We theoretically demonstrate that a single bare subwavelength metal slit without any surrounding corrugations can have a capability to steer the incident light into focusing patterns by introducing a high index in the transmission half-space. The focusing properties are identified to depend on both the slit width and the output permittivity. The underlying physics lies in the interference of quasi-cylindrical waves scattered from the slit, and our proposed model agrees well with the simulation results. This finding is believed to inspire some novel ideas for the nano-optics design.
©2013 Optical Society of America
Light propagating through a single aperture has been interesting over the centuries because of its great importance in both the scientific research and industry applications. According to the classical diffraction theory, the light emitted through a tiny aperture often spreads into a wide angle particularly for a subwavelength-scale structure [1,2]. To hurdle this basic problem, some researches have recently reported to control the beam divergence below the diffraction limit using novel technologies of nano-optics [3–13]. From the geometric aspect of the structure, related studies can be divided into two categories: a single slit (or hole) surrounded by some auxiliary grooves or gratings on the exit surface [3–5], and a bound collection of metallic waveguides on the nanoscale [8–11]. In the first type of system, the light beaming behavior is physically based on the phase delayed coupling of surface plasmon polariton (SPP) resonances in the periodically arranged side corrugations [6,7]. Such a beam controlling method has been also realized in the photonic crystals [12,13], in which the evanescent fields creeping on the material surface becomes dominant. In the second type of system, the phase retardations of the emitted waves from the waveguide arrays, whose mutual coupling results in a beam focusing phenomenon, can originate from structure parameters such as the slit length, slit width, and even the filling materials inside the slits [8,14,15]. Anyway, there are several common features among the two types of structures: (i) the precise fabrication of patches of hard substructures is substantially needed with time-consuming procedures; (ii) the light beaming behavior based on the SPP resonance is only operated at specific wavelengths; (iii) the subwavelength structures are usually immersed into the air transmission environment.
In the current study, the spatial transmission properties of a single bare subwavelength metallic slit with a high index of dielectric medium in the output half-space are reported. Numerical simulations demonstrate an unexpected phenomenon: the beam focusing is achieved especially with a permittivity of the transmission space. Underlying mechanisms can be essentially due to the modulation of the emitted wave fronts by the interference of quasi-cylindrical waves (QCW) scattered from the slit .
2. Simulation model
Our simulation model is shown by a diagram in Fig. 1, where silver metal film with several hundred nanometers of thickness is used to construct a single bare subwavelength slit. The geometric width of the slit is denoted by w. The dielectric response of the metal to an electric field is described by the Drude-model fitted experimental permittivities at the wavelengths of interest . A plane wave of transverse magnetic (TM) polarization is normally incident onto the slit entrance embraced by the air (Ɛ0 = 1). The transmission half-space is full of a transparent dielectric medium with a relative permittivity of Ɛd. To investigate the influence of output permittivity on the structural transmission, two-dimensional electromagnetic field distributions are simulated with the finite-difference time-domain (FDTD) method, in which the calculation region is bordered by perfectly matched layers to absorb scattering lights leaving the region of interest.
3. Simulation results
First, we compare the spatial transmission properties at different Ɛd for the given slit width of w = 100 nm. Under these circumstances, the incident wavelength is fixed at λ = 1 μm and the slit thickness is coordinated to obtain the first maximum transmittance. As seen in Fig. 2(a), for the free transmission space of Ɛd = 1, the spatial distribution of the magnetic field Re(Hz) emerged from the slit consists of uniformly circular wave fronts, resembling a single point source with a radial emission. With the gradual increase of the output permittivity, the field enhancement inside the slit turns to have an asymmetric distribution along the propagation direction, in which the maximum enhancement position moves closer to the slit exit opening. Figure 2(b) shows the field distribution patterns of Re(Hz) at the permittivity of Ɛd = 6, wherein the boundary between the radiative wave and the surface wave can be identified, and the energy distribution in each wave front seems to be nonuniform. Actually, this phenomenon will become more serious when the output permittivity increases to Ɛd = 10, and the resultant field Re(Hz) is plotted in Fig. 2(c). At this time the spatially confined surface wave is distinguished clearly from the cylindrical diffractive wave launched from the slit, or the light beam emitted from the slit tends to be angularly reshaped in the transmission space. This result reveals that the single bare slit especially with the high permittivity in the output space can possess the beam reshaping capability in a way similar to the traditional refractive optics, but associated with much different mechanisms.
Secondly, we investigate the influence of slit width on the beam reshaping at the given incident wavelength of λ = 1 μm. As a matter of fact, with gradual increasing the slit width, the beam reshaping is easily observed even with the low permittivity of transmission space. Figures 3(a)–3(c) demonstrate the spatial distributions of the magnetic field intensity |Hz| for two different slits at the same permittivity of Ɛd = 14. When the slit width is w = 200 nm, as seen in Fig. 3(a), the transmission light intensity is modulated into a unique pattern like a candle flame, or the beam focusing is evidently observed in the transmission space. In order to describe the ability of the single bare metal slit system to focus light, we obtain the curve of transmission intensity distribution along the y direction at x = 0, as shown by an inset diagram in Fig. 3(a), and consequently the focal length is here defined as the distance from the center of the slit exit to the maximum intensity point. Moreover, besides the strong light channel formation along the direction of normal to the slit surface, some growing sidelobes with relatively weak intensity also appear on two flanks, which is naturally reminiscent of the high-order diffraction features of an optical grating. For the case of w = 300 nm (see Fig. 3(b)), the transmission light is also focusing and super-directive with a slight divergence, the beam collimation of which can be extended to a distance of several wavelengths. Moreover, the focal region begins to move away from the slit accompanied by the growth of more sidelobes.
The beam focusing effect introduced by the single bare slit can also be accomplished with different incident wavelengths. Figure 3(c) depicts the diffractive light from the single slit at the output permittivity of Ɛd = 14 for the larger incident wavelength of λ = 2 μm, where the slit width is employed as w = 600 nm to keep the same ratio of w/λ = 0.3. In this case, the focal length of the transmission beam is measured about 740 nm, larger than that of Fig. 3(b). Correspondingly, the obtained electric field |Ex| distribution has the same focusing effect as that of the magnetic field (see Fig. 3(d)); whereas the simulated electric field |Ey| distribution displays the multiple beaming profiles (see Fig. 3(e)), which are completely different from those field distributions when Ɛd = 1. So far, we can recognize that beam focusing of the single slit actually depends on multi-parameters; that is, either the larger permittivity or the slit width, or the incident wavelength is ready to promote the spatial directivity of the transmission light.
For the purpose of quantitative description of the beam focusing properties, some characteristic parameters have to be defined. Figure 4(a) shows variations of transverse full-width at half-maximum intensity (FWHM) of the focal beam spot for several output permittivity values. Clearly, for the given slit width, the focal beam width decreases gradually with the increase of permittivity, or the large output permittivity situation prefers giving a tight beam focusing behavior. For example, with w = 200 nm, the obtained FWHM of focal spot is about 250 nm at the permittivity of Ɛd = 4, while at the higher value of Ɛd = 14 the FWHM decreases to only about 120 nm, which indicates that such beam focusing is beyond the diffraction limit (half of the wavelength in the transmission space), as represented by the dotted line in Fig. 4(a). On the other hand, for the given output permittivity, the FWHM of focal spot is increased with larger slit widths. These results provide an effective way to control the beam focusing of the transmission light through the single slit.
Figure 4(b) shows the permittivity dependence of the focal length for several slit widths. In the case of w = 200 nm, the resulting focal length increases from 30 nm (at Ɛd = 3) to 180 nm (at Ɛd = 14). The rising tendency of the focal length versus the output permittivity becomes more prominent with larger slit widths. In particular, for the case of w = 400 nm, the focal length increases to reach the scale of the incident wavelength for sufficient large permittivity. Namely, when either the slit width or the output permittivity becomes increased, the beam focal position moves away from the slit. To appraise the directionality of light beaming, the focal depths are also examined, whose variation tendency as a function of the output permittivity is found similar to that of the focal length, which implies that the transmission light can be channeled into a long distance.
Figure 4(c) illustrates the relationships between the focal length and the output permittivity for different incident wavelengths of λ = 0.7–5 μm but with the same normalized slit width (w/λ = 0.3). Clearly, the focal length can be engineered by tuning the incident wavelength. When the incident wavelength is smaller, the resulting focal length becomes shorter. Here we should point out that when the output permittivity is Ɛd = 14, the beam focusing can still be evidenced even for the mid-infrared wavelength incidence of λ = 10 μm, although for this case silver is close to a prefect electric conductor (PEC) and can hardly support SPP excitations. As a result, we can reasonably exclude the role of SPP excitation in the beam focusing of our cases.
4. Proposed physical model
To understand the physical origins of such a beam focusing from the single bare slit, the phase distribution of the magnetic field in the transmission space is firstly presented for the slit width of w = 300 nm. As shown by an inset picture on the upper right corner in Fig. 5(a), for the free transmission space (Ɛd = 1), a series of diffracted phase fronts displays smooth semi-spherical profiles, which suggests that the narrow slit reemit field in the same manner as a point source. However, when the permittivity of transmission space is set as Ɛd = 14, the emitted phase fronts seem to be modulated into the corrugated profiles (see Fig. 5(a)). In contrast to the formation of concave phase fronts during the conventional light focusing processes, here the diffracted phase fronts are seen to be disturbed spatially at some places, or each phase isoline can be viewed as the assembling of several segmental arcs with different radii. Especially, the flat-top profile of phase fronts close to the slit exit suggests that the field reemission can no longer be considered from a point source. In addition, Fig. 5(a) demonstrates that spatial distribution of fluctuation points on the phase fronts seem to be well defined in the radial directions. Inspired with this idea, we can deduce that each phase front is in fact joined together by several propagating spherical waves with variable time delays, whose point sources can be tracked onto the interface between the slit exit and the dielectric medium in the transmission space.
In fact, the above assumption can be further confirmed by the simulated phase map of Ey field around the slit with the output permittivity of Ɛd = 14, as shown in Fig. 5(b). Clearly, the phase of Ey field changes periodically in the vicinity of the slit exit opening, which indicates the propagation of surface wave along this interface. In addition, the spatial morphology of Ey phase distribution on both sides of x = 0 appear to be symmetric but with a π phase difference, and the amplitudes of Ey field on both sides of x = 0 are identical due to the symmetry of the structure. As a result, the Ey field distribution is expected to have an anti-symmetrical relationship of around the metal slit. According to the microscopic analysis of subwavelength metallic slits, a hybrid wave composed of SPP and QCW contributions can be scattered by the slit edges [16,18,19]. For the output interface on the right and left sides of the metal slit, both SPP and QCW components can be lunched due to the metallic properties. However, for the spatial region spanned by the slit opening, an interface between the dielectric medium and the air opening of the aperture is formed due to the absence of metal material. As reported by reference 20 and 21, SPP mode begins to disappear on the dielectric interface, but two QCW components excited by the slit edges can still remain associated with the wave vector of . And subsequently their optical interference will result in the periodic distribution of intensity fringes on the output interface of the metal slit, which can be confirmed by the simulation result in Fig. 6(a). Being similar to the optical behaviors of a finite array of hard nanoholes , the obtained constructively interfered intensity fringes on the dielectric interface over the slit opening can act as a finite array of “soft” nanostructures, whose reemitting electromagnetic waves will lead to the beam focusing of the transmission light. Based on the near-field interference of electromagnetic field radiation from the multiple intensity fringes distributed on the output interface of the metal slit, the resultant focal length of the nanograting-like arranged light sources can be obtained :Figs. 4(b) and 4(c), which appear to be consistent with the data of FDTD simulations. Furthermore, the beam focusing pattern of the magnetic field can also be retrieved with the present optical interference model for a finite array of point light sources, Furthermore, the beam focusing pattern of the magnetic field can also be retrieved with our present physical model by assuming a finite array of point light sources on the dielectric interface of the slit, which have an in-phase and periodic spacing within a spatial range equivalent to the slit width. The structural parameters for the analytical calculations are obtained from Fig. 3(c), and the achieved result is shown in Fig. 6(b). It is seen clearly that the two pictures have a good consistence, which in fact confirms the validity of our theory.
For a single bare slit with subwavelength width in the current study, when the transmission space is surrounded by the air environment, the wavelength of diffracted QCW component is as large as that of the incident light, so that the dielectric interface spanning over the slit exit does not hold sufficient length range for the QCW interference, or the periodic intensity fringes of electromagnetic fields cannot be formed on the slit-dielectric interface to modulate the emitted wave fronts, instead most of the emerging light from the slit is in a radiative mode to cause the beam scattering in wide directions. However, when the permittivity of the transmission space is increased, the wavelength of the scattered QCW component begins to reduce, which is equivalent to the increase of the slit opening width. As a result, the constructively interfered intensity fringes is allowed to form on the slit-dielectric interface, and the number of interfered fringes becomes large with increasing the permittivity value of the output space, which consequently leads to the longer focal length of the transmission light according to Eq. (1). On the other hand, as the slit width increases at the given output permittivity, the length of dielectric interface spanning over the metal slit can extend to support more numbers of the constructive interference fringe, which also results in the longer focal length of the transmission beam. In addition, when the incident wavelength increases, the periodic spacing of the constructively interfered fringes becomes large. However, if the ratio of w/λ is kept small, the longer focal length can still be obtained.
In summary, we have theoretically investigated the optical transmission properties of a single subwavelength slit without additional surrounded grooves, but with the high permittivity dielectric medium in the output half-space. Simulation results based on FDTD method have revealed that the transmission light can be engineered into a tight focusing beam, whose features are closely dependent on both the output permittivity and the slit width. Then we have proposed a physical model by considering QCW propagation on the slit-dielectric interface, and its optical interference acting as the secondary sources subsequently modifies the wave fronts of reemitted light into the space. All discussion and analyses are in consistent with the FDTD simulation results. The observed near-field beam focusing effect can be applied for either the metallic or the nonmetallic single subwavelength slit. Being analogous to the oil immersion microscopy, our current study discovers that the single bare subwavelength slit can easily manipulate the transmission light into the beam focusing with the subwavelength scale, just by filling the output space with some dielectric media such as semiconductors, oil, glycerin and liquid mixtures. Such kind of simple processing method will have potential applications in biological sensing , lithography , and nano integrated devices [26,27].
This work was supported by the National Science Foundation of China (grant no. 11274184), Tianjin National Natural Science Foundation (grant nos. 10JCZDGX35100 and 12JCZDJC20200) and Research Fund for the Doctoral Program of Higher Education of China (grant no. 20120031110032). The authors thank Philippe Lalanne and Haitao Liu for their helpful discussion.
References and links
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23. For a lineup arrangement of N point sources with a between spacing of d, the constructive interference of their emitted waves requires that the maximum phase difference among them should satisfy the relationship of , where k is the wavevector of the propagating wave, f is the distance along the central perpendicular of the source plane, n is an integer. From the above mentioned formula, we can derive f expression to represent the spatial position of the focal spot.