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Families of fractals are investigated as near-field aperture shapes. They are shown to have multiple transmission resonances associated with their multiple length scales. The higher iterations exhibit enhanced transmission, and spatial resolution exceeding the first order. Near-field enhancements of greater than 400 times the incident intensity and resolutions of better than λ/20 have been shown with apertures modeled after third iteration prefractals. Enhancements as large as 1011 have been shown, when compared with conventional square apertures that produce the same spot size. The effects of the complex permittivity values of the metal film are also addressed.
©2005 Optical Society of America
1. Introduction
Illumination through sub-wavelength apertures in opaque films is one method for achieving resolution smaller than the diffraction limit [1]. Many different methods have been used to overcome the rapid falloff of transmission with decreasing aperture size. Most of these methods rely either on engineering the shape of the aperture [2–5], or the spacing of an aperture array [6,7]. Both methods are able to greatly enhance sub-wavelength transmission, but due to their distinctly different mechanisms for eliciting this effect, they naturally lend themselves to different types of practical application. Single sub-wavelength apertures have been shown to collect light from an area no larger than (λ)2, and therefore benefit from diffraction limited illumination [8]. Aperture arrays, on the other hand, require several wavelength-sized periods of structure in order to operate optimally [9], and require a widely expanded incident beam. In many applications, diffraction limited illumination is not feasible. Conversely, many applications require high powers, so that illuminating an entire aperture array may be prohibitively inefficient. Therefore, a versatile structure, capable of collecting light from an area ranging from the diffraction limit out to several wavelengths would be a valuable asset in bridging the gap between single nano-apertures and aperture arrays. Fractals, by definition, are multifaceted structures that exhibit self-similarity in their geometric structure. Unlike Euclidean objects, they are defined by an iterative rule, rather than a formula [10]. Each successive iteration of this rule introduces a new length scale which provides an additional degree of freedom in engineering resonant structures. The self-similar, multi-resonant features of fractal geometries have been exploited in designing microwave antennas with broadband behavior, large gain, and sub-wavelength size [11,12]. They have also been used to make frequency selective surfaces with behavior analogous to photonic crystals, in a far more compact structure [13]. These results represent far-field microwave results, however, and do not address the issues of field confinement and enhancement which are of crucial importance in near-field optics. In the optical regime, random fractal aggregates of metal particles have been shown to produce the immense local fields needed for surface enhanced Raman spectroscopy (SERS) [14]. This is verification that fractal structures are able to produce a strong confinement and enhancement of the optical near-field. Due to their random nature, however, these structures cannot be engineered to have precise spectral properties, nor can the location of the enhancement be accurately controlled. Interestingly, several of the different aperture shapes used to obtain enhanced transmission very closely resemble the first or second iterations in different families of fractals [2,5,15]. Figure 1 shows the first few iterations of the Hilbert, Purina, Sierpinski Triangle and Sierpinski Carpet fractal families, respectively.
Fig. 1. First three iterations of a). Hilbert Curve, b). Purina Fractal, c). Sierpinski Triangle, and d). Sierpinski Carpet.
In this letter, we present numerical studies demonstrating that the fractal’s ability to simultaneously exploit both shape and spacing can be used with nano-apertures to increase near-field resolution and enhance transmission. By carefully choosing the fractal family and iteration, as well as including the dispersive properties of metals at optical and infrared frequencies, it is possible to design a structure that combines both the efficient collection and spectral control exhibited in the microwave regime with the highly localized near-field distributions of the fractal aggregates, all into one compact structure. Many of these fractal aperture structures are capable of outperforming many of the conventional optical nano-aperture shapes.
2. FDTD results
2.1 Spectral analysis
Finite Difference Time Domain electromagnetic modeling is used to examine the spectral behavior and to map out the transmitted near field profiles of the higher iterations of these families of fractals. This method has proven to be a useful and accurate tool in designing and predicting the behavior of sub-wavelength metal structures [2,16]. In the case of spectral characterization, an aperture can be viewed as a resonant structure. Its resonance peak is the wavelength at which its transmission function is highest. For two apertures of the same size, the aperture with the longer resonance wavelength will have greater resolution potential, because, due to the scalability of Maxwell’s equations, its size can be reduced to have the same resonance wavelength as the other, albeit at a smaller physical size. An excellent example of this is the C-aperture which has been shown to greatly outperform a square aperture of the same area and a square aperture modeled to produce the same near-field spot size, because its resonant wavelength occurs at a much longer wavelength [16]. In this study, the spectral transmission properties of the first three iterations of several families of fractals are modeled. An incident modulated Gaussian pulse with its spectral content spanning the visible and mid-IR wavelengths is used to probe the structures, and the Fourier transform of the time response of each aperture is used to calculate its spectral transmission properties or transmission efficiency function. For computational efficiency, this value is calculated as a running sum at discrete wavelengths as described in [17], and averaged at several common probe locations throughout the transmission region. It is normalized to the cross-sectional area of the aperture and the spectral power distribution of the incident pulse.
In each case, the minimum feature size of the aperture (d) is fixed at 140nm. These apertures are placed in an infinitesimally thin perfectly electrically conducting screen (PEC), in order to avoid probing thickness or material resonances. These effects are incorporated at a later stage of the design process and are addressed later in this paper. Figure 2 shows the results of these initial, broadband simulations. A notation is introduced here where (n,m) represents the mth order resonance of the nth fractal iteration.
Fig. 2. Caculated broadband spectral transmission efficiency of apertures modeled after the first three iterations of a). Hilbert Curve b). Purina Fractal, and c.) Sierpinski Carpet.
As the iteration number increases, two effects occur. First, a new resonance appears at a longer wavelength. Second, some of the resonances that exist for the earlier iterations become stronger or narrower (Δλ/λ). This allows the fractal structure to be of use in two regimes. In the case where the existing resonances are enhanced, the fractal structure can be used to increase transmission by collecting photons from a larger area. It is essentially a method of cascading similar apertures in an intelligent fashion in order to localize the fields to a central spot, while suppressing sidelobes. The degree to which this is achieved depends heavily on the fractal family that is chosen. This mode of operation is of particular interest where diffraction limited resolution is not possible, and one wants to collect the incident light in a more efficient manner. It is also feasible to operate at the new, higher order resonances that appear at longer wavelengths. These apertures can be rescaled to be resonant at a shorter wavelength, despite having a smaller physical size.
2.2 Near-field distribution
Simply decreasing the physical size of the aperture isn’t a sufficient condition for stating that it enhances near-field resolution. In order to quantitatively investigate the resolution of different aperture configurations, it is necessary to look at the transmitted near-field intensity patterns at resonance. Some of these families of fractals are very efficient in suppressing sidelobes, and are capable of producing a near field spot that is much smaller than the physical size of the aperture. Others, despite exhibiting enhanced transmission, distribute the transmitted light throughout the structure. Therefore, it is necessary to calculate the near-field distribution for each iteration, at each resonance order, to verify the confinement of the transmitted light. In order to make accurate comparisons of aperture performance, the apertures were rescaled to tune each of the resonances found in Fig. 2, to 1µm. This was achieved through an iterative process of narrowing the spectral range, increasing the resolution of the FDTD mesh, and introducing a finite 100nm film thickness. Once the dimensions are set, monochromatic excitation at 1µm is used to calculate the steady-state near-field distribution at a distance of d/2 away from the aperture, where d is the aperture’s minimum feature size. Figures 3(a) and 3(b) show the calculated near-field distribution for apertures modeled after the first two iterations of the Hilbert fractal at a distance of d/2 away from the metal surface. In each case the intensity gain was calculated at d/2, where d is the minimum feature size of the aperture. This distance was chosen because it marks the edge of the confined near-zone described by Leviatan, in which the fields remain largely collimated [18,19]. Beyond this region the fields rapidly diverge. Note that the aperture in Fig. 3(b) is a symmetric version of the C-aperture which has been shown numerically and experimentally to provide five to six orders of magnitude enhancements in transmission over conventional square apertures [16]. Figures 3(c) and 3(d) show the near-field distributions for the second and third order resonances of the third iteration Hilbert fractal aperture.
Fig. 3. Calculated electric field distributions of the first three iterations of the Hilbert fractal family at their resonant wavelengths. The second (c.) and third (d.) order resonance of the third iteration fractal aperture is shown.
It can be seen that the second order, due to its efficient suppression of sidelobes produces an improvement in resolution over the previous iteration (85.8nm×101.4nm vs. 121.0nm×132.0nm). The third order aperture, however, has a multi-lobed transmission mode, and does not confine light as well, but it produces a very strong enhancement in the average near-field intensity (441.0 vs. 87.5). The animation in Fig. 4 shows a pseudo-color plot of the time evolution of the field inside the aperture over several optical cycles. A vector plot of the pointing vector in that plane is overlayed to emphasize the power flow. Note that power circulates each of the C-shaped segments independently, but due to the self-similarity of their orientation, they effectively pump power into the central lobe. It is essentially cascading many apertures to collect photons from a larger area. However, due to the self-similarity of the assembly, the optical intensity remains well-confined.
Fig. 4. Animation of the transverse power flow at the input face of the second order resonance of the third iteration Hilbert fractal (3,2). The minimum feature size (d) is set to 54nm, and the wavelength is 1µm. Pseudocolor plot is of the relative intensity for unit input, and the vector flow diagram is of the in plane power flow (Sx,Sy). (2.29MB)
To operate at the highest order resonance, the aperture must be made smaller for it to be resonant at 1µm. Therefore, the minimum feature size decreases with each iteration. This provides enormous increases in near-field intensity, and also provides improved resolution if the structure suppresses sidelobes efficiently. This is not the case with the highest iteration of the Hilbert fractal, but other fractal families will be described later which do produce good confinement at the highest order resonance. Figure 5 shows a similar animation to the previous clip (Fig. 4). This animation shows the power flow of the highest order resonance of the third iteration of the Hilbert fractal aperture (3,3). Note that while the power circulated in individual units of the lower order Hilbert curve, in this higher order of operation, the power oscillates back and forth throughout the entire structure. This helps to elucidate the nature of these different resonances. The highest order resonance occurs when the whole structure is at resonance, and the lower order resonances occur when each successive sub-unit is at resonance.
Fig. 5. Animation of the transverse power flow at the input face of the third order resonance of the third iteration Hilbert fractal (3,3). The minimum feature size (d) is set to 18.5nm, and the wavelength is 1µm. Pseudocolor plot is of the relative intensity for unit input, and the vector flow diagram is of the in plane power flow (Sx,Sy). (2.12MB)
As many iterations of the different fractal rules occur, these different aperture shapes can become extremely different. Therefore, doing an exact, quantitative analysis and comparison depends to a large extent on the desired application and can be difficult. Two parameters that are of particular interest in most near-field applications, however, are the near-field spot size, and intensity enhancement. In several previous works [2,20], the quantity of transmission efficiency, or power throughput has been presented
Due to the fact that the near-field spot sizes of these apertures can be much smaller than their overall physical size, a more appropriate figure of merit is that of intensity gain, defined as,
which is essentially the average intensity enhancement within the near-field spot (FWHM). Also, due to the varying field distributions of the different fractal families and resonances, quantitative comparison of spot sizes is not entirely straightforward. For this reason confinement factor is defined as a figure of merit describing spot size,
where XFWHM and YFWHM are the full width half maximum sizes of the near-field spot in the x and y directions respectively. Figure 6 shows a distribution plot of the intensity gain and confinement factor for some of the higher fractal aperture iterations.
Fig. 6. Distribution of Igain and confinement factor values for apertures modeled after the second and third iterations of the Purina Fractal (diamonds), Sierpinski Carpet (squares), and Hilbert Curve(triangles). Square apertures (black line) providing the same spot size are shown for comparison.
It can be seen from this graph that the Hilbert fractal family produces the highest intensity gain of the shapes studied (Igain=441), and the Purina Fractal produces the smallest near-field spot size (FWHM=λ/20.04). Although much far-field work has been done in the microwave regime based on the Sierpinski Carpet structure [13,21], our studies show that this fractal family is not of particular use in the near-field. Its higher iterations are able to enhance transmission, but the structure is inefficient in suppressing side-lobes, and in all of the iterations and resonance orders investigated here, the intensity is widely distributed throughout the aperture. The solid line shown is the transmission value for conventional square apertures producing the same spot size as the fractal apertures. Comparing the two values shows the enhancement factor of the fractal aperture shapes which can reach 1011 at λ/20. Numerical models of a C-aperture have been reported which is able to achieve λ/10 resolution and a very large intensity gain of 138.5 [19]. It can be seen that this is a highly optimized version of the second order Hilbert curve, and similar methods can also be employed to further optimize the higher iteration fractal aperture shapes presented above.
2.3 Real metal analysis
Due to the broad treatment of the many different geometric resonances, and the dispersive behavior of metallic materials, the effects of metallic optical properties have been ignored until this point. A method has been presented for locating the multiple resonances of the fractal structures and determining their confinement capabilities. Once a proper fractal family and iteration has been chosen, as well as an operating wavelength, the optical properties of the metal must be taken into account in order to accurately determine the dimensions of the aperture and to assess the performance. A Drude model can be used to fit the complex permittivity function of a metal at optical and infrared frequencies. As a demonstration of this method, the second order, third iteration Hilbert fractal aperture (3,2) can be designed to be resonant for a 100nm thick Ag film at 1µm. A least squares fitting method has been used to fit the real and imaginary values of the permittivity function of Ag in the wavelength range of 750nm-1250nm to 0.2% error. The Drude parameters used are εinf=3.810, τc=8.96×10-15 s, and ωp=6.79×1015 rad/s for the long wavelength permittivity, relaxation time, and plasma frequency, respectively. An incident pulse simulation is first done to locate the new, shifted resonance of the aperture in the Ag film. The aperture is then rescaled accordingly, and a monochromatic illumination study is carried out. Figure 7 shows the near-field distribution at d/2 for this aperture in comparison with the ideal, lossless case.
Fig. 7. Near-field intensity distribution at resonance for the Hilbert (3,2) aperture in a). 100nm thick Ag film, and b). 100nm thick PEC film
The near-field distribution of the aperture in the silver film is quite similar to that of the PEC film, with a slightly higher background, and an approximately 50% lower intensity gain (31.3 vs. 61.6). This degradation in signal, however, is accompanied by a significant improvement in resolution (56.4nm×63.7nm vs. 83.8nm×101.4nm). This is due to the fact that incorporation of the optical constants of the silver film caused a red shift in the resonant wavelength of the aperture. This requires that the aperture must be made smaller in size than an aperture in a PEC film in order to be in resonance at 1µm. Also, note that since the size parameter d is reduced, so also is the distance at which these measurements are made (d/2). This example verifies that the non-idealities of real metals at optical and infrared frequencies can be taken into account to design a fractal aperture with high transmission for given experimental conditions.
3. Fractal choice
Numerous families of fractals exist, many of which can be used for enhanced near-field transmission. The fractal families studied here were chosen because they are well-known classical examples of fractals but they are by no means exhaustive. A family of fractals can be defined by a seed structure, and an iterative rule. Depending on the structure one starts with, and the rule one defines for successive iterations, widely varying structures can result. With so many options, it can be very difficult to choose which fractal family to use. Two properties are frequently used to describe a fractal’s geometry, which can be of use in choosing a fractal family. The first property used to describe fractals is fractal dimension defined as [22],
where N is the number of self-similar copies found in the next iteration, and 1/S is the factor to which the size of the copies is reduced. This number lies somewhere between 1 and 2, and essentially describes how efficiently a fractal fills space. In general, fractals with a higher fractal dimension, such as the Hilbert family (D=log(5)/log(7/3)=1.8995) will be more useful in enhancing transmission and producing intensity gain because they fill a large portion of space, and their scaling factor (S) is not as large. Fractals with a smaller fractal dimension such as the Purina family (D=log(5)/log(3)=1.465) tend to be sparser, branching structures, that can more efficiently suppress sidelobes and funnel light to a very localized spot. Also, due to the larger scaling factor, the minimum dimension size of the higher order resonances decreases rapidly with each iteration, giving structures of this type a further advantage in achieving improved resolution. A qualitative property often used to classify fractals is lacunarity. It is a measure of how much “open space” a fractal has [10]. Although its overall transmission might not be better, it is clear that a fractal with low lacunarity will be more efficient in localizing the transmitted light to a small near-field spot. The Sierpinski Carpet is a good example of a fractal family with high lacunarity that does not efficiently confine light within its structure. The Hilbert and Purina fractals, however, have low lacunarity, and are able to produce near-field spots that are much smaller than their overall physical size. The fractal families presented here are classical examples which appear exactly as described by theory. They are not yet optimized. Further work can still be done to vary or scale one or more of the dimensions to either tune the resonances to a desired range or to further optimize aperture performance. Deviation from the exact fractal structure to obtain specific spectral behavior has been demonstrated with fractal antennas for far field applications [21].
4. Conclusion
Fractal apertures provide two potential advantages over conventional near-field apertures. By increasing the resonance wavelength, while maintaining a small near-field spot size, these apertures are able to enhance resolution while being illuminated with diffraction limited beams. Also, by efficiently suppressing sidelobes, light can be collected from a larger area within the diffraction limited illumination beam and provide enhanced transmission into a more intense near-field spot. Depending on the application, the desired performance, and the fabrication limits, a proper fractal family and iteration number can be chosen. Apertures of low lacunarity are beneficial to use because they are able to tightly confine light in the near-field. Fractal based aperture shapes have been shown to produce near-field spot sizes smaller than λ/20, and enhancements of over 400 times the incident intensity. It is worth noting that these results represent preliminary results with classical fractal shapes. This class of structures introduces another degree of engineering freedom which allows for numerous methods of further optimization. With its ability to enhance transmission while confining light to sub-wavelength dimensions, the fractal aperture represents a new class of versatile, highly-efficient near-field probes with potential extensions and improvements on the use of nano-apertures for nano-lithography [23], data storage [5], and single molecule studies [24].
Acknowledgments
The authors would like to acknowledge the Wallenberg Global Learning Network for their support of this research.
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