## Abstract

We present the design of a plasmonic lens (PL) which is composed of pixelated nano-grooves on a gold film for the coupling and focusing of surface plasmon polaritons (SPPs) into multiple focal spots on the optical axis. The pixelated grooves are arranged along the *y*-axis and the *x*-position of each groove is optimized by the simulated annealing algorithm. PLs that implement two and three on-axis foci are presented and the designed structures have been validated with FDTD simulations. We also successfully constructed a long-focal-depth PL with a longitudinal FWHM of the focus that reached 25 plasmonic wavelengths, while its transverse field profile is maintained over 15 µm distance. The presented design method constitutes a new basis for plasmonic beam engineering, and the proposed particular SPP focal fields have potential applications in multiple imaging, particle manipulating, and plasmonic on-chip signal transmission.

© 2017 Optical Society of America

## 1. Introduction

Surface plasmon polaritons (SPPs) are combined excitations involving of coherently coupled photons and collective oscillating free electrons on the surface of a metal [1]. Because of their strong field confinement and enhancement, SPPs allow controlling of light in a nanoscale and lead to novel applications in photonics [2,3]. Various two-dimensional plasmonic optical components, such as reflectors [4,5], waveguides [6,7], interferometers [8,9] have been developed. Among these functional devices, plasmonic lens (PL), which converts large-scale incident light to focused plasmonic waves, has been extensively explored [10–13] since the obtained highly confined focal spot enables important applications including optical far/near-field conversion [14,15], super-resolution imaging [16], nanolithography [17], and near-field imaging and sensing [18]. So far, most existing plasmonic focusing devices demonstrate only one focus or deal with one image plane. In this work, we concentrate on the design of plasmonic lenses that can functionally excite and focus the plasmonic waves into multiple identical intensity spots along the optical axis. To our knowledge, such peculiar focusing properties of SPPs have not been reported in the literature.

In optics, several techniques have been proposed to generate multiple on-axis foci. Based on scalar approximations, diffractive optical elements can be used to focus light into a large number of discrete positions [19,20]. In the vectorial focusing regime the polarization state of a beam becomes a key factor, and a beam with radial-variant polarization or with spatially modulated radial polarization can be focused to produce two identical intensity spots along the optical axis [21,22]. Since the polarization of a SPP beam is inherently related to its propagation direction [1], we can only modulate the amplitude or/and phase of the excited wavelets to generate a desired pattern. In this paper, we proposed a plasmonic lens with spatially-modulated pixelated nan-grooves, which serve as local couplers to directly excite SPPs from illuminating light. The required phase front for focusing SPPs into multiple predefined longitudinal positions is encoded in *x*-positions of the grooves, which are optimized by simulated annealing algorithm [23].

## 2. Design theory

#### 2.1 Beaming structure and simulation method

The proposed PL with multiple on-axis foci consists of a series of pixelated nano-grooves along the *y*-axis fabricated on a gold film. As a unit cell, the dimensions of each pixelated groove dimensions along x and y directions are *w* = λ_{SPP}/2 and Δ*y*, respectively (see Fig. 1(a)). The *x*-position of each groove is chosen from *P* possible options of *x*_{n} = nλ_{SPP}/*P*, where integer *P* (*P* ≥2) denotes the modulation level corresponding to a phase step of 2π/*P*, n = 0,1,…, and *P*-1. When an x-polarized plane wave normally illuminates the PL, SPPs are excited at the grooves with the same initial phase but a groove-position-related phase difference, which provides a means of control over the phase fronts of the outgoing SPP waves. Therefore, the idea here is to modulate the phase front according to the needed field pattern by optimizing the x-position for all the grooves.

During the design process, it requires to repeatedly adjust the x-position of each pixelated groove and evaluate the resulting field distribution of target area, thus rapid calculations of plasmonic wavefields over extend two-dimensional region are prerequisite. In this study, an SPP point source model is used which assumes that every point on each pixelated groove functions as an SPP point source with a amplitude in proportion to the groove length Δ*y*. The radiation pattern of the plasmonic field has the following expression [24]:

**r**is the observation point,

**r**' the center position of the groove; A(

**r**') is a complex coefficient whose magnitude is proportional to the incident field amplitude at

**r**' and the phase distribution is the same as that of the incident light. For a normally incident plane-wave here, A(

**r**') is a constant;

*k*

_{SPP}the propagation constant of the SPPs at the air/gold interface;

*k*

_{z}the wave vector along the

*z*direction; and

*φ*

_{s}the azimuthal angle formed between the polarization direction ( +

*x*direction) and the vector

**r**-

**r**'. Then the total field is the vector sum of the contributions of all grooves. The propagation loss is already taken into account in Eq. (1) since

*k*

_{SPP}is a complex number. This model and its variants have been successfully applied to practice with very good accurate [25–27]. Compared to full-wave simulation methods (such as FDTD and FEM), the usage of this model significantly reduces time and computer memory required for simulation, thus provides the possibility for optimization with an iterative algorithm.

#### 2.2. Optimal design procedure

For a given length of *L* and a modulation level of *P*, a PL contains *L*/Δ*y* grooves with *P*^{(}^{L}^{/Δ}^{y}^{)} possible groove combinations. Here, we use a simulated annealing (SA) algorithm, which is a global optimization technique, to search for the very groove combination that produces the best approximation to the desired field pattern. The optimal procedure is similar to that used in [28] and is performed as follows: (1) Each pixelated groove is initiated by a random x-position of 0, -λ_{SPP}/*P*, −2λ_{SPP}/*P*,..., or -(*P*-1)λ_{SPP}/*P*, and the cost function *F*_{cost} is calculated, which has a general form as:

*I*

_{i}is the SPP intensity at the

*i*th focal point F

_{i};

*I*

_{total,i}the sum of the intensity of all the sampling points on the preset focal planes of F

_{i}; ${I}_{\text{aver}}=\frac{1}{M}{\displaystyle {\sum}_{i=1}^{M}{I}_{i}}$; and

*w*

_{i}the weight factor.

*I*

_{i}and

*I*

_{total,i}are evaluated with Eq. (1). The first term inside the bracket in Eq. (2) aims to achieve plasmonic focusing at designate locations, and the second term helps to obtain a uniform distribution of intensity among the

*M*focal spots. (2) The

*x*-position of a random groove is randomized to another allowed value, and the cost function

*F*

_{cost}

^{new}is evaluated. (3) If

*F*

_{cost}

^{old}<

*F*

_{cost}

^{new}, then the change is accepted; otherwise, the change is probabilistically accepted with a possibility specified by exp[(

*F*

_{cost}

^{old}-

*F*

_{cost}

^{new})/

*T*], where

*T*is a temperate parameter which is initially high and then slowly reduces as the algorithm runs. (4) For a given temperature

*T*, step (2)-(3) are repeated

*M*times, here

*M*= 1000, 500, and 250 for the case of

*T*≤ 0.01, 0.01<

*T*≤ 0.1, and 0.1<

*T*≤ 10, respectively. (5) Parameter

*T*is reduced to

*αT*(0<

*α*<1) and steps (2)-(3) are repeated with the new

*T*. (6) The algorithm terminates when no change is accepted at current temperature. Then the optimal solution is found. Note that

*T*plays a crucial role in controlling the evolution of the solution. For a large

*T*, the algorithm is likely to accept solutions that are worse than current solution. This probabilistic characteristic enables the algorithm to jump out of any local optimums. As

*T*is reduced slowly so is the chance of accepting worse solutions. Therefore, the search space is gradually pruned and converges to the optimum solution eventually. Throughout this work,

*T*is initially set to 10 and

*α*= 0.9.

## 3. Results and discussion

#### 3.1 PL with two on-axis foci

As the first demonstration, a PL with modulation level *P* = 2 is designed. The relevant parameters of the lens are as follows: The working wavelength λ is 830 nm, corresponding to plasmonic wavelength λ_{SPP} = 814 nm. The two preset foci F_{1} and F_{2} are located at (11µm, 0) and (17µm, 0), respectively. The length of the device L is 24µm and the groove size parameters w and Δ*y* are 407 and 300 nm, respectively. The PL is symmetric with respect to the *x*-axis; therefore there are 40 pixelated grooves that need to be optimized. Here, the cost function *F*_{cost} is specified with *w*_{1} = *w*_{2} = 1, and *I*_{total} is sampled over 61 points on the focal plane (from y = −18 to 18µm, sampling space = 600 nm). We run the SA algorithm 30 times and pick out the best structure, which is illustrated in Fig. 2(a). The corresponding longitudinal intensity profile of the optimized PL is presented in blue dotted lines in Fig. 2(c), where two intensity peaks are positioned at *x* = 10.9 and 16.4 µm, which coincides well with our objective.

To verify the performance of the designed PL, we utilize the three-dimensional finite-difference time-domain (FDTD) method to characterize the structure. In the FDTD simulation, pixelated grooves (groove depth = 40 nm) are etched on a 50 nm gold film which is sandwiched between air and a glass substrate. An *x*-polarized plane wave is used as the excitation source, which illuminates from the air side and covers the entire PL structure (inset of Fig. 1). The dielectric constants of gold and glass are set to ε_{gold} = −26.6 + 1.67i (interpreted from [29]) and 2.25, respectively. Perfectly matched layer absorbing boundary condition is used at all boundaries. The simulated FDTD intensity distribution |*E*|^{2} around the focal area is shown in Fig. 2(b), which clearly exhibits bifocusing of SPPs along the optical axis. Two intensity peaks are formed at (11.2 µm, 0) and (16.5 µm, 0) with nearly uniform peak intensity (intensity ratio = 1:0.98) as expected. The longitudinal (*y* direction) and transverse (*x* direction) intensity profiles through the foci are shown in Figs. 2(c) and 2(d), respectively, and both the intensity profiles are in excellent agreement with the SA results. The longitudinal and transverse FWHM are 660 and 2560 nm for focus F_{1}, and 673 and 3250 nm for focus F_{2}. The resultant field in the region x < 0 is a mirror image of that in the region x > 0 due to the symmetry, therefore there are another two foci on the –*x*-axis. The chromatic dispersion of the PL has also been studied by comprehensive investigation of the shape, intensity and focal length of the foci for excitation wavelengths ranging from 680 to 980 nm (in the step of 10 nm). Although the lens is designed at λ = 830 nm, it achieves a good on-axis bifocusing in the wavelength range of 780-960 nm, wherein the shapes and relative intensities of the foci are well maintained, and the transverse FWHM of the foci are kept as small as ~0.8λ (see Fig. 2(e)). Beyond the bandwidth of 780-960 nm, the shapes and intensities of the foci deteriorate severely. In particular, no prominent focusing phenomena are observed at the short wavelength end (not shown here). Figure 2(f) shows an inverse proportional dependence of the extracted focal lengths on the wavelength in the range of 780-960 nm.

#### 3.2 PL with three on-axis foci

A PL for producing three on-axis foci is also designed to manifest the feasibility of the proposed method. Three foci F_{1}, F_{2}, and F_{3} are set at (8µm, 0), (13µm, 0), and (18µm, 0), respectively. The cost function *F*_{cost} is tuned to be more sensitive to the intensity at the farther focus by setting *w*_{3} > *w*_{2} > *w*_{1} to balance the propagation loss of the SPPs as well as the reduction of the output numerical aperture (NA), defined as (λ/λ_{SPP})sin*θ*, where *θ* is the semi angle formed by the focal point and the most outer edges of the lens. Here, we use *w*_{1} = 2/3, *w*_{2} = 1 and *w*_{3} = 8/3. Other relevant parameters are the same with the PL of two on-axis foci. The structure of the optimized PL is sketched in Fig. 3(a). The FDTD simulated image shows three well-defined focal spots at (7.7µm, 0), (12.9µm, 0), and (16.8µm, 0) with a parasitic hot-spot near (6.0µm, 0), as seen in Fig. 3(b). The longitudinal and transverse intensity profiles through the foci are shown in Figs. 3(c) and 3(d), respectively. The intensity ratio of the three foci is 1:0.80:0.89. The longitudinal and transverse FWHM are 496 and 1500 nm for focus F_{1}, 604 and 2180 nm for focus F_{2}, and 673 and 2920 nm for focus F_{3}. The performance of the lens under different wavelengths is also evaluated by FDTD simulations, which reveal that the lens can only keep its performance in the wavelength range of 790-900 nm. Since the structure of this three-focus lens is more complex than that of the bifocusing lens, it is more sensitive to the working wavelength. Similar to the PL with two on-axis foci, the focal length of each focus is in inversely proportional to the incident wavelength.

#### 3.3 PL with an ultra-long focus

In principle, the preset number and the location of on-axis foci could be arbitrary prescribed with Eq. (2) as long as the optimum solution is included in the solution space, which is determined by the lens parameters of *P*, *L*, *w* and Δ*y*. This flexibility offers us a novel route to design an intensity controllable SPP beam. As we know, a SPP wave always suffers from large propagation loss and also undergoes diffractions in the plane of the metal/dielectric interface. To address this issue, nondiffracting SPP beam has been recently proposed [30–32]. To some extent, focusing SPPs with a preserved focal spot also can be regard as a particular kind of nondiffracting beam. Here, we propose a PL with an ultra-long focus which is established by allocating ten equidistant foci from *x* = 15 to 42 *µ*m, therefore focal point F_{i} (*i* = 1,2,3,…,10) is located at ((12 + 3*i*)*µ*m, 0). These intended foci are closely separated to enhance constructively interfere of focused beamlets near their vicinities, so as to result in an elongated focus instead of discrete spots. The weighting factors in the cost function are *w*_{1} = 1/3, *w*_{2} = *w*_{3} = *w*_{4} = *w*_{5} = 2/3, *w*_{6} = *w*_{7} = *w*_{8} = 1, and *w*_{9} = *w*_{10} = 4/3. The length of the lens L is 16 *µ*m, which is intentionally shortened to the advantage of longer focal extension as the result of reduction in NA. The modulation level *P* is set to 4 to achieve a refined phase modulation. Other relevant parameters are the same as previous cases. We run the SA algorithm 100 times and pick out the structure with a maximum longitudinal FWHM, which is illustrated in Fig. 4(a). To excite the SPPs effectively, another four layers with the same structures are placed after the first one to form a grating structure with period λ_{SPP} [33,34] and a groove-to-pitch ratio of 0.5 [35].

Subsequent FDTD simulation demonstrates formation of an ultral-long focus (see Fig. 4(b)). The longitudinal FWHM of the focus reaches as long as 20.4 *µ*m, which is about 25λ_{SPP} (see Fig. 4(c)). Moreover, transverse FWHM of the focus versus propagation distance are plot in Fig. 4(d), where the FWHM experiences an abrupt drop at *x* = 15 *µ*m (location of the first focus F_{1}) and maintains ~1.65 *µ*m within ~15 *µ*m distance till *x* = 30 *µ*m. More clearly, transverse intensity profiles at specific distances of *x* = 20 and 29 *µ*m are also shown in Figs. 4(e) and 4(f), respectively, manifesting a good preservation of the beam width. In this regard, the focused beam is able to work as a nondiffracting one within a certain range. It should be emphasized that, though this SPP beam has a nondiffracting appearance, it is not a real nondiffracting one but is a result of focusing to a line. The beam-produce mechanism totally differs from the case of the recently proposed cosine-Gauss plasmon beam formed by the interference of two intersecting SPP plane-waves [30,32,36,37]. However, the obtained ultra-long focus can mimic a particular cosine-Gauss beam in some extent. This could be verified by considering the evolution of a cosine–Gauss beam whose initial transverse profile (modeled by: cos^{2}[Re(*k*_{spp})•sin(*δ*)•*y*]•exp(−2*y*^{2}/*w*_{0}^{2}), here *δ* = 6.3° denotes half of the crossing angle between the two SPP plane-waves, *w*_{0} = 3*µ*m denotes the beam waist) is almost the same as that of the focus at *x* = 20 *µ*m (see Fig. 4(e)). The transverse profile of the beam at the propagation distance of 9 *µ*m (x = 29*µ*m) and the FWHM of the main lobe at varies *x*-positions (calculated by propagation integral) are shown in Figs. 4(f) and 4(d), respectively, which coincide well with the long-focus case.

## 4. Summary

To summarize, we have proposed the design of a plasmonic lens which is composed of pixelated nano-grooves on a gold film for coupling and focusing of SPPs into multiple focal spots on the optical axis. The constituent pixelated grooves are arranged along the *y*-axis and the x-position of each groove are optimized by simulated annealing algorithm. PLs that implement two and three on-axis foci are presented and the designed structures have been validated with FDTD simulations. We also successfully realize a PL with an ultra-long focus that can be considered as a diffraction-free beam within a given region. It is reasonably believed that the building-blocks of the lens can be coupling structures other than nano-grooves to achieve more complex functionalities, such as responding to a circularly polarized source by using polarization-sensitive elements. The presented design method constitutes a new basis for plasmonic beam engineering and the proposed lenses could find potential applications in multiple imaging, optical trapping, and plasmonic on-chip signal transmission.

## Funding

National Natural Science Foundation of China (61205051, 61275063); Natural Science Foundation of Fujian Province of China (2013J05097); Fundamental Research Funds for the Central Universities (20720150032).

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