We present a direct-method solution toward the general problem of plasmonic wavefront manipulation and shaping to realize pre-designated functionalities based on the surface-wave holography (SWH) method. We demonstrate theoretically and experimentally the design and fabrication of holographic plasmonic lenses over surface plasmons with complex wavefront profiles. We show that visible light at 632.8 nm transmitting through a high-aspect-ratio slit or a micro-rectangle hole in a silver film can be focused to a preset three-dimensional point spot in free space via appropriately manipulating the interaction of excited surface plasmons with the nanoscale groove pattern of the holographic lens. The experiment results of scanning near-field optical microscopy for measuring the three-dimensional optical field distribution agree well both with designs and with numerical simulations, and this strongly supports the effectiveness and efficiency of the SWH method in the design of plasmonic devices that can fulfill manipulation and transformation of complicated-profile surface plasmons.
©2013 Optical Society of America
Surface plasmon polaritons (SPPs) are propagating waves confined at the metal-dielectric interface [1–4]. Plasmonic nanostructures can manipulate the in-plane transport and scattering of SPPs in a controllable means. Thus they provide a promising route to construct ultracompact integrated micro/nano optical devices and systems [5,6]. In addition, plasmonic nanostructures can also be exploited to fulfill and model conventional optical transport control functionalities, but with a much smaller size scale. Patterning metallic surfaces to utilize the scattering of SPPs to realize conventional optical functionalities has attracted much attention in recent years [7–17]. The advantage is that these structured components have a subwavelength scale and the total structures can be fabricated in a small area on a flat metallic surface . Therefore, plasmonic structures provide an ultracompact integration solution to the exploitation of micro-size and nano-size optical devices. For example, light collimators [7–13], plasmonic wave plates , lenses [9, 15, 16], and focusing  have been designed to manipulate light waves above the metallic surface in the Fresnel region.
To fully realize device and system functionalities in the conventional optics, plasmonic devices must have the capability to address various complicated conditions and situations for both the input wave and the output wave. However, so far these plasmonic devices were designed with special skills, such as simulated annealing method, where complicated and time-consuming iteration algorithms and computations are requested to solve inverse electromagnetic problems  to obtain pre-designated devices and functionalities. Fairly speaking, these design methodologies are not universal enough to deal with very general systems and functionalities.
Recently a novel methodology called surface wave holography (SWH) [19, 20] was proposed as a universal method to manipulate light waves scattered by appropriately patterned grooves milled on the surface of a metallic plate. For a given light wave transportation, SWH allows one to directly determine the geometric morphology of the grooves without the need of complicated and time-consuming iterative computation to obtain inverse-problem solutions in the conventional wisdoms. In this regard, SWH belongs to the direct-method category of device and system design. It has been shown that complex output functionalities of light transmitting through a subwavelength circular hole perforated in an opaque metal screen could be achieved through the SWH method. These functionalities include focusing light waves to an arbitrary point in three dimensions , collimating light waves to an arbitrary solid angle , and shaping light waves to an “L” pattern and an “O” pattern at a given plane in the Fresnel zone after the metallic screen . As a counterpart of its powerful capability on shaping the output wavefront, we will show that the SWH method also has the equal powerful capability to allow the input plasmonic wave to take a complex profile pattern. The capability to manipulate both the input and output freedom of a plasmonic structure will significantly improve the design of plasmonic devices for application to modern optical devices and systems.
In this paper, we specify the output functionality as the focusing of light to demonstrate our idea. We build a plasmonic lens that can focus light transmitting through an aperture of complicated geometric shape (other than a simple circular subwavelength aperture) in an opaque metal screen. It is well known for some years that milling grooves on the outgoing surface of a metal film can focus and collimate light emerging from a subwavelength hole [7–13]. As an application, Capasso’s group integrated these structures on the output surface of semiconductor lasers and greatly reduced the light-emitting angular divergence [11,12]. The prospect of integrating plasmonic structures to shape the output beams of semiconductor lasers was therefore expected and demonstrated in Refs [11,12]. However, by now these researches are limited to apertures of the simplest shape, i.e., a narrow slit which was treated as a line, and a tiny hole which was treated as a point. For the case of a slit, the structure was usually taken as a two-dimensional (2D) case and only the light confinement along the short axis could be realized. In this paper, we show that the light from a slit, which is usually treated as a 2D structure, can also be confined in three dimensions. In other words, we can use a specially designed plasmonic lens to achieve three-dimensional (3D) focusing of light transmitting through such a slit on a spot. With this exotic lens placed at the exit plane of semiconductor lasers, the ordinary output slit beam can be shaped into a 3D focusing spot in the Fresnel zone.
In our experimental demonstration, we build these plasmonic samples with their surface morphology of grooves perforated on a silver thin film directly determined according to the SWH method, and the 632.8 nm light emerging from a slit (11 × 0.12 μm2) or a micro rectangular hole (3 × 0.12 μm2) can be focused to a point located at 7 μm above the silver surface by these plasmonic lenses. We perform the finite-difference time-domain (FDTD) simulations and the scanning near-field optical microscopy (SNOM) experiments to evaluate the performance of these exotic plasmonic lenses. Good agreement between both numerical and experimental results with the pre-designated functionality strongly supports the feasibility of our idea and also further confirms the efficiency of the SWH method in shaping complicated input wavefront of surface plasmons into a given optical functionality.
2. Principal steps of SWH to design holographic plasmonic lens
As has been discussed in Refs [19, 20], the SWH has borrowed the concept from the conventional optical holography and also involves the writing and reading processes of the object wave with the aid of the reference wave. But different from the conventional optical holography, the reference wave is now a confined plasmonic wave excited by illumination of the aperture and propagating outwards away from the aperture, instead of the usual plane wave. Figure 1 illustrates the three principal steps involved in the determination of the morphology of the perforated grooves for a complex input plasmonic wave emanating from the slit. As detailed in the literatures [19, 20], we need the information of the object wave and the reference wave to determine the plasmonic holographic structure.
As what we are concerned is the 3D focusing effect at a preset position, i.e., 7 μm above the metallic surface, so the object wave is the outgoing propagating wave of a point source located at z = 7 μm. Notice that for 2D focusing functionality, the source would be a line source. The field pattern of this object wave at the metal surface is determined by rigorous FDTD calculations. In our FDTD simulation, an x-polarized (the coordinates is defined according to the fabricated sample, as shown in Section 3) point source is placed at z = 7 μm and the field distribution of the x component of the electric field at z = 0 μm (the outgoing surface of metal plate) is stored as U0, as shown in Fig. 1(a). Then, we extract the phase distribution of U0 and denoted it as ϕ0(x,y). As for the reference wave, we use a light beam to illuminate a hole of complex shape (now a slit or a line aperture) to get the complex-pattern surface waves. Some structure details of such a hole must have the subwavelength scale in order to efficiently excite the surface waves.
In our FDTD simulation, we use a plane wave to shine the hole on a silver film and the field distribution (also the x component of electric field) immediately above the metallic surface is stored as the reference wave Ur, whose phase term is ϕr(x,y). In our numerical studies, the optical permittivity data of silver are taken from Ref . and input into the FDTD simulations. According to the SWH method, we pattern grooves at positions where ϕ0(x,y) + ϕr(x,y)-2mπ = 0 (m is an integer). In practice we divided the sample plane into 20 × 20 nm2 pixels and used the criterion | ϕ0 + ϕr-2mπ<0.25| to decide which pixel should be etched (more technical details can be found in the literatures [19, 20]). The hole-excited surface waves are scattered by these grooves and propagat in the free space above the silver plate. Due to the constructive interference of the scattered waves, a focal spot appear in the preset position if the morphology of the groove patterns is appropriate.
3. Results of the slit sample
The first example of our study is focusing the light emerging from an 11 × 0.12 μm2 slit into a point at 7 μm above the surface. In experiment, a silver film with a thickness of 240 nm is deposited on a SiO2 substrate (with a refractive index of 1.55) by magnetron sputtering, and then patterned by focused ion beam lithography. By controlling the dwell time of the focused ion beam milling, only the hole is penetrable, and the grooves are approximately 80 nm deep. The scanning-electron-microscopy (SEM) picture of the fabricated sample is shown in Fig. 2(a). We define that the x axis is parallel to the short axes of the slit, the y axis is parallel to the long axes [as depicted in Fig. 2(a)], z axis is normal to them with its positive direction pointing out of the page, and the coordinate origin (0, 0, 0) is located at the center of the slit. Usually, such a slit is treated as a 2D case in practice because the length is two orders of magnitude larger than the width. However, it can be seen in Fig. 2(b) that the surface waves excited by an 11 × 0.12 μm2 slit in a silver film has a complex distribution, and this makes the wavefront shaping into a 3D focusing spot a very difficult task in the framework of usual inverse problem solution, which would request very complicated and time-consuming electromagnetic computations. Yet, it is an ordinary processing in the framework of SWH to deal with these complex surface waves, showing the great advantage of this direct method.
According to our design, if we illuminate the patterned sample in Fig. 2(a) with a 632.8 nm laser from the side without grooves (the side where z<0), a focal spot located at (0, 0, 7) μm is expected. This is indeed confirmed by our FDTD simulation results as shown in Fig. 2(c). In experiment, in order to monitor the focusing effect of the structure, we use a SNOM, whose probe aperture is approximately 100 nm, to scan the field distributions at different heights above the sample surface. The sample is fixed in a stage which can move precisely in z direction, so that the SNOM probe which scans the signals at a fixed plane can give the field distributions at different heights. As the probe moves outward from the surface, we expect that the light is first focused to a point approximately at z = 7 μm (the designed position) and then is defocused.
These expected pictures are indeed observed in the SNOM experiment, with the measurement results of field intensity distribution patterns shown in Fig. 3. When the scanned plane is close enough to the sample surface, the field distribution is much the same as the shape of the etched slit (Fig. 3, z = 0.5 μm). Then the transmitted field is diffracted as the monitor plane moves outward from the sample surface. At these heights, the detected field is spread and therefore the maximal intensity is reduced (Fig. 3, z = 0.5 μm ~z = 4.5 μm). When z comes near to 7 μm, the constructive interference of the grooves-scattered waves becomes dominant and a clear focus can be recognized. It is interesting that this focus is confined more tightly in y direction than in x direction. It has a full width at half maximum (FWHM) of 0.58 μm in y direction and a FWHM of 1.63 μm in x direction. For reference, we write down the Rayleigh criterion of a conventional refractive lens: d = 1.22λf/D, where d is the diameter of the focal spot, λ is the wavelength of the laser, f is the focus length, and D is the diameter of the lens' aperture. Directly substituting the parameters of our planer lens, i.e., λ = 632.8 nm, f = 7 μm, and D = 12 μm, into this equation gives a diameter of the focal spot of our lens, which is d = 0.45 μm. It is also worth noting that the maximal intensity in z = 7 μm is even higher than the maximum intensity in z = 0.5 μm (where the detected plane is close enough to the surface). As the scanned plane continues to move outward, this tightly confined focus is spread in both the x and the y directions, as shown in Fig. 3, z = 10 μm. By now, we have demonstrated that the light transmitting through a slit whose width is much smaller than the wavelength in a silver film can be focused to a point with confinement in both the x direction and the y direction by designing and building a holographic plasmonic lens around the slit.
To provide more insight of this focusing effect, the field evolutions in the z direction, i.e., the field images at the x = 0 yz plane and the y = 0 xz plane, are shown in Fig. 4. In the simulation results as displayed in Fig. 4(a) and 4(b), a rice-shaped hot spot with its maximum intensity at z = 6.7 μm is clearly recognized, which fits quite well with our design (z = 7 μm). Similar images are also obtained in the SNOM experiment, as shown in Fig. 4(c) and 4(d). The focus in the SNOM experiment is located at z = 7.5 μm, which is slightly different from the simulated value (6.7 μm). This difference is attributed to the deviation of the fabricated structure from the simulated structure, especially the deviation in the width of the slit, whose value in the fabricated sample is 0.15 μm whereas in simulation is 0.12 μm (the simulated value is the same as the design value). These rice-shaped maxima in Fig. 4 indicate that the focusing depth of the fabricated structure is approximately 2.5 μm, which is relatively large when compared with the FWHM of the focus. Such an ability of plasmonic lenses to focus light in a range of distance has also been demonstrated in Ref [18, 19]. From Fig. 4(c) and 4(d), we can see that the experimental images are tilted approximately at an angle of 5 degrees, and this is ascribed to that the probe is not exactly vertical to the scanned plane (in another word, the probe is titled). Note that the scanned plane is well adjusted according to the sample surface so that the angle between the scanned plane and the sample surface is negligible, namely, they are parallel to each other. The good agreement between the experimental results and the simulated results confirms the focusing effect of the holographic plasmonic lens and again support the power of the SWH method in handling complicated wavefront shaping problems.
4.Results of the micro-rectangle-hole sample
To further demonstrate the power of the SWH direct method, we make another design, simulation and experiment, which show that the 632.8 nm light transmitting through a micro rectangular hole, whose size in design is 3 × 0.12 μm2 and in fabricated sample is 3.22 × 0.15 μm2, can also be focused to a preset point. Strictly adhering to the steps described in Fig. 1 of SWH and using a micro-rectangular hole on a silver film to calculate the reference surface wave Ur, we find the morphology of the perforated grooves on the silver film for the holographic plasmonic lens. Based on the design we fabricate the micro-hole sample, whose SEM image is shown in Fig. 5(a). We can see the complex profile of the surface waves excited by such a hole in Fig. 5(b). When patterned with the SWH structures, a focus is expected in z = 7 μm according to the FDTD simulation as shown in Fig. 5(c).
Figure 6 shows our SNOM-experiment results of this sample. Again, the light waves is first focused and then defocused and a focal spot can be recognized in z = 6 μm, which differs from the designed value z = 7 μm. Besides, the focal effect of this rectangular-aperture sample is not as good as the slit one (Fig. 3). The reasons of them are the deviation of the reference surface wave excited in the experiment structure (by a hole of 0.15 μm width in fabricated sample) and the surface wave excited in the design structure (by a hole of 0.12 μm width in simulation). In the slit sample, due to the much longer scale in y direction, the surface wave depends on the width weakly. However, if the y scale is shrunk to 3 μm, the change of the width (e.g., from 0.12 μm to 0.15 μm) will change much the distribution of the excited surface wave and, therefore, the focusing effect and performance. The difference between the measured focal height and the designed value is ascribed to the reference surface wave excited in the experiment (by a hole of 0.15 μm width in fabricated sample) is significantly different from that excited in the design (by a hole of 0.12 μm width in simulation). Thus far, we have demonstrated the power of the SWH method on designing a plasmonic structure that allows the 3D focusing of complex input surface waves emanating from a slit or rectangular hole with a large aspect ratio to a preset point. For the case of the light from a hole of other complicated shapes, e.g., a C-shaped hole , an H-shape hole  and a bow-tie aperture , we can still expect the similar functionality of 3D focusing by fabricating the corresponding SWH structures.
In summary, we have demonstrated both theoretically and experimentally the power of the SWH method on shaping the complex input surface wave. This direct method allows us to design and build holographic plasmonic lenses around a slit or large-aspect-ratior rectangular hole to realize 3D focusing into a pre-designated spot upon the light transmitting these highly anisotropic holes. Our 632.8-nm-laser experimental results agree well with the designs and the simulations, showing the success of the design. Together with the previous work that demonstrates the power of the SWH method on meeting the complex output demands , the current work may pave the path to design and build optical devices and systems in the micro-scale Fresnel region with the conventional optical functionalities that transform a given general input wave into a preset general output wave. In addition, this SWH method has been proven to be applicable from the microwave , the infrared light  to the visible frequency. With the advantage of being able to fully address complex input and output conditions by 2D SWH structures build in a thin metal plate, these holographic plasmonic devices can be used for ultracompact optical integrations. We expect this method will find a wide range of applications, such as the design of the holographic antennas in the microwave --regime, the shaping of the output light of semiconductor lasers, the planar beam transformer between free-space beams and surface waves [25, 26], and the communication between two flat optical systems.
This research was supported by the 973 Program of China (Grant No. 2013CB632704), the Knowledge Innovation Program of the Chinese Academy of Sciences, and the National Natural Science Foundation of China (Grant No. 61177089 and 61227014).
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