## Abstract

We propose a structureless method for focusing surface plasmon polaritons (SPPs) on a flat metal film under illumination of radially polarized cogwheel-like structured light beams. Without metal structures, the locally induced SPPs can further be propagated following the predefined patterns to form symmetric focal spots with dimensions beyond diffraction limit. Benefiting from the radial polarization, this method can be employed to pattern various center-symmetric evanescent distributions for generating SPPs reconfigurably. The SPPs will be propagating and focusing in radial directions.

©2010 Optical Society of America

## 1. Introduction

Surface plasmon polaritions (SPPs) [1] were highly patterned and confined near the structured metal surface and intrinsically localized in a small volume in many previous works [2]. Recently, study of focusing SPPs has attracted great attention due to the important focusing properties. Tightly focused laser beams and appropriate plasmonic lens such as nanometer-sized protrusions or holes have been applied for focusing SPPs [3–7]. As pioneering work, SPPs excited by a radially polarized beam created an evanescent Bessel-like standing wave on the axially symmetric plasmonic structure. The degrees of freedom for dynamic SPPs patterns were also an important point in recent research on SPPs focusing. The focused beam technique of localizing and controlling SPPs are potentially utilized for various applications such as optical probe for high-resolution imaging, sensing, and optical tweezers [8,9].

Complementary to the patterned metal surface technique, in this paper, we propose a structureless method of focusing dynamic SPPs patterns on a homogeneous gold (Au) metal film by a highly focused radially polarized structured light, “cogwheel” beam. The structured light with modified wave front (amplitude or phase) offers new opportunities for subwavelength resolution nanooptics [10]. The proposed model was described in detail based on the focused beam technique. We theoretically discuss the physics involved in the beyond-diffraction-limit focusing phenomenon and the unique characteristics of dynamic focused SPPs patterns using the vectorial diffraction theory. By performing the 3D-FDTD simulations, the proposed focusing model was verified by comparing the difference of field distributions with and without the metal film.

## 2. Generation and confinement of focused SPPs by structured light

Solving the Maxwell’s equations under the appropriate boundary conditions yields the SPPs dispersion relation, i.e. the frequency-dependent SPPs wave-vector,

*ω*is the frequency of incident light. Surface plasmon resonance (SPR) excitation occurs when the wave vector of the evanescent wave

*k*

_{light}matches that of the SPPs, which can be expressed as:

*θ*

_{sp}is the resonance angle, ${\epsilon}_{\text{d}}$is the dielectric constant of the targeted medium. In this work, a 50 nm thickness Au film (${\epsilon}_{m}=-5.588+\text{2}.\text{215i}$ at 532 nm) was deposited on the surface of glass substrate (

*n*

_{2}= 1.5). The medium above Au layer was chosen to be air (

*n*

_{1}= 1). The gap between the objective lens and the glass cover slip was filled with an index matching immersion oil (

*n*

_{2}= 1.5). The SPR angle

*θ*

_{sp}is about 47°.

In this paper, the focused SPPs patterns are generated by a radially polarized “cogwheel” beam. The structured light, “cogwheel” beam is the collinear superposition mode of Laguerre-Gaussian (LG) modes described by [11,12]

where |*u*

_{m}> and |

*u*

_{n}> are the fields of LG

_{0m}and LG

_{0n}modes and

*φ*’ is the relative phase of the interfering modes. The optical vortices mode with a phase singularity exp (-

*ilφ*) has a helical wave front, where

*l*is named as the topological charge. The “cogwheel” mode is a superposition of two vortices with equal but opposite topological charges (

*l*=

*m*= -

*n*), resulting in a periodic intensity modulation around the ring circumference with a large number of intensity maxima corresponding to 2

*l*and can propagate without changing the form in space.

In our proposed model as shown in Fig. 1
, the SPPs with complex structures can be engineered on the Au surface with a focused, radially polarized “cogwheel” beam, instead of protrusions or holes patterned on the metal film. The radial-polarized “cogwheel” beam can be produced by passing a radially polarized beam through an azimuthally modulated phase mask or a computer-generated hologram [11,12]. To a given topological charge, the radius of “cogwheel” ring can be modulated to satisfy the SPR angle requirement [13]. When the normal incident “cogwheel” beam converges toward the geometric focus, it gives rise to many diffraction-limited spots in “cogwheel” form, which contain large spectrum of wavevectors limited by the numerical aperture (NA = 1.49) of the lens. At resonant angles ± *θ*_{sp}, the wavevector-matched SPP waves can be generated and propagated towards the geometric center at all azimuthal directions, and the counterpropagating SPP waves in opposite directions interfere with each other to form the SPP standing waves. The focused incident beam is entirely *p*-polarized, thus providing cylindrical symmetric focusing. The SPPs interference patterns in the shape of “cogwheel” beam are excited at the focal plane of Au surface. Existence of these counterpropagating waves was recently shown by recording the fluorescence from single fluorophores by epiillumination scanning confocal microscope [14] and using direct measurement method aperture-type scanning near-field optical microscopy [7].

## 3. Intensity distribution of dynamic focused SPPs patterns

In a high NA system, a full vectorial formulation is derived by using vectorial Debye integral of Richards and Wolf [15,16], which inspired follow-up work in the study of high NA focusing property of doughnut beams, Bessel beam, and cylindrical vector beams in the past [17,18]. We investigate the focusing property of a monochromatic radially polarized “cogwheel” beam. The diffracted field distribution near the focal plane above the metal film is expressed by [4,17,18]:

*l*=

*m*= -

*n*, from Eq. (3) we can derive ${E}_{0}(\theta ,\varphi )\propto (\mathrm{exp}(il\varphi )+\mathrm{exp}(-il\varphi ))RD(l,\theta )=\mathrm{cos}(l\varphi )RD(l,\theta )$,

*RD*(

*l*,

*θ*) represents the radial distribution of “cogwheel” mode.

*r*,

*ψ*, and

*z*are the cylindrical coordinates of an observation points.

*k*

_{1}and

*k*

_{2}are the wave vectors in media above and below the Au layer.

*φ*is the angle with respect to the polarization direction. The upper and lower limits of the integral angle

*θ*denote NA of the objective lens${\theta}_{\text{max}}={\text{sin}}^{-\text{1}}(\text{NA})$and the critical angle$({\mathrm{sin}}^{-1}(\sqrt{{\epsilon}_{1}/{\epsilon}_{2}}))$, respectively. ${t}^{p}$is the transmission coefficient of the

*p*-polarized light for three-layer configurations, and can be define as [1]

Here, *d* is the thickness of the Au thin film,${t}_{ij}^{p}$, ${t}_{jk}^{p}$, ${r}_{ij}^{p}$, ${r}_{jk}^{p}$ are the transmission and reflection Fresnel coefficients for the respective two layer interfaces, *i* = glass/index-matching material, *j* = Au and *k* = air. ${k}_{zj}$ is the longitudinal wave vector.

According to the Eq. (4) and Eq. (5), the electric field distribution above the Au layer, which takes into account of transmission factors for glass/Au/air configuration, can further be expressed as

Integrations over *φ* have been accomplished using the identity

*(x) is the Bessel function of the first kind with order*

_{l}*l*. Under the SPR condition,${k}_{1}^{2}-{({k}_{2}\mathrm{sin}{\theta}_{sp})}^{2}<0$ thus it indicates the radial and longitudinal components become evanescent along the

*z*direction.

Intensity distribution of the focused SPPs patterns is proportional to the modulus squared of Eq. (6). The symmetric SPPs field can be focused and structured from all azimuthal directions above the flat metal layer and its shape is perfectly maintained with incident “cogwheel” beam due to cylindrical symmetry polarization as shown in Fig. 2(a)
. The number of SPPs spots is different depending on the topological charges. For *l* = 1 and *l* = 2 as examples, there are 2*l* SPPs spots periodically arranging on the primary intensity ring circumference. Due to the dynamic character of optical vortex, it is only necessary to modulate phase values of the incident beam for desired SPPs landscapes. The fringes period is measured as 240 nm ± 5 nm originating from the interference of SPPs with opposite wavevectors, in good agreement with half wavelength of SPPs (λ_{sp}/2≈241 nm) by solving the Maxwell’s equations [Eq. (1)] under appropriate boundary conditions. FWHM of the SPPs spot on the primary intensity ring is calculated as being 0.3*λ*_{0}. The peak intensity of |*E*_{z}|^{2} component is about six times stronger than |*E*_{r}|^{2} component, and plays a major role in shaping the overall intensity distribution. The reason is that an angular selectivity near SPR angle is made after passing through the metal film via SPR excitation, and |*E*_{z}|^{2} component is predominantly enhanced due to the field enhancement effect compared to |*E*_{r}|^{2} component.

The field strength transverse profiles are plotted in Fig. 2(b). In the previous works on evanescent zeroth-order Bessel beam excited by radial polarized beam, SPP wave is focused into an intense spot on the focal plane [4,5]. Whereas, in our case the focused SPPs field can be modified into the special patterns and corresponds to non-zeroth order Bessel function J* _{l}*(

*k*

_{2}

*r*sin

*θ*

_{sp}) derived by Eq. (7). The longitudinal component

*E*

_{z}with topological charge

*l*mainly contributing to the SPPs field distribution is approximately proportional to J

*(*

_{l}*k*

_{2}

*r*sin

*θ*

_{sp}). It also can be seen from Fig. 2(b) that the focused SPPs patterns give zero field intensity in the central region except for

*l*= 1, this is because in Eq. (7)

*J*

_{l}_{+1}(0) = 0 and

*J*

_{l}_{-1}(0) = 0 for all integer

*l*except for

*J*

_{0}(0) = 1 when

*l*= 1. From Eq. (6), it is seen that the focused SPPs components with the phase factor cos(

*lψ*) retain the phase characteristic of “cogwheel” mode formed by two vortex beams with the same topological charge but opposite helicity, resulting in cosine-dependent intensity distribution (2

*l*hotspots) in the azimuthal direction.

In the same focusing condition, the field distributions under linear polarization illumination are inhomogeneous and much weaker as shown in Fig. 3
[19]. Since SPPs can only be excited by *p*-polarized incident beam, the SPPs patterns appear in only one direction and the homogeneous SPPs spots cannot be obtained. So complementary to the linear-polarization, the constructive interference launched by a certain radial-polarized interfered vortex beam allows an SPPs to be focused into the evanescent interfered vortex beam and remain the phase characteristic of the incident beam. The concept presented in this paper would be extended to generate various plasmonic structured light patterns due to symmetric polarization. We can directly modify the phase structure to induce the change and rotation of SPPs patterns in a controlled manner continuously, which has more freedom than the method of patterning metal film.

## 4. Comparison of the field distributions with and without the metal

To confirm the formation of an effective focal region for SPPs focusing, we carried out FDTD simulations of SPPs propagation through flat metal film by using the FullWave module of commercial RSOFT software [20] and the results are shown in Fig. 4
. The field intensity upon the metal surface (*z*_{0} = 0) is strongly modulated [Fig. 4(a)] and the field pattern decays exponentially in the longitudinal direction [Fig. 4(b)], indicating that counterpropagating SPPs are excited. The counterpropagating SPP waves propagate towards the geometric centre and interference fringes are visible. As mentioned above, owing to the focused SPPs maintaining the singular phase character of optical vortex, the centre region is dark. The period of SPPs interference is measured as 240nm in good agreement with the vectorial diffraction theory results (Fig. 2). As depicted in Fig. 4(b), the total field strength (red curve) along the *z* axis above the Au film obtained from the Fig. 4(a) shows the field amplitude decreases exponentially as exp(−2|*k*_{z}||*z*|). The decay length is estimated at about 182 nm by finding the 1/e^{2} point.

We also evaluated the electric field intensity distributions along *xy* plane at *z* = 50 nm with or without the Au film. It is seen from Fig. 5(a)
that the SPPs spots are shaped like many concentric “cogwheel” rings of diminishing intensity. Figure 5(b) shows the focal intensity distribution at the glass/air interface without metal film. FWHM of the spot on the primary intensity ring without the metal film is about 0.6λ while with the metal film it is only 0.3λ which is beyond the conventional diffraction limit. The reason is that the electric field contains both propagating and evanescent components in the case without the metal film, the incident energy below the critical angle can entirely transmit, but with the metal film, the propagating part of the incident energy below the SPR angle is reflected back into the objective lens and is efficiently blocked. The excitation scheme can easily pattern each of the SPPs spots in an area approximately 0.09λ^{2} from a single beam and has potential applications in biological molecule manipulation and subwavelength optical sorting, since evanescent field including SPPs contains high spatial frequencies and reduced lateral size in comparison with propagating fields.

## 5. Conclusions

In conclusion, the structured light beams are versatile sources for generation of desired focused SPPs patterns confined beyond the diffraction limit, which no protrusion or hole is necessary. The focusing SPPs by “cogwheel” beam in radial polarization can be structured into the shape of concentric “cogwheel” rings at a planar dielectric-metal interface. We demonstrate that cogwheel-like structured light for dynamic SPPs array is a reconfigurable solution among the contenders. Benefiting from the symmetric polarization, the concept presented in this paper could be extended to generate various shapes of plasmonic structured beams through designing different phase masks, which may offer the new opportunity for the high resolution imaging microscopy and optical tweezers. The focused SPPs spots field gradient force for the dielectric particle will be our interest for investigation in future.

## Acknowledgements

This work was partially supported by the National Natural Science Foundation of China under Grant No. (10974101 and 60778045), Ministry of Science and Technology of China under Grant no. 2009DFA52300 for China-Singapore collaborations and National Research Foundation of Singapore under Grant No. NRF-G-CRP 2007-01.

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