1. Introduction

Plasmonic structures which can manipulate light at subwavelength scale have attracted considerable interests in recent years [1–3]. Many studies have shown that due to the interactions between the surface plasmons (SPs) and the metallic nanostructure, the propagation of light can be fully controlled [4–10]. For example, by using the groove-on-film structures, the wavefront can be reshaped, and a series of optical phenomena such as beaming, focusing, splitting and beam shaping have been demonstrated theoretically and experimentally [4–7]. At the same time, the phase of the wavefront can be tuned point by point with properly designed array of nanoparticles. Based on this, anomalous reflection and refraction phenomena have been observed [8–10].

By combining these wavefront controlling technique with the concept of optical holography, plasmonic holography has been realized by Ozakis group [11]. In their work, the holographic plate is fabricated by exposure to the interference between object light and reference light, like the way used in the traditional holography. Another way to realize holography is by using computer generated holograms and micro/nano-fabricating techniques. This way is more feasible, and has received much attention [12–14]. Recently, this type of plasmonic holographic imaging is realized by Chen’s group and Dolev’s group [7, 15]. However, in their work the amplitude is binary modulated (using metal gratings), which causes the loss of details and results in the mismatch between the original object and the image. In this article, we will realize plasmonic holography with higher imaging quality by using nanoantenna array to modulate the phase of the wavefront. In section 2, we will use the weighted Gerchberg-Saxton (GSW) algorithm to generate optimized hologram of the preset letters “NANO”. In section 3, we will design suitable nanoantennas, which are the unit cells of our plasmonic array, to realize phase shift from 0 to 2π. In section 4, we will orient these nanoantennas according to the optimized hologram, and realize high quality plasmonic holography. Finally, we will conclude this article.

2. The design of the hologram using GSW algorithm

There are various methods of designing hologram for a preset shape. The simplest method is setting a light source with the preset shape and recording the phase of electric field at the hologram plate. However, this method ignored the information of the amplitude of the electric field, and hence caused low efficiency and uniformity of the image [16]. To overcome this weakness, some hologram generation methods are proposed, such as Curtis-Koss-Grier algorithm [17], simulated annealing [18] and Gerchberg-Saxton (GS) algorithm [16]. The GSW algorithm, which is a variation of GS method with bias to the spatial uniformity, has been widely used in the hologram generation for its good balance between efficiency and image quality [19].

To apply this method, we first discretize the preset shape into grids, and regard them as M light sources with random phases ${\theta }_m^{\left(0\right)}$. We calculate the wavefront in the hologram plane of light radiated by these sources at first. The total electric field at the hologram plane is the superposition of the electric fields radiated by every sources, so the phase of the electric field at jth pixel ( ${\phi }_j^{\left(k\right)}$, the superscript (k) denotes the kth iteration) is

Next we carry on a backward calculation from the hologram φ^(k) to the sources, and we can obtain the amplitude of the electric field at the position of the mth source:

Then we update the phases of the sources ${\theta }_m^{\left(k+1\right)}$ to the argument of $V_m^{\left(k\right)}$, and recalculate the weight $w_m^{\left(k+1\right)}$ for every sources:

Successive application of Eqs. (1), (2), (3) and (4) comprises one iteration of the GSW algorithm. And usually, the convergence can be achieved in tens of iterations.

With the GSW algorithm, we design the hologram for a preset letter “NANO” shown in the inset of Fig. 1(a). In our work, the wavelength of the incident light is 1550 nm, and the distance between adjacent pixels is 750 nm. Figure 1(a) shows the designed hologram with 400 × 400 pixels. At the same time, Fig. 1(b) shows the corresponding image directly calculated from the hologram by using the Fraunhofer diffraction formula [20]. From the image we can find that the shape of the diffraction pattern is nearly the same as the preset shape, and the uniformity is also very good.

 

Fig. 1 (a) The 400 × 400 hologram for the preset shape “NANO”; (b) The Fraunhofer diffraction pattern of the hologram shown in Fig. 1(a).

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We should notice that the hologram mentioned above contains continuous phase shifts from 0 to 2π. However, designing and fabricating a series of nanoantennas with continuous phase shifts is impractical. So in our work, the phase shifts are limited to several equally spaced levels. To decide the suitable number of phase levels (N), we study the relationship between N and the image quality, which is characterized by the efficiency (e) and the relative standard deviation (RSD) of the intensity:

The results are showed in Fig. 2. We can find that the efficiency is not very sensitive to N. And when N is more than 8, the efficiency is close to 1, which indicates nearly all the energy is in the preset region. We can also find that the RSD is approximately inverse proportional to the number of levels, i.e. the uniformity of the intensity in the preset shape is higher for larger N. In the following, we choose to use 16 levels in our design to achieve a balance in the image quality and the difficulty of fabrication.

 

Fig. 2 The efficiency and RSD of the holographic imaging when adapting different phase shift levels. The dashed lines are the efficiency (black) and the RSD (red) for continuous phase shifts.

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3. Optical properties of V-shaped nanoantenna

According to Ref. [9], the V-shaped nanoantennas can tailor the phase of the scattered light from 0 to 2π. Here, we also adopt the V-shape Au nanoantennas, which are placed on the Si substrate and comprised of two nano-rods with semicircle ends, shown in the inset of Fig. 3(b). We use finite-difference time-domain (FDTD) technique to simulate the phase shifts of the scattered lights of the V-shaped nanoantennas. The simulation scheme is shown in Fig. 3(a). The nanoantenna is placed in the xy-plane, and the incident light is along z-axis from bottom to top. The angle between x-axis and the symmetric axis of the nanoantenna is 45°. We use an x-polarized light to stimulate the nanoantenna, and record the y-component of the electric field above the nanoantenna, which is perpendicular to the polarization of the incident light. This arrangement assures that only the scattered light is recorded. Then we can obtain the phase shifts by comparing the phase of the scattered light among different nanoantennas.

 

Fig. 3 (a) Scheme of the simulation to get the phase shifts for nanoantennas with different geometric parameters. The nanoantennas are placed on a Si substrate, and the angle between the symmetric axis of each nanoantenna and x-axis is 45°. The incidence is x-polarized, and the y-component of the scattered electric field is recorded. (b) The phase shifts and amplitudes of the scattered electric field for nanoantenna #1 to #16.

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We calculated the phase shifts of varies of V-shaped nanoantennas with different lengths (L) and angles (θ), and find 8 V-shaped nanoantennas with phase shifts from 0 to π. The nanoantennas with phase shifts from π to 2π can be obtained simply by flipping these 8 nanoantennas with respect to x-axis, since the y-components of the scattered electric field of the flipped nanoantennas are just opposite to the original ones, which means an additional phase shift of π[8]. The geometrical parameters of the 16 nanoantennas which we use in the following of this article are showed in Table 1. At the same time, the amplitudes (black squares) and phase shifts (red circles) of the scattered light are also depicted in Fig. 3(b). We can see that the phase shifts increases linearly with the No. of the nanoantennas, and the amplitudes of the scattered lights for all V-shape nanoantennas are within 3.44 ± 0.20. The evenly spaced phase shifts and near constant scattering amplitudes will guarantee the good performance of the nanoantenna array hologram.

Table 1. The arm lengths and angles for the 16 nanoantennas. The arm widths of all nanoantennas are 50 nm.

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4. Nanoantenna array hologram

In the above discussions, we solved the two prerequisites required by our nanoantenna array hologram: the hologram generated by GSW method and 16 nanoantennas with equally increasing phase shifts from 0 to 2π and constant scattering amplitude. And in this section, we will build the nanoantenna array hologram. However, the simulation of an array containing 400 × 400 nanoantennas is very memory and time consuming. But fortunately, due to the unique property of holography, only a small piece of the hologram still has the ability to reconstruct the whole image. The hologram shown in Fig. 4(a) is 1% of the whole hologram shown in Fig. 1(a), which only has 40 × 40 pixels. We also calculated its Fraunhofer diffraction pattern, which is shown in Fig. 4(b). We can see that although the pixels are much less than the whole hologram, the preset “NANO” shape can still be clearly recognized.

 

Fig. 4 (a) The 40 × 40 hologram, and the part of the corresponding nanoantenna array; (b) The Fraunhofer diffraction pattern of the hologram shown in (a); (c) The simulation scheme; (d) The intensity distribution in the focal plane simulated by FDTD technique.

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In the following, we will build the holographic plate with the 16 kinds of nanoantennas. We use nanoantennas #1 to #16 to realize the phase shifts in the hologram. Take the hologram in the red square in Fig. 4(a) for example, the phase shifts of the pixels at the top-left corner are 11π/8, 5π/4, 5π/8, . . . , so we put nanoantennas #12, #11, #6, . . . in the corresponding positions, respectively. Part of the nanoantenna array are also shown in the figure detailedly. The distance between adjacent nanoantennas is 750 nm, which can ensure that the interaction among the nanoantennas is weak enough to be neglected [8].

Then the plasmonic holographic imaging with above holographic plate is studied by FDTD method. The simulation arrangements are shown in Fig. 4(c). The stimulating light whose wavelength is 1550 nm is from the bottom to the top of the nanoantenna array, and the image is in the focus plane of a positive lens. We record the near-field electric field distribution above the nanoantenna array. By using near-to-far-field transformation, we can obtain the angular distribution of the scattered electric field [21]. Then the electric field distribution in the focus plane of lens can be simply calculated by the lens equation for collimated light

Finally, we obtain the intensity distribution in the focal plane of the lens. The result is shown in Fig. 4(d). We can see that the quality of the image is surprisingly good, and agree very well with the results directly calculated from the hologram using Fraunhofer diffraction equations (shown in Fig. 4(b)). This indicates that the design of our plasmonic holographic plate with nanoantenna array is feasible. Additionally, it should be noticed that in our simulations we only have 40 × 40 nanoantennas in the array due to the limitation of our computer hardware. However, with recent micro/nano fabricating techniques such as UV lithography and electron beam lithography (EBL), nanoantenna array with much larger size can be fabricated, and holographic imaging with higher quality can be achieved. Currently, the major drawback of this type of plasmonic holography is the relatively low total efficiency, which is about 0.2%. The low total efficiency is mainly caused by two reasons: One is the low fill factor of the nano-antennas array, so most part of the incidence is not scattered by the nano-antennas; another reason is the low coupling coefficient between the two perpendicular polarizations. Based on this, several methods to improve the total efficiency are implied, such as: decreasing the distance between adjacent nano-antennas, increasing the scattering cross-section of the nano-antennas, and enhancing the coupling coefficient between different polarizations.

5. Conclusion

In this article, we demonstrated a novel high-quality holographic imaging with nanoantenna array. By using the GSW algorithm, we designed a hologram of a preset shape “NANO” with high efficiency and good uniformity. We also give 16 kinds of V-shaped nanoantennas with phase shifts ranged from 0 to 2π. By orienting these nanoantennas, we obtain the holographic plate for the preset shape “NANO”, which contains 40×40 nanoantennas. We calculated the electric field distribution of the holographic plate under plane-wave illustration by using FDTD technique and near-to-far field transformation. A clear “NANO” shape appears in the focal plane. This simulation result agrees very well with the result by the Fraunhofer diffraction equation directly from the hologram, which indicates the hologram is implemented by nanoantenna array correctly and effectively. And since our method allows more levels, the phases of the wavefront is tuned more precisely, and higher imaging quality is obtained. We believe this technique can be helpful in beam shaping, holographic printing and holographic detection.