Multi-focus plasmonic lens with metallic nanoslits of variant widths have great potential applications in optical interconnection, integrated optics and nanophotonics. But the design method with simulated annealing algorithm or Yang-Gu algorithm requires complex calculation and multi focuses are limited to be set on the same output plane. In this paper, we propose a design method based on holography. The desired light field distribution and the incident plane wave can be treated as object wave and reference wave, respectively. So the calculation is relative simple and multi focuses can be located in different output plane. Numerical simulation of multi-focus lens design is performed through finite-difference time-domain (FDTD) method and the result confirms the feasibility of our method.
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Surface plasmon polaritons (SPPs) are a special kind of electromagnetic wave propagating along the interface between the dielectric and noble metal with amplitude decaying exponentially into the space perpendicular to the surface and have maximum in the surface [1,2]. The unique properties of SPPs have shown its potential to develop miniaturized and integrated photonic devices that could manipulate photons in sub-wavelength scale. Plasmonic lens [3–5] which consists of an array of slits with variant widths in opaque metallic film for SPPs focusing or imaging  shows great convenience for miniaturization and integration.
Usually, plasmonic lens is designed with the equal optical path principle , which is effective to design one-focus lens. But this method is difficult to obtain the phase distribution of the lens with multi-focus which is important in coupling the incident plane wave to multi channels in optical interconnection. Most recently, Zhu et. al. reported their study of multi-focus plasmonic lens design by using simulated annealing (SA) algorithm  and Yang-Gu algorithm . To the best of our knowledge, this is the first work to introduce optimization algorithm into plasmonic lens design. However, these two methods require adjusting the width of each slit and then calculating the light field distribution at the lens plane and output plane again and again. Therefore, it is a tremendous and very time-consuming work and the focuses are limited to be set on the same output plane.
In this work, by introducing the concept of holography into the plasmons [9–12], we present a simple method to design multi-focus plasmonic lens. The desired output light field distribution and the incident plane wave are treated as object wave and reference wave, respectively. Then the amplitude distribution of interference light field is discarded and only the phase distribution is recorded. Finally the width of each nanoslit is determined according to this phase information. Compared with the existed design method, our method doesn’t need complex computation and iteration. Furthermore, multi focuses are not limited to be set on the same output plane. Illustrative design examples with two focuses and three focuses are given and simulated through FDTD method. The result shows that our method is effective and has potential application in designing plasmonic devices with sub-wavelength scale focuses.
In theory, holography can reconstruct arbitrary wave front through two steps: First, consider the desired wave front as object wave and record the interference pattern of object wave with reference wave. Second, reconstruct object wave by illuminating the interference pattern with the same reference wave. Based on this idea, desired light field distribution and incident plane light can be treated as object wave and reference wave respectively, during the multi-focus plasmonic lens design. The detailed of the design is shown in Fig. 1.
Figure 1(a) shows the record procedure of the hologram. Taking triple-focus plasmonic lens design as an example, the light field distribution formed by three point light sources in the positions where the focuses of the plasmonic lens should be, are considered as the object wave, and the incident plane wave is treated as the reference wave. Then the interference light field which is also called hologram can be derived. Usually, hologram only records the intensity distribution of interference light field, so the conjugate image and low diffraction efficiency could not be avoided. Kinoform, which is a special kind of hologram, regards the amplitude of interference light as constant and only records the phase distribution, can eliminate the conjugate image and get high diffraction efficiency close to 100% theoretically [13,14].
Figure 1(b) presents the schematic of multi-focus plasmonic lens where a metallic film is perforated with a great number of nanoslits with specially designed widths. For the particular structure of the plasmonic lens, modulating the phase and amplitude of light in each nanoslit at the same time is difficult. It shows that when the depth of nanoslit is chosen reasonably, the amplitude modulation of nanoslits with different widths can be neglected. So the plasmonic lens can be regarded as a pure-phase modulated element and the calculation procedure of phase distribution at the lens plane is equivalent to the record procedure of Kinoform.
The detailed phase derivation is described as following: the light field distribution at the lens plane is interference of the spherical waves originated from the focuses with the incident plane wave. The object wave U(x = 0, y) at the lens plane could be written as
Then we can fix the width of each nanoslit according to the phase distribution .When the TM polarized plane wave impinges on the metal surface, surface plasmon polaritons (SPPs) can be excited at each nanoslit entrance. For slit width is much smaller than the incident wavelength, it is reasonable to only consider the fundamental SPPs mode. And the complex propagation constant of it can be calculated from the equation
3. Design and result
To confirm our method, two kinds of plasmonic lenses to generate two focuses and three focuses have been designed. The incident wavelength is 650nm with TM-polarized and the permittivity of Ag used is −17.36 + 0.715i. The lens thickness d is 500nm, slit interspacing (center to center) is 100nm which is larger than the metal’s skin depth to prevent the interaction of light and each slit width is determined by the phase distribution. From Eq. (4), it can be known that slit width w ranges from 10nm to 70nm could ensure the phase modulation from 0 to 2π. In order to appraise the performance of multi-focus plasmonic lens, commercial software Lumerical FDTD solutions is used to numerically demonstrate the validity of the designed structure. A 2D simulation is chosen and the simulation region is bounded by perfect matched layer (PML) with uniform cell of Δx = Δy = 1nm.
First, plasmonic lens which could generate two focuses with equal intensity in the same output plane at Cartesian coordinates of (3um, 2um) and (3um, −2um) is designed and simulated. The diameter of the lens aperture is 10um and the simulation region of FDTD is 12um × 10um. Figure 2(a) shows the amplitude distribution of magnetic field Hz, through which we could see two focuses at Cartesian coordinates of (2.97um, 2.01um) and (2.97um, −2.01um) clearly which deviates from our objective very little. Figure 2(b) depicts the phase distribution of Hz. The change of concave wave front to convex wave front indicates the location of the focus which agrees well with the result in Fig. 2(a). The cross section of the generated focuses in the x direction is given in Fig. 2(c), indicating a full-width at half-maximum (FWHM) of 432nm which is smaller than the incident wavelength. The calculated slit widths distribution of the lens is shown in Fig. 2(d).
To further verify our method, plasmonic lens with three focuses of equal intensity in three different output plane at Cartesian coordinates of (6um,-5um), (7um, 0um) and (8um, 5um) is designed. Diameter of the plasmonic lens aperture is 20um and the calculated slit widths distribution in the Ag film is shown in Fig. 3(c). FDTD method is also utilized to character the structure with simulation region size 12um × 20um. The obtained amplitude distribution of Hz is presented in Fig. 3(a). It achieves three focuses at focal length of 6.07um, 7.07um and 8.09um, whose Cartesian coordinates are (6.07um, −5um), (7.07um, 0um) and (8.09, 5um) respectively. The phase distribution of Hz in Fig. 3(b) indicates the location of three focuses which agrees well with the result in Fig. 3(a). The FWHM of three focuses are 345nm, 355nm and 375nm which are all smaller than the FWHM in Fig. 2(c), for the size of lens aperture is increasing.
One may consider using the equal optical path length principle  to design multi-focus plasmonic lens: adopt single lens to achieve single focus and then combine several lenses side by side. For comparison, plasmonic lenses with two focal points at (3um,-2um) and (4um, 3um) which lie on different output planes and have asymmetrical y coordinate are designed by the equal path length principle and our method respectively. In both cases the lens aperture D is 10um and the simulation region is 10um × 12um.
Figure 4 presents the FDTD simulation results of the designed plasmonic lens by two methods. Two focuses can be observed clearly for both cases. Compared with Fig. 4(c) which is achieved by the equal optical path length principle, we can see that two focuses in Fig. 4(a) produced by our method show better focusing capability and more concentrated energy. This is due to the fact that in our method we have already considered the influence introduced by the interference of light emitted from all the slits to each focus. For the structure in Fig. 4(c), we first design a one-focus plasmonic lens with lens aperture D = 5um based on the equal optical path length principle and then combine two lens side by side to form D = 10um lens. When a plane wave incident normally, each part could generate one focus. But for the interference of light field from the other lens, focus turns to be terrible focusing capability and less concentrated energy. This interpretation can also be validated by comparing the phase distribution of Hz in Fig. 4(b) and in Fig. 4(d). The phase distribution around the focuses in Fig. 4(b) shows more round and this corresponds to a better focusing behavior. In addition, the coordinates of focal points are (2.99um,-1.98um) (3.99um, 2.99um) in Fig. 4(a) and (2.60um, −2.08um) (3.90um, 2.98um) in Fig. 4(c). The maximum focal length deviation is 0.33% with our method and 13.3% with the original method (the equal path length principle). Actually, focal-shift occurs when the diffraction effect of the incident wave at the boundary of the aperture becomes significant and the resulted focal length is to a large degree determined by the lens size [15,16]. It concludes that the larger the lens size, the better the quality of the focus, and the closer the agreement between design and simulation. The focal-shift in Fig. 4(c) is more significant, for the equivalent aperture is 5um, but in Fig. 4(a) is 10um.
Although above simulations have confirmed the feasibility of our method, someone may want to know the effect of amplitude discarding analytically. The detailed procedure of rebuilding Eq. (1) from Eq. (3) can be found in  and only the result is given following:Eq. (6), which takes up 78% of the total radiance in the output plane, represents the original object wave. The other terms, which take up 22% of the total radiance, are convolutions of the original object wave and will deteriorate the quality of the restored object wave. It concludes that the main effect of amplitude discarding is the introducing of some noises to the reconstructed object wave. To further eliminate these noises, one may consider using phase retrieval techniques such as multi-plane iterative Fourier transform algorithm [18–20] to derive the phase distribution of plasmonic lens.
It should be notable that the coordinates of focal points can be chosen arbitrary and the y coordinates are not limited to be set symmetrically with our method. However, the number and energy ratio of focal points are limited by aperture size of plasmonic lens, number of slits, and achieved phase level of nanoslit. These are all consistent with the limiting of Kinoform, for our design method is based on the theory of Kinoform. For fixed space of slit, number of focal points the plasmonic lens can achieve, mainly depend on the aperture size. Larger size corresponds to more slit number and more design freedom. To adjust the energy ratio of focal points freely, more phase level of nanoslit achieved and large lens size are required.
In conclusion, a method for optimal design of multi-focus plasmonic lens is proposed. The idea of holography is introduced to carry out the design. Two kinds of lens which contain two focuses and three focuses have been designed and the FDTD simulation is adopted to appraise the structure. It is found that the designed lenses can achieve the objectives well. The method presented here can also be applied to construct a wide range of plasmonic devices for beam shaping, pattern generation, and wavelength multiplexing, and so on.
This research was supported by the National Nature Science Foundation of China (No.11174281, 61275061, 11074251); the Graduate Student Innovation Foundation of the Institute of Optics and Electronics, Chinese Academy of Sciences. The authors thank their colleagues for their discussions and suggestions to this research.
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