## Abstract

We present a detailed investigation of the effect of lens size on the focusing performance of plasmonic lenses based on metallic nanoslit arrays with variable widths. The performance parameters considered include the focal length, depth of focus (DOF), full-width half-maximum (FWHM) and the maximum intensity of the focal point. 2D FDTD simulation was utilized. The results show that all the lens parameters are greatly affected by the lens size. A larger lens size, with a total phase difference of at least 2π, will produce a better focusing behavior and a closer agreement with the design. The Fresnel number and diffraction theory can be used to explain the effect of lens size. Suggestions are provided for realization of a practical plasmonic lens using the existing nanofabrication techniques.

© 2011 OSA

## 1. Introduction

Since the first discovery of extraordinary optical transmission (EOT) through metallic nanohole arrays [1], surface plasmons (SPs) or plasmonics, which exploits the unique optical properties of metallic nanostructures to route and manipulate light at nanometer length scales, has attracted considerable interest in the scientific community [2–6]. As an important category of plasmonic devices, plasmonic lenses based on thin metallic films have been proposed, designed and developed [7–14], as an alternative to the conventional dielectric-based refractive lenses, but enabling superfocusing beyond the diffraction limit and realizing all-optical or opto-electronic single-chip systems. They reveal great potential for applications ranging from ultrahigh-resolution imaging, single-molecular biosensing, optical data storage, to nanolithography. Though various design principles (such as corrugation structures [7,10], variable depths [8,10] / widths [9,13] / periods [11] / geometries [12,14], and materials [15,16]) have already been reported to implement the focusing capability of plasmonic lenses, most research is at an early stage. A number of basic technical problems still remain to be solved, concerning the design, fabrication and characterization of a practical plasmonic lens.

We consider here plasmonic lenses based on the metallic nanoslit arrays with variable widths. As we can see from a survey of the literature, this type of plasmonic lens usually shows a large difference in focal length between the theoretical design, numerical simulation, and experimental measurement (see Table 1 ). The latter two show a better agreement with each other, according to the one experimental result known to the authors [13]. Although some previous reports mentioned that such a large difference was mainly caused by the lens size, where “size” refers to the lateral extent of the structure along the width direction of nanoslits, no pertinent research and reasonable suggestions have been found to address this problem.

Therefore, we first investigate in detail the influence of lens size on the focusing performance of plasmonic lenses, including focal length, full-width half-maximum (FWHM), depth of focus (DOF), and maximum intensity of the focal point. Based on the results achieved by MEEP [17], a publically-available software package to perform electromagnetic simulations by the finite-difference time-domain (FDTD) method, some suggestions for the optimized design of a practical plasmonic lens to achieve the required focal length are provided. Two factors are taken into account: the effect of lens size and the practical fabrication capability. The conclusions we derive here are also applicable to other types of plasmonic lenses, since the size effect is a general problem.

## 2. Design principle

For the plasmonic lenses considered here, a single nanoslit surrounded by metallic walls acting as a metal-insulator-metal (MIM) waveguide, as illustrated in the inset of Fig. 1
, forms the basic compositional element. When a transverse-magnetic (TM) polarized plane wave (parallel to the width direction) is incident upon such a structure, surface plasmon polaritons (SPPs) will be excited at the slit entrance. Then, SPPs propagate inside the slit in specific waveguide modes until reaching the exit where they radiate into free space again. SPPs function as a form of slow light inside the slit [5], thus inducing a phase retardation which is very sensitive to the slit width.It is a complicated problem to accurately describe the SPPs’ excitation, propagation and coupling in a MIM plasmonic waveguide. However, if the metallic wall between any two adjacent slits is larger than the metal’s skin depth, ~25 nm for gold at visible wavelengths [18], and the slit width is much smaller than the operating wavelength, it is reasonable to consider only the fundamental mode in the slit [19]. This hypothesis had been applied to design the plasmonic lenses based on metallic nanoslit arrays with variable widths [9]. Accordingly, the complex propagation constant *β* is determined by the following dispersion relationship,

*k*

_{0}is the wave vector of light in free space,

*ε*is the permittivity of the metal and

_{m}*w*is the slit width.

Figure 1 gives the calculated results of propagation constant *β* for different slit widths using gold at a wavelength of 650 nm. The permittivity of gold at this wavelength is *ε _{m}* = −12.8915 + 1.2044i, obtained by using the fitted Drude-Lorentz model from [20]. The real and imaginary parts of

*β*represent the phase velocity and propagation loss of SPPs inside the metallic slit, respectively. The phase retardation, important for realizing a plasmonic lens, is primarily defined by the real part, expressed as Re(

*βd*), where

*d*is the thickness of the metallic film.

According to geometric optics, in order to realize a focusing effect, the plasmonic lens based on a gold nanoslit array (as shown in Fig. 2
) must be designed by defining a specific slit width for each nanoslit, so that a curved wavefront is generated, and the waves emerging from the slits are in phase at the focal point. The required phase retardation as a function of distance *y* from the lens’ central line (*x*-axis) can be calculated as

*λ*is the wavelength and

*n*is an integer. For the work presented here, the thickness

*d*of the gold film was 450 nm, the wavelength

*λ*was 650 nm, and the desired focal lengths

*f*were 3 µm and 5 µm.

Therefore, the required phase retardation can be obtained from Eq. (2) and varies with *y* as shown in Fig. 3
, where the solid line is for *f* = 3 µm (*f*_3) and the dashed line for *f* = 5 µm (*f*_5). Also shown in Fig. 3 are the discrete positions of nanoslits (triangles for *f*_3 and squares for *f*_5) with appropriate widths to approximate the required phase curves. To investigate the effect of lens size on the focusing performance of the designed plasmonic lens, four different cases for each focal length were taken into account, described in Fig. 3 as cases *a*, *b*, *c*, and *d* for *f*_3 with a total phase difference of 0.6π, 1.28π, 1.94π, and 2.51π, corresponding to a lens size of 2.28, 3.31, 4.13, and 4.78 µm, respectively, and cases *a’*, *b’*, *c’*, and *d’* for *f*_5 with a total phase difference of 0.58π, 1.59π, 2.58π, and 4π, corresponding to a lens size of 2.99, 4.72, 6.19, and 7.66 µm, respectively.

## 3. Simulation results and discussions

All the designed plasmonic lenses were verified by employing the 2D FDTD simulation, assuming the slit length to be infinite, which is reasonable for a slit length larger than 15 µm [21]. A TM-polarized incident plane wave was defined by setting the electric field component *Ey*. The grid size was set to 2 nm in both *x*- and *y*-direction for *f*_3 and 5 nm for *f*_5. The Drude-Lorentz model was utilized to accurately model the dielectric function of gold. A single simulation took about 5 hours for cases *d* and *d’* on a personal computer (CPU: Dual-Core, 2.7 GHz; RAM: 4 GB).

Figure 4
shows representative simulation results for the plasmonic lenses of *f*_3, from which we note the clear focusing of the wave. To show the focal point clearly, the electric-field intensity inside the nanoslits is saturated. For cases *a* and *d*, when the lens size increases from 2.28 to 4.78 µm, the focal point shifts away from the lens structure, and the maximum intensity also increases. Both Figs. 4(a) and 4(c) show a regular curved wavefront as designed, by which a focal point can be formed. On the other hand, as illustrated in Figs. 4(e) and 4(f), we can further quantitatively analyze the specific performance parameters, such as the focal length, DOF and FWHM, using the A-A and B-B cross-sections (see Fig. 4(d)) of the electric-field intensity pattern at the focal point. For the case of *f*_5, the similar simulation results are obtained. The derived results for all the cases are given and compared in Tables 2
and 3
, for the designed focal length of 3 µm and 5 µm, respectively. We can see from Table 2 that, as the lens size increases with the total phase difference from 0.6π (case *a*) to 2.51π (case *d*), the focal length increases from 1.820 µm to 2.806 µm, approaching the designed value of 3 µm gradually with an ultimate deviation of ~6.5%. When the total phase difference increases further to 4π (case *d’*), as indicated in Table 3 for *f*_5, the achieved focal length will be much closer to the designed 5 µm, with a difference of less than 1%.

On the other hand, when the lens size increases from case *a* (*a’*) to case *d* (*d’*), the maximum intensity of the focal point will also be enhanced greatly, while both the FWHM and DOF decrease. Taken together, these results reveal that a more concentrated focal point, or a better focusing capability, can be achieved for an increased lens size. As a result, we propose that, for the design of a plasmonic lens based on a metallic nanoslit array with variable widths, the total phase difference of the nanoslits should be at least 2π to yield the required focal length, leading to a deviation between theoretical and simulated behavior of less than 13.3% for *f*_3 and 15.7% for *f*_5, as indicated by our results.

## 4. Discussions

Although the large effect of the lens size on the ultimate focusing performance of the designed plasmonic lenses is demonstrated above, as mentioned by other researchers [13,22,23], the underlying mechanism is still not clearly understood. According to the design principle of the lens, derived from geometric optics as described in **SECTION 2**, all the cases for either *f*_3 or *f*_5 should present the same focal length, because they are derived from the same curve of phase retardation. However, as is the case for conventional microlenses, diffraction effect may also play a significant role in the ultimate focusing behavior, especially for the cases with small lens sizes [24].

As employed in [24], a useful parameter to characterize the role of diffraction is the Fresnel number *FN*, defined as

*λ*is the wavelength,

*ρ*is the radius of the lens, and

*f*

_{0}is the designed focal length. For a large

*FN*, geometric optics is suitable to realize the focal length desired. However, for a small

*FN*, the focal length will be shifted towards the lens due to the influence of diffraction at the lens stop. In this case, diffraction theory should be employed to solve the position of the focal point. When a plane wave radiates on a circular aperture with radius

*ρ*, the output intensity

*I*along the optical axis can be approximately calculated using the Rayleigh-Sommerfeld integral, which giveswhere

*I*

_{0}is the initial intensity of the wave and

*Z*is the distance from the aperture. Accordingly, the maximum intensity, indicating the position of the focal point, can be derived at a position

*Z*as Equation (5) implies that for a small

_{m}*FN*, the focal length is mainly determined by the radius of the aperture and the operating wavelength, and it is not sensitive to how the aperture is composed of, i.e. the shape, number, or specific arrangement of sub-apertures. For the plasmonic lens designed in this paper,

*ρ*was taken to be a half of the lens size.

Based on the above analysis, if we just consider diffraction theory, the focal length and *FN* for all the cases can be calculated, as listed in Table 4
. When the *FN* is less than 1, as presented for cases *a* and *a’*, the focal lengths calculated by diffraction theory are much closer to the simulation results (see Tables 2 and 3) than the focal lengths designed by geometric optics, which reveals that the diffraction effect dominates the ultimate focusing in such circumstances. Then, as the lens size increases, *FN* becomes larger than 1, and all the focal lengths calculated by diffraction theory are much larger than the simulation results. On the contrary, the simulation results gradually approach the focal lengths designed by geometric optics, and the larger the *FN*, the closer they are (for cases *d* and *d’*), which indicates that geometric optics dominates the focusing eventually. For cases in between, both the diffraction theory and geometric optics function. It is astonishing that the design rules of conventional microlenses can also be applied to plasmonic lenses. However, as the plasmonic lenses are composed of an array of metallic nanostructures, different from the conventional dielectric-based microlenses, discrepancies also exist. For example, for conventional microlenses, the *FN* to make the geometric optics dominant is 35.7, far larger than 1, to make a 2% mismatch as shown in [24]. In our research, for case *d’*, a *FN* of 4.514 can result in a mismatch of less than 1%. We thus see that, for a practical plasmonic lens based on a metallic nanoslit array with variable widths, a larger total phase difference (> 2π) is advantageous. The resulting designs are suitable for long operating wavelengths, e.g. THz and millimeter wave, as the fabrication techniques available are sufficient to realize the minimum width of nanoslits, e.g. tens of micrometers as in [23]. However, when operated in the optical regime, advanced nanofabrication techniques should be utilized, typically focused ion beam (FIB) milling or electron bean lithography (EBL). Considering practical fabrication capabilities, it is still challenging to fabricate a nanoslit with a width as small as 12 nm or 15 nm.

To circumvent this limitation, another plasmonic lens (case *e*) for *f*_3 with exactly the same lens size of 4.78 µm as in case *d* was designed, but only consisting of nanoslits of widths larger than 20 nm. The simulation results are shown in Fig. 5
, and a clear focusing of the wave can be still observed. Compared with case *d*, the focal length shifts slightly, to 2.874 µm (a deviation of 4.2% from the designed value). Meanwhile, both the FWHM and DOF also change slightly, but the maximum intensity of the focal point is decreased from 0.5224 to 0.3233, as two thick metallic walls of 652 nm in width exist to replace the nanoslits of widths less than 20 nm. To enhance the light intensity, we can also change the slit width to 20 nm for all the nanoslits of widths less than 20 nm, the simulation results (denoted as case *f*) are given in Fig. 6
. The achieved maximum intensity of the focal point is now 0.3609. Thus, we can see that plasmonic lenses may also be fabricated by using the lower resolution techniques, with some degradation in performance. Cases *e* and *f* demonstrate the concept, and further optimization can be made to improve the ultimate results.

Another important result is that, for a specific plasmonic lens, designed by the method described in **SECTION 2**, changing the film thickness while maintaining all the other structural parameters invariant can still focus the incident wave. Figure 7
shows some representative FDTD simulation results of the electric-field intensity for the plasmonic lenses with different film thickness based on cases *d* and *f*, and Tables 5
and 6
give detailed performance parameters of the achieved focal points. For the film thickness of 200 nm, the FWHM and the maximum intensity of the focal point are slightly degraded. For film thicknesses of 350 nm and 400 nm (cases *h*, *i*, *l* and *m*), the maximum intensity increases compared with the original case. This phenomenon may be due to the enhanced EOT for these two thicknesses, and a detailed technical investigation of the role of the metallic film thickness on the enhanced EOT can be found in [25].

This result suggests another possibility for the fabrication of a practical plasmonic lens with reduced demands on the nanofabrication process. For the purpose, we designed another plasmonic lens of *f_3* (case *p*), similar to case *e*, with a total phase difference of 2.51π. Its original thickness was 400 nm, and the minimum width of the composed nanoslits was 40 nm, increasing in steps of at least 10 nm, compared to 2 nm for case *e*. Then, in a modified version (case *q*), a film thickness of 200 nm was also analyzed. In this latter case, the maximum aspect ratio (depth-to-width ratio) of the structure was reduced to 5, which is practical for FIB milling as utilized in [13].

Figure 8
presents the FDTD simulation results of the electric-field intensity for cases *p* and *q*, and Table 7
gives the detailed performance parameters derived. The focusing effects can be observed for both cases. Compared with case *e*, the focal lengths are slightly decreased, with a deviation of 13.5% and 11.7%, respectively, from the designed value of 3 µm. The maximum intensity of the focal point is increased, and the FWHM is somewhat decreased.

## 4. Conclusion

We have undertaken a detailed investigation of the effect of lens size on the focusing performance of plasmonic lenses based on metallic nanoslit arrays with variable widths. The results show that performance parameters, such as focal length, FWHM, DOF, and maximum intensity of the focal point, are all significantly affected by the lens size. The larger the lens size, the better the quality of the focal point, and the closer the agreement between design and simulation. We find that reducing the metallic film thickness of a plasmonic lens allows fabrication with reduced demands on established nanofabrication techniques.

These analyses were based on the results from the 2D FDTD simulation. However, other factors, such as the finite slit length and the spacing between two adjacent slits, can also influence the ultimate focusing behavior. A more accurate investigation can be made with the 3D FDTD simulation, albeit at a cost of enormously increased computing time.

## Acknowledgment

This work was supported in part by the Postdoctoral Research Fellowship from the Alexander von Humboldt Foundation, Germany.

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