The electromagnetic eigenmodes of and light transmission through a C-aperture to the far field, and to a storage medium, have been studied based on the full vectorial finite difference method. It is found that the cutoff wavelength of C-aperture waveguides in a gold film is much longer than that in a perfect electric conductor, and the fundamental mode is confined in the gap and polarized with the electric field along the gap. The light transmission resonance through C-apertures to far field and to a storage medium occurs at wavelengths below the waveguide cutoff wavelength. Measurements on the fabricated C-apertures confirm the mode confinement and transmission resonance.
©2008 Optical Society of America
Conventional optical data storage and heat-assisted magnetic recording use a focused light spot to raise the local temperature of a medium during recording. The storage density is determined by the size of the optical spot, which is limited by the far-field diffraction of light to ~λ/2. (λ denotes the wavelength of the light source in free space.) Use of an optical near field could potentially increase current recording densities many fold, which surmounts the diffraction limit. In order to do so, subwavelength metallic apertures [1–5] or optical antenna [6–10] could be used to produce an optical spot much smaller than λ. For instance, marks with diameters of about 60-nm have been written on a magneto-optical or a phase change medium using a tapered optical fiber ; marks down to 40-nm were recorded using a beaked metallic plate near-field optical probe .
The usefulness of metallic apertures in near-field optical devices is limited by their poor transmission efficiencies. In a perfect conducting metal sheet, the incident power is reflected from the aperture as a result of mode cutoff, for which there can be no propagation of light through a hole when the wavelength of light is more than twice the hole-length across. In plasmonic metals, recent studies showed that a cylindrical hole always supports propagating modes near the surface plasmon frequency, regardless of how small the hole is [11,12]. For a rectangular aperture in a real metal, the cutoff wavelength increases due to the increased coupling between evanescent fields on the long edges inside the hole and penetration of the electric field into the metal [13,14].
The transmission of light through an aperture depends on the hole shape, the state of polarization of the incident beam, and the Fabry-Perot or cavity resonant wavelength [15–18]. The optical near field localized in the metal apertures is strongly enhanced by the excitation of localized surface plasmon around the metal aperture and surface plasmon polaritons in the presence of periodic corrugations surrounding the aperture, which can triple the light transmission in the far field .
Among various apertures, the C-shaped aperture, a type of ridge waveguides, has been extensively studied [15, 5, 19–24]. The ridge waveguide has been widely used in microwave and antenna systems because of its low cutoff frequency, wide bandwidth, and low impedance . The ability of C-shaped apertures to support a guided mode that can be excited by the incident polarization is believed to be critical for achieving large transmission, especially for the thicker films. Moreover, C-shaped apertures generate intense fields in the gap between the ridge and the opposite surface of the waveguide, which confines light to a spot size of λ/10 or below.
Cutoff wavelength plays a significant role in the transmission through an aperture perforated on a real metal. In a perfect conducting sheet, modeling showed that the C-aperture transmission resonance is always slightly longer than the cutoff wavelength of the ridge waveguide of the same cross section . In this paper, we present a numerical investigation of dispersion for a C-aperture perforated in a real metal (i.e., a gold film) in optical frequencies and evaluate the light transmission through the C-aperture with and without the presence of a storage medium. It is found that the mode cutoff still exists for a real metal but the cutoff wavelength is significantly longer than that in a perfect electric conductor. The light transmission resonance to far field or to a storage medium is found at a wavelength always shorter than the mode cutoff.
2. Cutoff wavelength and mode confinement of C-apertures in a gold film
To analyze the eigenmodes of a C-shaped waveguide perforated in a real metal, we extend the compact 2D finite-difference frequency domain technique developed for evaluating the leaky modes in photonic crystal fibers [26, 27]. In this mode solver, the discretization scheme is based on the two-dimensional Yee’s lattice , which is widely used in the finite-difference time domain (FDTD) analysis to directly solve the two Maxwell electromagnetic curl equations numerically . To truncate the computation domain, anisotropic perfect matched layers (APML)  are used to enclose the computation domain, which absorbs any electromagnetic waves outward without reflection. For a waveguide normal to xy plane and uniform along z direction, assuming that the fields have the dependence of exp[j(βz-ωt], (β denotes the propagation constant of a mode along z direction,) the Maxwell equations with APML boundaries at a given frequency ω are given by:
where E⃗(H⃗) represents the electric (magnetic) field, εr(µr) denotes the relative permittivity (permeability) of the waveguide materials, Z 0 stands for the free-space impedance, and k 0 for the wave number in free space. sx (sy) is 1 if x(y) is within the computation domain, and is given:
if x(y) is within the AMPL region. Here kx, σx, ky, and σy are parameters to be optimized to yield minimum reflection from the APML, and ε 0 is vacuum permittivity. In Eq. (1), the derivatives to z are replaced by jβ, and those to x and y are approximated by the central finite difference in Yee’s lattice .
At a given frequency ω, the complex propagation constant β is found from the resulting eigen equation, based on the two transverse electric or magnetic components, using the shift-invert power method .
Figure 1 shows the cross-sectional viewing of a single-ridged waveguide in xy plane perforated in gold. It is composed of a rectangular guide of sides, a, and, b, with a ridge of width, w, and gap, g, on one of its wide sides. In this study, we set a=200 nm, b=80 nm, g=w=40 nm. (These parameters were chosen to yield confined optical spot in the presence of a magnetic storage medium with good transmission efficiency at λ≈830 nm.) Fig. 2 shows the calculated complex effective index neff(=β/k 0) of the fundamental mode versus free space wavelength λ. In our calculations, we have used the εr value for gold as tabulated in Ref. 32 and assumed µr=1. The Yee cell size is Δx=5 nm, Δy=10 nm, and the number of points is 120 along both x and y direction. At the interface of two materials, the tangential electric displacement εr E⃗ is taken as the average of that on either side of the materials. (The normal electric displacement εr E⃗ is not sampled in Yee’s lattice.) It is seen that the real part of neff decreases rapidly as λ>0.7 µm; it is nearly 0 at λ=0.85 µm. Interestingly, the imaginary part of neff first decreases with increasing λ; reaches minimum at λ=0.75 µm, and increases rapidly at λ>0.82 µm. Based on the dependence of both real and imaginary parts of neff on λ, we may define λc=0.85 µm as the cutoff wavelength, which is much longer than the perfect conductor value λc=0.55 µm, based on Ref. 25.
Figure 3 plots the computed E field amplitude and the z-component of the Poynting vector of the fundamental mode in xy plane at λ=775 nm. Note that the x direction is normal to the long sides of the waveguide where the ridge is located. Among the three components of the E field, Ex is the largest. Comparing to Ex and Ez, the y-component of the E field is negligible and is therefore, not displayed. From Fig. 3 it is seen that the E field of the fundamental mode is transverse to and confined in the gap. The strong Ex in the vicinity of the ridge indicates the presence of significant amounts of induced electric charges on the waveguide wall in the ridge. On the side opposite to the ridge, Ex disperses slightly and the induced electric charge density on the waveguide wall is much lower than that in the ridge. The profile of the energy flow is slightly different from the E field and it is more concentrated in the gap, and maximized near the ridge.
To confirm the mode confinement, we fabricated a C-aperture by focused ion beam milling in a 122-nm thick gold film on a glass substrate and measured the light transmission by scanning an aperture on a cantilever over the film surface in contact mode, using a scanning near-field optical microscope. The aperture probe for this measurement is 100 nm by 140 nm in the opening. Fig. 4(a) shows the SEM image of the fabricated C-aperture, and Fig. 4(b) displays the measured near-field intensity profile. The optical spot size is measured 90 nm by 88 nm in the full-width-at-half-maximum (FWHM), which is limited by the size of the probing aperture. Apparently, the measured spot is much smaller than the overall C-aperture size, which demonstrates the light transmitted through the C-aperture is well confined below the rectangle footprint.
3. Light transmission through the C-aperture into free space
Figure 5 shows the C-aperture transmission versus the gold film thickness coated on a glass substrate at various wavelengths, computed with the FDTD technique. The incident beam is a gently focused Gaussian beam with a focusing lens of numerical aperture NA=0.2 and polarized along the x-direction. The FWHM of the incident beam is 2.60 λ along x direction and 2.57 λ along y direction. The light transmitted through the aperture is collected with an objective lens of NA=0.9 in the far field, which is calculated by integrating the Poynting vector normal to the exit pupil. The transmission displayed is normalized by the incident optical power on the rectangle of the C-aperture. In the calculation, The Yee cell size is Δx=5 nm, Δy=5 nm, and the number of points is 200 along both x and y direction. As expected, the transmission is maximized at certain film thickness at each wavelength, corresponding to the Fabry-Perot resonance. As λ is close to λc, the transmission peak shifts to thick film and is also broadened, which is resulted from the fact that the mode propagation constant β→0. It is interesting to see that the transmission is maximized at λ=775 nm, which is below λc and falls in the valley of the imaginary part of neff (see Fig. 2). At λ=689 nm, the transmission is much lower than that at λ=729 nm, which is explained by the high absorption in the waveguide (see Fig. 2).
At a given film thickness, the transmission through the C-aperture is a strong function of λ as a result of wavelength-dependent mode effective index neff. Figure 6 shows the measured far field transmission spectrum of a C-aperture fabricated on a 200-nm gold film on a glass substrate. A beam of light, exiting from a tunable, continuous wave (CW) Ti: Sapphire laser is coupled into an optical fiber. The beam exiting from the optical fiber is re-collimated and gently focused by an achromatic doublet lens of 75 mm in focal length onto the sample through the substrate. (The maximum incident optical power is ~10 mW.) The incident beam is chopped by an optical chopper at 1KHz and the transmitted light at the exit pupil of the collection lens is detected with a silicon detector and a DSP lock-in amplifier.
In Fig. 6 the transmission resonance is clearly exhibited and the peak transmission is consistent with modeling. The resonant wavelength, however, is longer than modeling. It may be due to the non-vertical wall in the fabricated C-aperture. It is also interesting to see that the transmission is not symmetric at the resonant wavelength: it falls faster in the short wavelength side than in the long wavelength side, which is consistent with our modeling.
Apparently the C-aperture transmission behavior in the far field can be understood based on the property of waveguide mode.
4. Light transmission in the presence of a storage medium
To evaluate the near-field recording performance of the C-aperture, we place a magnetic storage medium in the proximity to the C-aperture and calculate the light absorption (the time averaged Joule heat per unit volume) in the storage layer at various excitation wavelengths. In Fig. 7(a), a x-polarized collimated beam is focused by an objective lens of NA=0.9 onto a C-aperture perforated on a gold film coated on a glass hemisphere. A magnetic storage layer is placed 6-nm below the C-aperture, and a storage medium consists of a 15-nm thick Co layer, a 10-nm thick ZnS-SiO2 dielectric layer, and a gold heat sink layer on a glass substrate. In the modeling we assume that the optical power of the incident beam is 1 watt; the refractive indices of the glass hemisphere and substrate are 1.5, and that of ZnS-SiO2 is 1.7. The complex refractive indices of gold and cobalt are obtained by interpolation from Ref. 32. The Yee cell size is Δx=5 nm, Δy=5 nm, and the number of points is 160 along both x and y direction. Δz may be different for different layer and is chosen to yield integral samplings. Specifically it is 5 nm for the C-aperture layer, 3 nm for the spacer layer between the C-aperture and the magnetic layer, 5 nm for the magnetic layer, the ZnS-SiO2 layer, and the gold layer. Based on the Debye approximation [33, 34], the focusing field by the objective lens is calculated and the FWHM of the focused spot is found to be 0.37 λ.
Figure 7(b) displays the maximum absorption in the Co layer as a function of gold film thickness at various light wavelengths. The absorption is defined as σ|E|2. σ denotes the electric conductivity, which is extracted from the complex refractive index of the magnetic layer: σ=ωε 0 imag (εr). Similar to what we have observed in the far-field transmission, the absorption is maximized at certain gold film thickness at each excitation light. As λ increases, the optimal absorption increases, reaches maximum at λ=826.6 nm, and then decreases. Substantial reduction in absorption occurs from λ=858 nm to 910 nm, at which point the propagation of light through the aperture becomes evanescent.
Comparing to the case without the presence of a storage medium, the optimal gold film for transmission resonance is approximately halved, due to the nearly π phase shift in the reflection from the storage medium back into the C-aperture. In particular, the optimal wavelength for maximum absorption is pushed closer to λc of the C-aperture waveguide from 775 nm without the presence of the storage medium, but it is still below λc. In other words, the propagating waveguide mode dominates the optical power delivery from the C-aperture to the storage medium, which is the same as the observation on the far field transmission. The red-shift in resonant wavelength is expected, due to the absorbing nature of the storage medium.
Figure 8 shows the absorption profile at the middle plane of the storage layer. The absorption profile is similar to the mode profile shown in Fig. 2 and is confined to be a spot of FWHM=89 nm along the x direction and 75 nm along the y direction. Further reduction of the hot spot for high areal density near-field optical data storage or heat-assisted magnetic recording will require the down scaling of the dimension of the C-aperture, such as the ridge width, gap size and the optimization of sides length as well as the film thickness. The cutoff wavelength of the C-aperture waveguide should indicate the optimal wavelength for the best near-field recording performance.
5. Concluding remarks
The eigenmodes, the light transmission to far field, and to a magnetic storage medium of a C-aperture are calculated based on the full vectorial finite difference method. It is found that the cutoff wavelength of the C-aperture waveguide in a gold film is much longer than that in a perfect electric conductor and the fundamental mode is confined in the gap and polarized with the electric field along the gap. The light transmission resonance to far field and to a storage medium occurs at wavelengths below the cutoff wavelength, and the light transmission is through the excitation of the propagating fundamental mode. Measurements on the fabricated C-apertures confirm the mode confinement and transmission resonance.
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