There is considerable current interest in shaping the spatial dependence of the intensity of light transmitted through a metal film pierced by a 1D array of slits with subwavelength widths. Particular attention has been paid to the use of such structures as metallic nanoscale lenses. In contrast to conventional dielectric refractive lenses, whose focusing ability degrades as their dimensions decrease to the operational wavelength level, nanoscale slit lenses retain their focusing ability in this limit. Because of their planar geometries, they are simpler to fabricate at these dimensions than dielectric lenses, and use of metal rather than a dielectric provides a higher index contrast that leads to a stronger focusing capability. Finally, they focus light in the far rather than the near field, unlike, e.g., perfect (negative-index metamaterial) lenses.
However, it is not necessary to have slits that completely pierce the metallic film to achieve such focusing. A suitable modulation of the film thickness can produce the same effect. We have demonstrated this by calculating numerically the spatial distribution of the intensity of light transmitted through a metal film with surfaces modeled by two aligned, reversed arrays of nanogrooves of finite depth, sandwiched between vacuum and a dielectric substrate (see Figure 1). One of the motivations for adopting this structure was the opportunity it provides to examine the role played in its focusing capability by surface-plasmon polaritons propagating through the slits of a metallic lens, since such surface electromagnetic waves do not exist in these structures. However, the structure modeled does support coupled surface-plasmon polaritons propagating along the vacuum/metal-film and metal-film/dielectric interfaces. A second reason for using this structure is that it provides an additional degree of freedom in comparison with arrays of slits, namely the ability to change the shape of each nanogroove to improve the focusing capability of the resulting lens.
Figure 1.The structure studied. n: Index of refraction. x, z: Spatial directions. ζ(x): Surface profile function. L: Thickness of the unpatterned gold film.
Our setup consists of vacuum, metal (gold), and a dielectric at z>ζ(x), −L−ζ(x)<z<ζ(x), and z<−L−ζ(x), respectively. The surface profile function ζ(x) has the form . Here, Lis the thickness of the unpatterned gold film, 2M+ 1is the total number of grooves, t< L/2 the groove depth, and d the distance between the centers of consecutive grooves. The exponents controls the shape of each groove and (we adopted s =2 and 4), while bj=α+β |j|+ γ j2 (where α, β, and γ are constants) defines the width of each groove. The metal film was illuminated in the xz plane from either the vacuum or the dielectric side by light of p polarization and we calculated the spatial distribution of the squared modulus of the transmitted magnetic field, |Hy(x, z)|2, by a Green's function approach.
If a periodic nanogroove array is illuminated by p−polarized light, surface−plasmon polaritons are excited at the vacuum−metal and metal−dielectric interfaces with an efficiency that depends on the groove spacing, width, and depth, as well as the dielectric constants of the metal and dielectric.1,2 The resulting 2D patterns of near− and far−field intensity have been studied at different wavelengths for cases in which the width is the same for all grooves.2 The transmission through gold and silver films as a function of wavelength1 shows sharp peaks and dips associated with the resonant excitation of surface−plasmon polaritons with wavenumbers in the vicinity of the boundary of the second Brillouin zone.
If the groove pattern is only a few wavelengths in length, and the distance between adjacent grooves is considerably smaller than the wavelength of the incident light, the transmitted−intensity pattern exhibits a diffractive focus and has rotated or angled diffraction lobes,3,4 which can be used as an alternative to a dielectric refractive lens for imaging.3,5,6 Both p− and s−polarized fields exhibit a focus in this case, but because of the enhanced transmission mediated by surface−plasmon polaritons, the intensity of the transmitted p−polarized field is several orders of magnitude higher than that of the s−polarized field. The position of the maximum of the field intensity (the focal point) is determined by the diffraction by the aperture of the groove pattern, as long as the wavelength of light in the substrate is small enough to ensure that the focal point lies in the far field. Thus, there is no need to have slits that pierce the film and surface−plasmon polaritons that propagate through them across the film to achieve focusing of the transmitted field. The additional phases that these surface waves introduce do not play any significant role in forming the focused spot and do not determine the focal distance.
We undertook a simulation study to search for the best way to improve the quality of the focus by changing the groove profile and width variation. We found that varying the groove width quadratically while keeping the groove separation constant shortens the depth of focus, increases the field intensity at the focus, and slightly decreases the focal distance. Overlapping the grooves at the edges of the array (while keeping the depth of the array constant) further improves the quality of the focused spot: the transverse and longitudinal spot sizes decrease while the maximal field intensity increases.
The transmission of light through the array of grooves relies strongly on excitation of surface−plasmon polaritons and their conversion into transmitted light by the array. We found that when light is incident on the film from the vacuum and transmitted into the dielectric substrate, the focal distance is inversely proportional to the wavelength of the light in the dielectric substrate, in accordance with Fresnel diffraction by the aperture. Figure 2 shows the magnetic−field intensity |Hy(x, z)|2 for two wavelengths of incident light, λ = 630nm and 1μm. For shorter−wavelength incident light, surface−plasmon polaritons associated with the metal−filmdielectric−substrate interface, which have considerably shorter wavelengths than the vacuum wavelength, are scattered strongly by the surface structure. With the increase of the wavelength of the incident light, their scattering (as well as their excitation) by the same structure becomes weaker (see Figure 2).
Figure 2.Color−level plots of the transmitted magnetic field, |Hy(x, z)|2, in the dielectric substrate when light is incident from vacuum onto the gold film, showing the potential for wavelength−tunable focusing using these structures. The parameters of the surface profile function are L = 400, t = 196, and d = 250nm, s = 4, α = 40nm, β= 0, and γ = 0.98×10−3.
In the opposite geometry, when light is incident from the dielectric substrate onto the gold film and is transmitted into the vacuum, the wavelengths of the surface−plasmon polaritons on the gold−vacuum interface are very close to the vacuum wavelength, and their scattering (and excitation) by the same spatial structure is weaker. As a result, the intensity of the transmitted light is weaker, and practically wavelength independent. The longer wavelength of light in vacuum shifts the position of the maximum of the field intensity closer to the film surface, into the region where it becomes wavelength independent (see Figure 3).
Color−level plots of |Hy
in vacuum when light is incident from the dielectric onto the gold film, showing the potential for dispersiveless focusing using these structures. The parameters of the surface profile function are the same as those used in Figure 2
Planar nanolenses will have numerous applications in polarimetric imaging devices, solar cells, LEDs, and nanophotonics systems. When the total number of grooves in a metallic planar lens is increased, the transverse spot size approaches the diffraction limit, which could facilitate an even smaller pixel size for a photodetector focal−plane array to improve its responsivity and detectivity, as well as suppression of cross−talk between different pixels. In addition, the wavelength−tunable focal distance for normally incident light from the vacuum side provides us with a new approach for designing multicolor photodetectors. On the other hand, we find that the focal distance remains a constant for light emitted from the substrate with different wavelengths to ensure a dispersiveless strong focusing of light. In conclusion, our approach allows us to study the transmission of light by a finite array of grooves that constitutes the planar nanolens, as well as design the arrays to obtain desirable characteristics of planar nanolenses. In continuing work, we intend to seek designs of nanopatterned metallic films that produce focal spots with reduced longitudinal and transverse dimensions.
We would like to thank the Air Force Office of Scientific Research for support.
Danhong Huang, L. David Wellems
Air Force Research Laboratory Space Vehicles Directorate
Kirtland Air Force Base
Tamara A. Leskova, Alexei A. Maradudin
Department of Physics and Astronomy, University of California
3. L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, S. Fan, Planar lenses based on nanoscale slit arrays in a metallic film, Nano Lett. 9, pp. 235, 2009.