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Focal point shift in a solid immersion mirror of a high numerical-aperture is experimentally demonstrated with a scanning near-field optical microscope. The solid immersion mirror focuses light by a two-dimensional parabolic reflective surface integrated in a planar waveguide. The focal point shifts inward along the optical axis for metallized surface. The amount of shift from its geometrical node depends on the wavelength of the incident light and is determined to be roughly one-fifth of the wavelength.
© 2015 Optical Society of America
1. Introduction
Sharply focused beam of light has applications in lithography, confocal microscopy, optical data storage, and micro-particles trapping and manipulations. It is well known that the focused spot size brought by an objective is diffraction-limited and that the resolution of such optical systems in Rayleigh’s criterion is 0.61 λ/NA. Here λ denotes the wavelength of light and NA is the numerical aperture of the objective. Improvement in the resolution has been the aim of many research efforts in the last decades. In the 1950’s, F. Zernike invented the phase contrast microscope. Coherent spatial image filtering with a spiral phase element leads to a strong edge contrast enhancement of both amplitude and phase objects [1]. Amplitude and/or phase masks in the beam path of illumination narrow the central lobe of the focal intensity distribution [2, 3]. A radially polarized incident beam can be focused to a spot size significantly smaller than that for linear polarization [4]. One of drawbacks in these techniques is that it increases the sidelobes in the airy pattern and decreases the encircled energy in the central lobe.
A solid immersion lens (SIL) [5–8] or a solid immersion mirror (SIM) [9–15] is also used to decrease the spot size. Placing a truncated dielectric sphere between a focusing objective and a sample forms a SIL. The wavelength inside the sphere is reduced by the high refractive index of the truncated sphere, leading to a reduction in the diffraction limited spot size. By placing the lens in close proximity to a sample those optical rays at angles greater than the critical angle for exiting the high index material at the base of SIL or SIM can tunnel through the air gap between the lens and the sample, forming a small optical spot in the sample.
2. Focal point shift in solid immersion mirror
A SIM uses a curved mirror for bringing light into focus inside a dielectric material. Coating a metal film of good reflectivity, such as aluminum, on the curved surface, or making use of total internal reflection may achieve the focusing. It is well known that reflection changes the phase ϕ of the incident light and the amount of change (Δϕ) depends on the angle of incidence. For a p-polarized plane wave incident on a flat boundary between two homogeneous media of different optical properties, the complex amplitude of the reflected waves rp is given by Fresnel’s formula:
Here θi denotes the angle of incidence from the boundary normal and θt the angle of refraction. Figure 1 shows the phase of the reflected beam as a function of θi as a plane wave is reflected from a gold film (a) or from free space (b). In the calculation it is assumed that the medium of incidence has index of refraction n = 1.729, that of gold n = 0.188 + i 5.39 at λ = 830 nm. In curve (a) the phase ϕ increases slowly as θi < 45°, but drastically as θi > 45°; the amount of change Δϕ reaches ~145° as θi changes from 0 to 90°. In curve (b) there is a π phase jump at θi = 35.3°, which indicates the occurrence of total internal reflection. Δϕ, the Goos-Hänchen shift, reaches 180° as θi changes from 35.3° to 90°.
Fig. 1 Phase of a reflected plane wave from a gold film (a) and from free space (b) as a function of the angle of incidence. Calculation assumes that the medium of incidence has an index of refraction n = 1.729 and that of gold film n = 0.188 + i 5.39.
Figure 2(a) shows a SIM using a parabolic mirror. To obtain the least spot size, a SIM usually has θ close to 90°. Here θ denotes an optical ray angle making with the z-axis, which is equal to π - 2 θi. If this SIM collects light rays from 14° to 90°, the corresponding angle of incidence θi is from 83° to 45°. The amount of change in phase could shift the focus from its geometrical node appreciably. Interestingly, it is also expected that the focus shifts in the opposite direction between metallic reflection and the total internal reflection. In this article only the case of metallized SIM is addressed.
Fig. 2 Focusing of a planar solid immersion mirror. A two-dimensional parabolic mirror (a) is fabricated by cutting through an optical planar waveguide (b) and metallization on the sidewall. Light, launched into the waveguide by grating coupler, propagates in the waveguide, enters into the solid immersion mirror, and is brought to focus. XYZ is a right-handed rectangular Cartesian coordinate system with the z-axis along the optical axis of the mirror. (x, z) = (0, 0) is at the geometrical focal point of the parabolic mirror.
The observation of focal shift is performed on a planar solid immersion mirror (PSIM) [13–15], as shown in Fig. 2.
Light, propagating in a slab optical planar waveguide, is brought to focus by a two-dimensional parabolic mirror etched into the waveguide vertically from the waveguide film plane. Light reflection from the parabolic sidewall can be achieved with a metallized sidewall. The waveguide studied is fabricated by thin-film deposition on an Al2O3-TiC substrate. This substrate is commonly used for recording head sliders in the magnetic data storage industry. The waveguide consists of a 125-nm thick high-refractive-index core layer, Ta2O5, sandwiched between two cladding layers of lower-index, Al2O3. This thickness of the core layer is chosen to allow only the propagation of the fundamental transverse electric (TE0) mode and optimized to yield good field confinement. The diffraction-limited focused spot size for a PSIM is ~λ/(2β), which is determined by propagation constant β of the TE0 mode.
The fabricated PSIM for our experiments has a top opening of 50 μm and a bottom opening of 6.2 μm. The total length of the PSIM is 100 μm. The paraxial focal length of the SIM is f = 1.539 μm. Light propagating in the planar waveguide enters the PSIM from the top parallel to the optical axis. The central portion of the light, which is 6.15 μm wide in our case, is obscured and does not propagate through the SIM.
The vectorial electric field at the vicinity of the focus has both x- and z- components. The spatial field distribution can be approximated as a product of , where f is a function of x and z only, and g is the modal field profile of the slab planar waveguide. Based on the classical work of Wolf [16] and Richard and Wolf [17] who employed the Debye approximation, the focusing field f of the PSIM may be written as:
where θm is the maximum ray angle making with the z-axis, see Fig. 2, is the weighting vector for an optical ray from direction θ. is polarization dependent; for the focusing of a TE mode it can be written aswhere l0 is amplitude factor of the incident beam, and are unit vectors in the direction of x and z respectively, the last multiplier, , is the apodization factor. Note that in formulating Eq. (3), the waveguide has been approximated as a medium of refractive index β and that the mode reflection from a curved boundary is identical to Fresnel’s reflection from a flat boundary [18]. For an uniform incident beam (along the x-direction), l0 = 1; for a Gaussian beam,Here w0 is half-width of the beam at its 1/e amplitude.Figure 3 shows the calculated intensity distribution as a function of z in the neighborhood of the geometrical focal point, z = 0. ( denotes the transverse component of .) In the simulation the PSIM uses an Au coated sidewall, 2w0 = 50 μm, and λ = 830 nm. It is seen that the peak intensity occurs inward at 0.18 λ from the geometrical focus, which is about one-half of the focal depth.
Fig. 3 Intensity distribution along the optical axis z in the neighborhood of the geometrical node, z = 0, focused by a two-dimensional parabolic mirror coated with a gold film at wavelength of 830 nm.
3. Experiment and result
To observe the focal shift experimentally, the PSIM is truncated at different z positions by lapping and an aluminium-coated hollow aperture is scanned in contact mode on the xy plane for collecting light with a scanning near-field optical microscope (SNOM) (Witec Alpha). Thedevices were lapped using standard magnetic recording head fabrication procedures, which are accurate within 10 nm and leave a polished surface. The size of the SNOM aperture is typically 100 - 150 nm. Figure 4 shows the experimental setup. Light is launched into the waveguide by grating coupling [19, 20] and condensed by a parabolic mirror. The PSIM is placed on a flat mirror that is situated on a XYZ piezo-electric stage. The incident beam is s-polarized and its 1/e2 diameter is 50-60 μm.
Fig. 4 Experimental setup for measuring the intensity distribution of focused beam. S-polarized beam of light is coupled into the waveguide and focused by the PSIM. The diameter of the incident beam is 50 μm, which is obtained by focusing a collimated beam of ~5 mm by a focusing lens. Scanning an aperture over the surface normal to the waveguide plane and collecting the transmitted light through the aperture, the near-field light intensity distribution is obtained.
Figure 5 shows the series of intensity distribution versus z at λ = 830 nm. The geometrical focal plane is at z = 0. Also shown are the corresponding line profiles cross the middle along the x direction. At a light wavelength of λ = 830 nm, Ta2O5 has refractive index n = 2.10, while that of Al2O3 layer is n = 1.65. The thickness of the core layer is 125 nm. The propagation constant of the TE0 mode is calculated to be β = 1.776. It is seen that there are three spots in the 2 μm by 2 μm scan range: a central lobe and one sidelobe on either side of the central lobe. Clearly, the intensity profile changes appreciably every 50 nm in z. (Note that the electric field near the focus exhibits both x- and z-component. The x-component is focused and the z-component is nullified at the middle of the micrograph. The measurement, however, only shows the transverse component, i.e., the x-component, because the SNOM aperture does not interact with the longitudinal component, i.e., the z-component.)
Fig. 5 Measured intensity distribution at truncation z = −70, −125, −170, and –295 nm at light wavelength λ = 830 nm. The line profiles are the middle cross along the x direction, i.e., the horizontal direction. (The vertical direction is normal to the waveguide plane.) Note that the geometrical focal plane is at z = 0. Each frame on the left is 2 μm by 2 μm.
Figure 6 shows the analysis on the focusing characteristics. With decreasing z, both the intensity at the valley between either of the sidelobes and the central lobe and the peak-to-peak distance between two sidelobes decrease and reach a minimum at z ~-180 nm = 0.21λ. At this position, the intensity at valley is nearly zero. Full-width-at-half-maximum (FWHM) of the central lobe also decreases and reaches the minimum (~195 nm, below one-quarter of wavelength) at nearly the same z position. At the geometrical focal position, z = 0, the central lobe spot size is 240-260 nm in FWHM, which is 20-30% greater. In practice, this amount of change could lead to significant loss in resolution if not corrected. For instance, one application uses the PSIM to focus light onto an optical antenna in the heat-assisted magnetic recording [15]. The optical antenna has lollipop shape, with a peg attached to a disk. The disk size is only about 200-nm in diameter and we need to focus the light at the center of the disk to excite the resonance.
Fig. 6 PSIM focusing characteristics as a function of a. (a) Ratio of intensity at the valley, Ivalley, between either of sidelobes and the main lobe, to that of the main lobe. (b) Peak-to-peak distance between two sidelobes. (c) Full-width-at-half-maximum, FWHM, of the central lobe. (d) Comparison of intensity distribution between the theory, represented by the solid line, and the experiment, denoted by the dash lines, at truncation z = −180 nm.
There is only slight discrepancy between theory and experiment in focal position. The semi-analytical theory yields the best focus at −0.18 λ = 149 nm, while the experiment shows it is at −180 nm, which might be caused by non-perfect PSIM sidewall etch. There are a few degrees tilt in SIM sidewalls from vertical position. To see how well the theory predicts the experiment, the normalized intensity distribution is plotted along x direction at the focal position, see Fig. 6(d). The theory, represented by the solid line, almost overlaps the experimental data, denoted by the dash line, in the central lobe. But they slightly differ in the position of the sidelobes. The sidelobes mainly result from the optical rays of high angles, which is near the bottom of the SIM, z = 0. Since the SIM has only 6.15 μm wide at the bottom, the diffraction from the parabolic mirror is expected to be quite different from the stationary phase approximation assumed in the theory.
4. Conclusion
SIM focal shift with a scanning near-field optical microscope has been observed. The experiments were performed on a planar solid immersion mirror. The focus shifts inward for the metallized parabolic mirror and the amount of shift is about one-fifth of the wavelengths. The theory is consistent with the experiment in the central lobe and the intensity of the sidelobes but they slightly differ in the position of the sidelobes.
Acknowledgments
Author would like to thank Edward Gage, William A. Challener, Christophe Mihalcea, and Mike Seigler for their support of this work and many helpful discussions.
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