We investigated the process of focusing a radially polarized (RP) light beam through a sub-wavelength annular aperture (SAA). We found that the result was a non-diffraction doughnut-shaped light beam which propagates in free space. After analyzing the electric field component of the focus generated by the SAA structure, we identified the relationship between the focal field generated by the SAA. We then compared it to a case with a traditional objective lens. From our findings, we propose that a SAA structure can be viewed as a continuous numerical aperture optical element.
© 2009 Optical Society of America
The optical phenomenon of metallic nanostructures has been previously broadly discussed. One special phenomenon of metallic nanostructures is the existence of extraordinary transmission. In addition, the directionality of emitted light from metallic nanostructures known as directional beaming has also been studied. Another noteworthy research during the last few years relates to the study of the optical characteristics of metallic sub-wavelength annular apertures (SAA). In a near field, the SAA was found to excite surface plasmon (SP) at the metal-dielectric interface. At a far field region, a non-diffraction beam similar to the above-mentioned directional beaming can also be generated when a linearly polarized (LP) beam impinges onto the SAA structure. Non-diffraction beams, also called Bessel beams, were first proposed by Durnin et al. 
In addition to linearly polarized beams, there are many other types of polarized beams which exist in free space. One type is called a cylindrically polarized beam. A generalized cylindrically polarized beam is composed of a linear superposition of radially and azimuthally polarized beams. Radially polarized beams are one special state of generalized cylindrical polarized beams. In 1972, radially polarized (RP) beams were found to be initially generated when the resonant cavity of the laser was modified. Since that time, various other ways have been found to generate RP beams. For example, an interferometric technique and liquid crystal device[8, 9] were used to convert LP beams directly into RP beams. Herein, we utilized a liquid crystal device to generate the RP beams needed for our experiments.
When RP light beams are focused onto the SAA structure, we can then locate the nondiffraction doughnut beam. This non-diffraction doughnut beam matches the characteristics of a high-order Bessel beam. We found that the focal spot of the non-diffraction doughnut beam generated by the SAA structure was much smaller than the one generated by a Mach-Zehnder interferometer. From previous studies, we know that the polarization of light influences the transmission of the sub-wavelength metallic structure. In addition, the electric field at a focal plane of the RP beam is different from a LP beam when a traditional objective lens is used.
To investigate and understand the effects relating to SAA generated Bessel beams, the electric field at the focal plane of the RP beam was separated into longitudinal and transverse components and their corresponding features. Different intensity distributions were also examined by using a traditional optical perspective related to the numerical aperture (NA) of the focal lens. We found that when the SAA was illuminated by the RP beam, we can treat the SAA structure as a special lens. When we compared the focal characteristics of the Bessel beams generated to the focal characteristics of a traditional objective lens, we found that it is possible to regard the SAA structure as a continuous NA optical element.
2. Generating non-diffraction doughnut shape beam by a sub-wavelength annular aperture
To observe the transmission light of the SAA structure, a SAA structure made on silver film was prepared. We used a focus ion beam system (FEI Nova 200) to fabricate the SAA structure. Several configurations, which differed only in the corresponding ring diameters, were made on 250nm thick silver film (Fig. 1). We fabricated three ring diameters (6µm, 9µm, and 12µm) which were all equipped with a 150nm width slit. Figure 2(a) shows our experimental set-up. We found that the linearly polarized light beam emitted from the He-Cd laser with 442nm wavelength passed through a liquid crystal polarizer which was composed of a polarization converter, TN cell, and a variable phase retarder (Fig. 3), which together transformed the LP beam into a RP beam. The polarization was checked by using polarizer. Figure 2(b) shows the intensity distribution after passage through a polarizer with a vertical direction. The RP beam was first focused using a 5X objective lens with a 0.13 numerical aperture to form a doughnut-shaped focus with a 16.3 µm spot size (Fig. 2(c)). After focusing with the objective lens, the RP beam was aligned by a microscope in order to impinge onto the SAA structure precisely. Since the SAA structure can be viewed as a plasmonic lens, the RP focus beam propagating in free space was found to be a non-diffraction doughnut beam as shown by previous researchers.
Figure 4 shows the images measured at different distances away from the SAA surface when the RP beam was incident onto the SAA structure with a 12µm diameter. The FWHM of the smallest central spot size located at 14µm above the exiting surface was found to 0.729µm (Fig. 5.). We found that the central spot size can be maintained below 2µm for a distance up to 25µm above the SAA exiting surface. In addition to observing the small focal spot size, we also used other SAA structure diameters and analyzed their depth of focus (DOF). We normalized the intensity profile of the focused beam with respect to the maximum intensity. Figure 6 shows the normalized light intensity at different distances away from the focal point. We found that the length of the DOF depends on the diameter of the SAA structure itself. The non-diffraction characteristics were demonstrated by the slow energy decay rate of the focal spot along the z axis when compared to that of the Gaussian beams. This long depth of focus demonstrates the non-diffraction characteristics. As such, we can conclusively say that the SAA can be considered to be a new method of generating a nondiffraction doughnut shaped beam.
3. Continuous numerical aperture proposition to explain the non-diffraction doughnut beam of a sub-wavelength annular aperture
To provide insight into our experimental results, a finite-difference time-domain (FDTD) simulation was applied to compare the focus characteristics. The dielectric constant of silver at 442nm was set to be -5.735+j0.7536. Our total simulation dimension was 20µm×20µm×25µm and the maximum mesh size in the three-dimensional simulation region was 10nm. By using this material, we found that the surface plasmon interference occurred on the surface of SAA structure, which corresponds to the characteristics of silver. Figure 7 shows the polarization of generalized cylindrically polarized beam. The unit electric field of the cylindrically polarized beam can be expressed as follows:
Where e⃗r is the unit vector in a radial direction, e⃗φ the unit vector in an azimuthal direction, and P the relative amplitude of the field. The polarization of each position in the beam was rotated ϕ 0 from its radial position.
The long depth of focus (DOF) feature was found when the generalized cylindrically polarized beams were incident onto the SAA structure. Figure 8 shows the focusing feature when generalized cylindrically polarized beams were incident onto the SAA with different ϕ 0 values. The variable ϕ 0 was dependent on the electric field intensity ratio of the radial direction and the azimuthal direction. If the radial component is stronger than the azimuthal component, a point-shaped focal pattern can be observed near the SAA structure. In contrast, a doughnut-shape can be seen near a SAA structure.
To verify our experiment, a radially polarized beam was generated when ϕ 0 had a value of zero. An interesting phenomenon at the focal point (see Fig. 9) shows that the focal pattern will be point-shaped near the SAA surface, but in far field, the focal pattern will become more of a doughnut shape. Results show that the electric component is split into an Er and an Ez component when the RP beam is incident to the SAA structure. In Fig. 10, the focal intensity contributed by the Er component was doughnut-shaped in size but was point-shaped in size when contributed by the Ez component. Due to the non-propagating nature of the light beam z component, it cannot be observed and measured accurately by using microscopy. We determined that the shape of the focal plane depends on the intensity ratio of the Ez and Er components. Thus, a long DOF can be assumed to be focused by adopting several lens equipped with different NA. Based on this perspective, we can then derive the corresponding NA at different focused positions. Results show that the focal plane away from the SAA structure can be focused by a low NA lens (z>10µm). In addition, the focal plane near the SAA structure can be viewed by focusing with a high NA lens (z<4µm).
Fig. 11 shows the maximum intensity ratio of the Ez and Er components at the focal plane for different derived NA values of the SAA structure by analyzing the simulation results. A continuous relationship was found between the maximum intensity ratio and the derived NA. To obtain more optical properties on the metallic SAA structure, two SAA structures of 6µm and 9µm diameters were fabricated and their characteristics were measured. We found the relationship was identical as that shown in Fig. 11. This result shows that the relationship seen in Fig. 11 will always hold no matter the diameter size of the SAA structure. Therefore, based on this observation, we propose that we can consider a SAA structure as a continuous NA lens.
To prove the feasibility of a continuous NA proposition, we calculated the electric component of the focal plane numerically by focusing the objective lens and comparing the focal field of the SAA structure to a traditional objective lens. The unit electric field of the RP beam can be expressed as follows:
To compare the focal properties of the SAA structure to the objective lens, we put in an annular diaphragm before an objective lens (see Fig. 12). In this way, the electric component on the focal point was contributed by a certain angle. After the RP beam was focused by the objective lens, the focal electrical field components can be expressed as follows [12, 13]:
where A is the constant which can be set to unity, P(θ) is the pupil apodization function, θmax and θmin are derived by the slit width, r the position in the focal plane, k the wave vector, θmax=sin-1 (NA), θ min=θ max-(R 2-R 1)/R 1cscθ max, β 0 is the radius ratio of the lens and incidence light, and J n(x) represents the Bessel function of the first kind of order n. All measurements are in units of wavelength, therefore β 0=1.5, A=1, k=2π/λ, R1=6, R2=5.85.
The intensity of the focal point is a combination of the Ez and Er components. The Er component can be described in the first order of the Bessel function, so that the focal intensity contributed by the r component shows up to be shaped like a doughnut. The Ez component at zero order of the Bessel function is such that the focal intensity contributed by the z component shows up as a point shape. We calculated the focal pattern focused by different NA lenses (see Fig. 13). The focal pattern, as determined by the intensity ratio of the r and z component, was a doughnut-shaped for the case where there was a low NA lens and was point-shaped for the case with a high NA lens. These focal patterns correspond to the focused beam pattern of the SAA structure for different derived NA values.
We found that as the NA value increases, the Ez component becomes stronger than the Er component. The relationship can be seen in Fig. 14 which shows the ratio of the maximum intensities of the Ez and Er components at different NA values. We chose the NA value to be in the range of 0.18 to 0.94. When comparing the relative curves to the one obtained from the SAA structure, we see the same tendency or characteristics in all the curves. Thus, it appears that both the focal pattern and the electric component distribution can support a continuous NA proposition of the SAA structure.
We propose a new method to generate non-diffraction beams with a cylindrically polarized beam incidence, using a SAA structure. Both simulation and experimental results show that a sub-wavelength focal spot and long depth of focus can be achieved by using a SAA structure. A comparison of the focal field characteristics between a SAA structure and an objective lens by a RP beam incidence supports the idea of a continuous NA in a SAA structure. Moreover, these results provide new insight into SAA structures which will be useful for future studies.
This work was financially supported by the Materials & Chemical Research Laboratory of the Industrial Technology Research Institute (ITRI) and Taiwan’s National Science Council under Grant No. 97-2221-E-002-159-MY3. We appreciate C.Y. Wang and other group members for their assistance with this work.
References and links
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