Engineering Foundation(KOSEF) grant funded by the Korea government(MEST) (No. R17-2008-040-01001-0).
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