1. Introduction

Surface plasmon polaritons (SPPs) are light waves confined to the interface between metal and dielectric medium, which have the application prospect for ultracompact optical integration based on the unique property of strong confinement of light [1–4]. The manipulation of SPPs on metal surface is of vital importance to construct ultracompact integrated micro/nano optical devices and systems, and thus has become a long fascinating theoretical and technological target of research [5–8]. Many plasmonic structures have been proposed for intensity manipulation and wavelength management. The SPPs intensity manipulations mainly exhibit two aspects: SPPs channels and in-plane SPPs scattering control. For channeling SPPs, many subwavelength plasmonic waveguides have been proposed as primary optical integration components, such as the metal-insulator-metal waveguide [9], V-shaped groove [10] and slot waveguide [11]. For SPPs scattering, nanostructure arrays are designed to scatter SPPs to form new traveling path on metal surface by many method, such as the simulated annealing method [12], the nonperfectly matched Bragg diffraction method [13] and surface electromagnetic wave holography [14]. A reflective-type metahologram is proposed to reconstruct off-plane images [15]. The dipole antenna arrays are applied to create spatially varying abrupt phase discontinuities for controlling light propagation [16]. The control of SPPs is more flexible through scattering structure.

Plasmonic demultiplexer is a key element for SPPs wavelength management in integrated SPPs system. In the SPPs channel systems, many demultiplexer structures have been developed to achieve multichannel wavelength selecting based on the plasmonic waveguide with different kinds of resonators, such as microring resonators [17], nanodisk resonators [18], nanocapillary resonators [19] and slot cavities [20]. However these resonators consisting of complex resonator geometry and tiny gaps are difficult to manufacture.

In SPPs in-plane scattering control, nanostructure arrays are applied to scatter SPPs. SPPs routing can be realized through designing the nanostructures distribution patterns. Based on the nonperfectly matched Bragg diffraction method, nanometer structures have been designed to focus the multiple-wavelength SPPs to different destinations located on metal surface [21]. Based on plasmonic crystals, two-dimensional optical wavelength demultiplexers and multiports for SPPs are realized [22]. Plasmonic demultiplexers for SPPs consisting of concentric grooves on a gold film have been proposed and experimentally demonstrated to realize multiple-channel SPP guiding [23]. A nonperiodic nanoslit array have been designed by the simulated annealing method and applied to couple free-space light into SPPs and simultaneously focus different-wavelength SPPs into various predefined locations [12]. A multiple-wavelength focusing and demultiplexing plasmonic lens is designed by manipulating the widths of the slit arrays through simulated annealing method [24]. The design of these nanostructures array appears complicated and needs massive calculations.

In Ref [14], we have developed the SWH method to determine the distribution pattern of subwavelength grooves for in-plane SPP manipulation. Simple example functionalities such as focusing a Gaussian and cylindrical SPP beam to a single point have been demonstrated. In the designing works, the method exhibits simple, easy and direct to use characteristics. It was shown that the SWH methodology is powerful for the SPPs intensity manipulations. In the common holography, more than one hologram is generated on the same region in the recording medium for wavelength-multiplexed holographic data storage [25]. In this paper, we will excavate the applications of SWH method from the SPPs intensity manipulation to the SPPs wavelength in-plane management based on the wavelength-multiplexed holography. Groove structures are designed to realize broadband focusing and demultiplexing of SPPs on metal surface.

2. Multiple-wavelength surface electromagnetic wave holography

The SWH method is applied to design the groove patterns, which involves writing and reading processes [14,26–28]. In the writing process, the object wave interferes with reference wave and the light intensity distribution is obtained. The holograms are produced through etching grooves on those positions with maximal intensity. In order to realize wavelength management, more than one hologram is etched in the same region on metal surface, as shown in Fig. 1(a).The subsidiary holograms corresponding to each wavelength are produced alone, and then they are written on the same region of metal surface. Here multiple-wavelength surface wave holography is introduced, with two wavelength holograms chosen as the example considered.

 

Fig. 1 Schematic of multiple-wavelength surface holography, (a) writing process, (b) reading process.

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Firstly, the individual hologram of wavelength ${\lambda }_1$ is written. The reference wave is a SPP with the complex amplitude $U_{W1}=A_{W1}(x,y)\exp [-i{\psi }_{W1}(x,y)]$ propagating on the metal film. A object on metal surface radiates SPPs with wavelength ${\lambda }_1$ on metal surface, and the corresponding wave is expressed as $U_{O1}=A_{O1}(x,y)\exp [-i{\psi }_{O1}(x,y)-i{\phi }_{01}]$. Then the object wave $U_{O1}$ interferes with the reference wave $U_{W1}$. The intensity distribution of the interference pattern on the metal film is $I_1=\left(U_{W1}+U_{O1}\right){\left(U_{W1}+U_{O1}\right)}^{*}$. The maxima occur at

After the writing process of hologram, the reading process is followed to reproduce the object wave [Fig. 1(b)]. According to the SWH methodology [14], the reading wave $U_R$ should be conjugated to one of the reference waves, namely $U_R=U_{Wn}^{*}=A_{Wn}(x,y)\exp [i{\psi }_{Wn}(x,y)]$ and it’s wavelength is identical to ${\lambda }_n$. Under these conditions, the corresponding object wave $U_{On}^{*}$ can be reconstructed. For two holograms discussed here, when the reading wave $U_R$ is conjugated to the reference wave $U_{W1}$ or $U_{W2}$ and it’s wavelength is identical to ${\lambda }_1$ or ${\lambda }_2$, the corresponding object wave $U_{O1}^{*}$ or $U_{O2}^{*}$ can be reconstructed.

3. Simulation results for demultiplexing

We consider two wavelengths of ${\lambda }_1=1.1\mu m$and ${\lambda }_2=1.0\mu m$. The corresponding dielectric constant of gold is ${\epsilon }_1=-\text{52}\text{.5}+3.97i$ and ${\epsilon }_2=-\text{41}\text{.9}+2.95i$, respectively. For ${\lambda }_1$, $k_{SP1}$ is the wave vector of SPPs on metal surface with $k_{SP1}=n{}_{eff1}{k}_{01}$, where $n{}_{eff1}$ is the effective index of SPPs and $k_{01}$ is the wave vector in vacuum. For ${\lambda }_2$, $k_{SP2}$ is wave vector of SPPs on metal surface with $k_{SP2}=n{}_{eff2}{k}_{02}$, where $n{}_{eff2}$ is the effective index of SPPs and $k_{02}$ is the wave vector in vacuum. For the interface between gold and vacuum, the effective index of SPPs is $n{}_{eff(1,2)}=\sqrt{{\epsilon }_{(1,2)}/({\epsilon }_{(1,2)}+1)}$. The interaction of the incident wave with grooves is simulated by the three-dimensional finite-difference time-domain (3D FDTD) method. The multiple-wavelength SWH method is employed to determine the groove patterns for wavelength management, where the methodology involves writing and reading processes, as depicted in Fig. 1(a) and (b).

In the functionality we struggle to handle, we are aiming to focus aside two plane SPPs beam with different wavelengths into two points. At first, the subsidiary hologram corresponding to ${\lambda }_1=1.1\mu m$ is obtained. The reference wave with wavelength ${\lambda }_1$ is a plane SPP beam traveling along the -y direction with complex amplitude $U_{W1}=A_{W1}\exp (i{k}_{SP1}y)$, as shown in Fig. 2(a).The object wave with wavelength ${\lambda }_1$ comes from the $P_1$ at $r_1(x=8\mu m,y=-2\mu m)$. The object wave has complex amplitude $U_{O1}=A_{O1}\exp (-i{k}_{SP1}|r-r_1|)/\sqrt{|r-r_1|}$, as shown in Fig. 2(b). For the high contrast of interference fringes, the amplitude of reference wave $A_{W1}$ is set to be identical to that of object wave $A_{O1}$. The reference wave interferes with the object wave, resulting in the intensity distribution as shown in Fig. 2(c). The groove region, surrounded by dashed lines shown in Fig. 2, is supposed to be confined within a finite space of $-10\mu m<x<0$ and $-10\mu m<y<10\mu m$, which is $8\mu m$ in distance from the object points. In this area, we etch grooves of $0.05\mu m$ in depth and $0.12\mu m$ in width at those positions with maximal intensity of the interference fringe, resulting in the individual plasmonic holographic pattern, as shown in Fig. 2(d). Secondly, the hologram corresponding to ${\lambda }_2$ is obtained. The reference wave with wavelength ${\lambda }_2$ is also a plane SPP beam traveling along the $-y$ direction, similar to that in Fig. 2(a). The object wave with wavelength ${\lambda }_2$ come from the $P_2$ at $r_2(x=8\mu m,y=2\mu m)$. The object wave has complex amplitude $U_{O2}=A_{O2}\exp (-i{k}_{SP2}|r-r_2|)/\sqrt{|r-r_2|}$, similar to that shown in Fig. 2(b). The grooves are etched at those positions with maximal intensity of the interference of the reference wave with the object wave, resulting in the individual plasmonic holographic pattern within the same region of $-10\mu m<x<0$ and $-10\mu m<y<10\mu m$, as shown in Fig. 2(e). At last, above two individual holograms are etched on the same region within the finite space of $-10\mu m<x<0$ and $-10\mu m<y<10\mu m$, as shown in Fig. 2(f).

 

Fig. 2 Design of two-wavelength metallic holographic pattern for SPPs demultiplexing on metal surface. (a) Field pattern of the reference wave. (b) Field pattern of the object wave. (c) Interference fringe pattern. (d) Geometric configuration of holographic groove pattern corresponding to ${\lambda }_1$. (e) Geometric configuration of holographic groove pattern corresponding to ${\lambda }_2$. (f) The composite holographic groove patterns.

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On the holography reading process, the SPPs plane-wave beam conjugated to the reference wave of the writing process illuminates the holographic pattern shown in Fig. 2(d-f). The interactions of the incident wave with grooves are simulated by the three-dimensional finite-difference time-domain (3D FDTD) method. In simulations, a Gaussian beam with waist radius of $4\mu m$ in free space is incident on metal grating to excite the SPP Gaussian beam traveling along the y-axis direction. The wavefronts of the SPP Gaussian beam are straight, nearly identical to that of SPP plane-wave beam. Therefore the SPP Gaussian beam can play the role of reconstruction wave to read the holograms [shown in Fig. 2(d-f)]. It passes through the groove area and is scattered. Firstly, the reading processes for the individual holograms shown in Fig. 2(d,e) are simulated. For the individual hologram shown in Fig. 2(d), the scattering field intensity pattern of SPPs with ${\lambda }_1=1.1\mu m$ is illustrated in Fig. 3(a), where only the focusing region is shown. In this paper the field intensities are normalized to the maximum of the shown region and scale bars are shown. Obviously a focus spots is reconstructed with their centers located at $x=8\mu m$ and $y=-2\mu m$. For the hologram shown in Fig. 2(e), the scattering field intensity pattern of SPPs with ${\lambda }_1=1.0\mu m$ is illustrated in Fig. 3(b). Clearly a focus spots is reconstructed with their centers located at $x=8\mu m$ and $y=2\mu m$. The positions of two focus spots agree well with the pre-designated values. For the multiple-wavelength hologram shown in Fig. 2(f), the waves with ${\lambda }_1=1.1\mu m$ and ${\lambda }_2=1.0\mu m$ are incident into the multiple-wavelength holography, respectively. The scattering field intensity patterns of SPPs are illustrated in Fig. 3(d) with ${\lambda }_1=1.1\mu m$ and Fig. 3(e) with ${\lambda }_2=1.0\mu m$. In Fig. 3(d), a focus spots is reconstructed with their centers located at $x=8\mu m$ and $y=-2\mu m$. In Fig. 3(e), a focus spots is reconstructed with their centers located at $x=8\mu m$ and $y=2\mu m$. Comparing Fig. 3(a) with Fig. 3(d), one can find that there is a little difference between scattering field intensity patterns of SPPs, but the focus spots in two figures are equally apparent and their positions are the same. Comparing Fig. 3(b) with Fig. 3(e), one can find that there is no difference between two scattering field intensity patterns of SPPs. Therefore it can be found that the presence of grooves for manipulating one wavelength does not affect much the reconstructions of SPP object wave of a different wavelength. For the reading process of the composite hologram shown in Fig. 2(f), the intensity distribution patterns in the $yz$ plane with $x=8\mu m$ are also shown in Fig. 3(g) with ${\lambda }_1=1.1\mu m$ and Fig. 3(h) with ${\lambda }_2=1.0\mu m$. Obviously, the focused spots of SPPs are confined to the metal surface.

 

Fig. 3 Simulation results of SPPs reconstructions of individual holograms and composite holograms for demultiplexing applications. Intensity distribution of SPPs in the focusing region on metal surface for (a) the individual hologram shown in Fig. 2(d) with incident wavelength $\lambda =1.1\mu m$and (b) the individual hologram shown in Fig. 2(e) with incident wavelength $\lambda =1.0\mu m$. (c) The spectra of coupling efficiency of focal areas with $-3\mu m\le y\le -1\mu m$for individual hologram shown in Fig. 2(d) and $1\mu m\le y\le 3\mu m$ for individual hologram shown in Fig. 2(e). Figure 3(d-h) SPPs demultiplexing with two-wavelength composite hologram [shown in Fig. 2(f)]. Intensity distribution of SPPs with $\lambda =1.1\mu m$ in the focusing region (d) on$xy$ plane and (g) on focal plane $yz$ plane with $x=8\mu m$. Intensity distribution of SPPs with $\lambda =1.0\mu m$ in the focusing region (e) on $xy$ plane and (h) on focal plane $yz$ plane with $x=8\mu m$. (f) The spectra of coupling efficiency of two focal areas with $-3\mu m\le y\le -1\mu m$ and $1\mu m\le y\le 3\mu m$.

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The spectra of coupling efficiency are also considered, which is defined as the ratio of total energy flux entering the focal area (with a cross-sectional size $2\mu m\times 2\mu m$ in the $yz$ plane) over the total energy flux of SPP Gaussian beam incident upon the groove region. For the individual hologram shown in Fig. 2(d), the coupling spectrum of the focal area with $-3\mu m\le y\le -1\mu m$ is calculated for a broad bandwidth of incident wavelengths and the result is represented by the red line in Fig. 3(c). Obviously, The peak at $\lambda =1.02\mu m$ appears with about 74 nm full width at half maximum (FWHM) in the coupling spectrum, which is deviated slightly from the pre-designed ${\lambda }_2=1.0\mu m$. But the grooves still can focus the SPP with $\lambda =1.0\mu m$ to the pre-designed point due to the broad FWHM of the peak. For the individual hologram shown in Fig. 2(e), the coupling spectrum of the focal area with $1\mu m\le y\le 3\mu m$ is calculated and the result is represented by the black line in Fig. 3(c). One peak at $\lambda =1.1\mu m$ appears in the coupling spectrum, which agrees very well with the pre-designed value. For the multiple-wavelength hologram in Fig. 2(f), the coupling spectra of two focal areas with $-3\mu m\le y\le -1\mu m$ and $1\mu m\le y\le 3\mu m$ are calculated, respectively. The results are illustrated in Fig. 3(f) for a broad bandwidth of incident wavelengths. The peak at $\lambda =1.02\mu m$ appears with about 90 nm FWHM in the coupling spectrum of the focal area $1\mu m\le y\le 3\mu m$, which is deviated slightly from the pre-designed ${\lambda }_2=1.0\mu m$. But the grooves still can focus the SPP with $\lambda =1.0\mu m$ to the pre-designed point due to the broad FWHM of the peak. One peak at $\lambda =1.1\mu m$ appears in the coupling spectra of the focal area $-3\mu m\le y\le -1\mu m$, which agrees very well with the pre-designed value. In Figs. 3(c) and 3(f), the wavelength deviation from the pre-designed ${\lambda }_2=1.0\mu m$ may be due to the depth of the grooves, which has difference influences on different wavelengths. When the grooves are shallow, the influence is small relatively. This leads us to set the groove depth as $0.05\mu m$. Comparing Fig. 3(c) with Fig. 3(f), one can find that the coupling efficiency decreases and the FWHM increases slightly for the composite hologram, but the spectra curves are almost the same to that of the individual holograms. The results show that the presence of grooves for manipulating other wavelength just reduces the coupling efficiency and does not affect the wavefront reconstructions of SPP object wave of a different wavelength. When the object wave of one wavelength is reconstructed, the scattering SPPs of these grooves corresponding to other wavelengths do not contribute to the reconstruction of the wavelength. This leads to the coupling efficiency decrease in the multiple-wavelength holography.

4. Simulation results for focusing

The multiple-wavelength SWH can also be applied to design metallic holographic patterns for focusing broadband SPPs to the same point. This is done by just setting the object points corresponding to different wavelength to the same point in every subsidiary-holograms $H_n(x,y)$ producing process. Here we set three object points corresponding to ${\lambda }_1=1.1\mu m$, ${\lambda }_2=1.0\mu m$ and ${\lambda }_3=0.93\mu m$ at the same point $r(x=8\mu m,y=-2\mu m)$ when three subsidiary holograms $H_1(x,y)$, $H_2(x,y)$ and $H_3(x,y)$ holograms are produced. For ${\lambda }_3=0.93\mu m$, the dielectric constant of gold ${\epsilon }_1=-\text{35}\text{.2}+2.36i$. The process is identical to that shown in Fig. 2. At first, the hologram $H_1(x,y)$ corresponding to ${\lambda }_1=1.1\mu m$ is obtained with object point at $r(x=8\mu m,y=-2\mu m)$. Second, the hologram $H_2(x,y)$ corresponding to ${\lambda }_2=1.0\mu m$ is obtained with object point at $r(x=8\mu m,y=-2\mu m)$. Third, the hologram $H_3(x,y)$ corresponding to ${\lambda }_3=0.93\mu m$ is obtained with object point at $r(x=8\mu m,y=-2\mu m)$. At last, the composite hologram is obtained through etching the above three holograms on the same region within a finite space of $-10\mu m<x<0$ and $-10\mu m<y<10\mu m$, as shown in Fig. 4(a).The scattering field intensity patterns of SPPs at the metal surface (i.e., z = 0) for the incident wave with${\lambda }_3=0.93\mu m$ as calculated by the 3D FDTD method are illustrated in Fig. 4(b). Obviously a focus is reconstructed with their centers located at $x=8\mu m$ and $y=-2\mu m$, which agree well with the pre-designated values. Likewise when the SPPs with wavelength ${\lambda }_2=1.0\mu m$ and ${\lambda }_1=1.1\mu m$ are incident upon the holographic patterns, the SPPs are focused at the same point with $x=8\mu m$ and $y=-2\mu m$, as shown in Fig. 4 (c) and (d), which agree well with the pre-designated values.

 

Fig. 4 Holographic patterns designed to focus SPPs with three wavelengths to the same point on metal surface. (a) The composite groove pattern. Intensity distribution of SPPs in the focusing region on metal surface with wavelength (b) $\lambda =0.93\mu m$, (c) $\lambda =1.0\mu m$, (d) $\lambda =1.1\mu m$. (e) The spectrum of coupling efficiency of the focal area with $-1\mu m\le y\le 1\mu m$.

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The coupling spectrum of the focal area of $-3\mu m\le y\le -1\mu m$ is calculated. The results are illustrated in Fig. 4(e). The spectrum appears a broad wavelength range ranging from $0.9\mu m$ and $1.15\mu m$ with high coupling efficiency, which indicates SPPs at other non-designed wavelengths are also focused at the point with $x=8\mu m$ and $y=-2\mu m$. When other SPPs with non-designed wavelengths are incident into the holographic patterns shown in Fig. 4(a), the scattered SPPs intensity distributions are displayed in Fig. 5.One can see that these SPPs with wavelength with $\lambda =0.90\mu m$, $\lambda =0.95\mu m$, $\lambda =1.05\mu m$, and $\lambda =1.15\mu m$ are focused around the point with $x=8\mu m$ and $y=-2\mu m$. Therefore the broadband focusing of SPPs is realized through the composite holography.

 

Fig. 5 The focusing of other non-designed wavelength through the holographic patterns shown in Fig. 4(a). Intensity distribution of SPPs in the focusing region on metal surface with wavelength (a) $\lambda =0.90\mu m$, (b) $\lambda =0.95\mu m$, (c) $\lambda =1.05\mu m$, (d) $\lambda =1.15\mu m$.

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As the composite groove pattern shown in Fig. 4(a) can focus broadband plasmonic wave on the same position on metal surface, we expect that the holographic structure can be applied to focus an ultrashort plasmonic pulse to a point. To show this, an ultrashort plasmonic pulse shown in Fig. 6(a) is incident into the holographic patterns, which covers a broad frequency range [shown in Fig. 6(b)].The pulse interacts with the holograms and the animatic process as calculated by the FDTD simulation is shown in Media 1. Four typical frames of plasmonic field pattern excerpted from video are displayed in Fig. 6(e-h) to illustrate the focusing characteristics. The white rectangle denotes the hologram region and the white points “F” indicate the position of the pre-designed focus spot. In Fig. 6(e) at $t=26fs$, the pulse is incident into the hologram region. In Fig. 6(f) at $t=69fs$, the pulse travels through the groove region and a new pulse as scattered by the holographic nanostructures appears, which is denoted by “1”. In Fig. 6(g) at $t=95fs$ the new pulse 1 focuses to the pre-designed point and the second new pulse is following (denoted by “2”). In Fig. 6(h) at $t=121fs$, the new pulse 1 travels away from the point F and begins to extend, meanwhile the second new pulse 2 travels toward the pre-designed point and the third new pulse (denoted by “3”) is forming. The three new pulses scattered away from the hologram focus at the pre-designed point. One can see the details of pulse transportation in Media 1. The temporal behavior and spectrum of the focused pulses are shown in Figs. 6(c) and 6(d), respectively. There are mainly three pulses in Fig. 6(c), which correspond to pulses in Figs. 6(e)-6(h). The spectrum of the focused pulsed also covers a broad range of frequency, similar to that of the incident pulse. However, the spectrum profile of the focusing pulse is different from the incident pulse, which means that the coupling efficiency for each spectral composition is different. The existence of the strong dispersion of the plasmonic hologram can be clearly recognized, and it inevitably induces distortion of the focusing pulse in its temporal profile from the incident pulse. Although the focusing of ultrashort plasmonic pulse by the designed plasmonic hologram is not perfect, the capability to focus a broadband plasmonic wave has once again been confirmed by this temporal study.

 

Fig. 6 The focusing of an ultrashort plasmonic pulse to a point through the holographic patterns shown in Fig. 4(a). (a) The incident pulse, (b)he incident spectrum, (c) the focal pulse, (d) the focal spectrum, (e-h) four frame excerpted from video in turn (see Media 1). Here the white point F indicates the position of the pre-designed focus spot and the rectangular region denotes the hologram region.

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5. Conclusions

In summary, we have expanded the surface electromagnetic wave holography (SWH) method to multiple-wavelength holography. Several subsidiary holograms corresponding to different wavelength are produced respectively through their object waves interfering with reference wave for the same wavelength. These subsidiary holograms are etched on the same region to form a composite groove pattern. If the SPP incident into the composite hologram is conjugated to the reference wave of one subsidiary hologram, the object of the subsidiary hologram can be reproduced. The composite hologram can realize SPPs demultiplexing through setting the objects of different wavelength at different positions. Broad band focusing can be realized by setting all objects of each wavelength to the same point. We show the two-wavelength demultplexing and 250nm-band focusing, which are fully achieved by the designed plasmonic holograms according to the FDTD simulations. These results strongly indicate that the SPPs demultplexing and broad band focusing can be realized through the multiple-wavelength SWH method designed groove pattern. The method has broad applicability and can become a universal and powerful tool in SPPs intensity manipulations and wavelength management for building novel plasmonic devices and circuits.