1. Introduction

Slowing down the propagation of light in order to coherently stop and store optical signals has attracted a lot of attention for its potential applications in optical memory [1], nonlinear optics [2, 3] and optical buffers [4]. Methods for confining electromagnetic (EM) waves into the smallest possible volume on a chip or device have been pursued for a long time. Typical examples include the use of quantum interference effects [5], photonic crystals [6] and stimulated Brillouin scattering [7]. Surface plasmon polaritons (SPPs) assisted metallic structures have the capability of spatially confining EM waves to spot sizes considerably smaller than the excitation light wavelength since SPPs are not restricted by the light diffraction limit [8]. Recently, the “rainbow trapping” effect has attracted great attention because of its ability to slow down and trap light signals with different wavelengths at different positions in matematerials [1] and in plasmonic graded grating structures [9–11]. For example, “rainbow trapping” in planar adiabatically chirped metal nanogratings was experimentally demonstrated across the entire visible spectrum [12]. More recently, highly efficient rainbow trapping based on a multi-layered metal-dielectric film stack has been proposed [13]. Previous investigations, however, have mainly focused on metallic gratings [9, 10], metal films coated with graded or chirped dielectric gratings [14, 15], or plasmonic graded air waveguides [11, 16, 17] in 2D planar systems. With present nanofabrication technologies nanostructures are difficult to achieve in the core of conventional single mode optical fibers due to the sufficiently thick cladding around the core. For this reason, rainbow trapping in cylindrical optical fibers has not been reported.

In 2003, Tong and Mazur demonstrated low-loss optical microfibers with diameters close to or smaller than the wavelength of the guided light [18]. Optical microfibers have been attracting more and more attention [19–22] due to numerous extraordinary optical and mechanical properties, including strong confinement, large evanescent fields, and flexibility. Nanofabrication technologies (like focused ion beam (FIB) milling) have been used to develop micro-devices and components in silica microfibers, including microcavities [23, 24], nano-resonators [25] and compact microfiber Bragg gratings [26–29].

In the present paper, in-line rainbow trapping is demonstrated in a cylindrical optical microfiber with a plasmonic graded metal grating. Dispersion and the rainbow trapping effect of the proposed waveguide are numerically investigated by 3D finite element method (FEM). Results show that the proposed structure can strongly slow down the propagation velocity of the SPP wave and that the different spectral components of near-infrared light can be trapped in different spatial positions along the microfiber. The in-line rainbow trapping is important for potential applications including optical storage, slow light, and enhanced light-matter interactions in fibers; the use of microfibers allows for a reduced footprint and an enhanced bandwidth.

2. Structures and model

Figure 1 shows the schematic diagram of an optical microfiber with a plasmonic graded grating, which is composed of grooves with increasing depth filled with gold. Here, $\Lambda$ is the grating period, while $w$, $h$ and $d$ represent the thickness, depth and width of the grating element, respectively. The frequency-dependent relative permittivity of gold ${\epsilon }_g$ is characterized by the well-known Drude model [30]:

 

Fig. 1 Schematic diagram of the optical microfiber with a graded metal grating (a). Cross section of the proposed device in the x-y plane (b) and in the y-z plane (c).

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3. Simulation results and discussions

3.1 Dispersion analysis

We first analyze the dispersion properties of the microfiber with a uniform grating with a constant depth. The analytical solutions are difficult to achieve due to finite width of the grating and the lack of symmetry in cylindrical coordinates, unlike the proposed 2D planar systems [9, 14]. Here, the dispersion relations of SPP modes are obtained by 3-D eigen-frequency solver of the commercial FEM software, COMSOL Multiphysics 4.3a. The unit cell of the grating is marked by a dashed black box in Fig. 1(c). The Bloch boundary condition is used in the z direction. In the calculation, $\Lambda =260\text{nm}$, $w=130\text{nm}$, and $d=250\text{nm}$, the diameter of the silica microfiber is $2\mu \text{m}$, and the refractive index of silica 1.45. Multiple SPP modes can be excited in the microfiber due to the finite width of the grating element when the plasmonic grating is launched by x- and y- polarized core modes. The field distributions of the lowest order SPP modes excited by y-polarized and x-polarized modes (defined as Mode I and Mode II, respectively) are shown in Fig. 2(a) and 2(b) ($h=400\text{nm}$), respectively. The lowest higher-order y- and x- polarized SPP modes (Mode III and IV) are also shown in Fig. 2(c) and 2(d). The energy of plasmonic modes is highly confined along the metallic element surfaces, rather than in the metal. As most of the energy of Mode II is concentrated inside the optical microfiber rather than at its outer surface, optical coupling to neighboring waveguides is weak; however, because of the strong confinement, nonlinear effects involved in the microfiber can be increased. This paper will focus more on Mode I, where nearly half of the energy is concentrated at the microfiber surface.

 

Fig. 2 Electric field distributions of the lowest order y- polarized SPP Mode I (a) and x- polarized SPP Mode II (b), and the lowest higher order y- polarized SPP Mode III (c) and x- polarized SPP Mode IV (d). The electric fields in the x-y plane and the y-z plane are in the planes described by pink dashed lines in Fig. 1(b) and 1(c). $\Lambda =260\text{nm}$, $w=130\text{nm}$, $d=250\text{nm}$, and $h=400\text{nm}$.

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The dispersions of the two lowest order SPP modes of different polarized states for different grating depths $(h=250\text{,}300,500\text{nm)}$ are presented in Fig. 3, where HOSM denotes the lowest higher order SPP mode. The dispersion of Mode II is basically unchanged by increasing the grating depth due to the constant width $d$ of the grating element; however, the metallic grating dramatically modifies the dispersion relationship of Mode I when the grating depth is changed. Dispersion curves show that when $h>300\text{nm}$ (defined as threshold value $h_{\text{t}}$), Mode I is the fundamental SPP mode, otherwise Mode II is the fundamental SPP mode. When $250\text{nm}<h<500\text{nm}$, there is only one y-polarized mode and one x- polarized SPP mode in the microfiber grating at frequencies lower than 292THz. When $h_{\text{t}}<h<500\text{nm}$, there is only one y-polarized SPP mode at frequencies lower than 260THz. The value of $h_{\text{t}}$ will increase if the width $d$ of the grating element increases, which will reduce the frequency bandwidth of single SPP mode operation. Therefore, selecting appropriate geometry parameters of the grating is necessary to ensure a broadband single SPP mode. An increase of the grating depth leads to an obvious red-shift of the cut-off frequency (edge of the band gap, $k_z=0.5$) for Mode I. It can be seen that at $h=500\text{nm}$the cut-off frequency is about 173 THz (1.73μm), while at $h=250\text{nm}$, the cut-off frequency shifts to 292 THz (1.03μm). At the approaching cut-off frequency, the SPP mode dispersion is very flat, which implies that the group velocity $v_g(=d\omega /d{k}_z)$ of the SPP mode significantly slows down. Therefore, a slow light waveguide can be realized. The density of SPP modes [32] at the edge of the band gap is high, corresponding to a high field enhancement close to the metal surface. Figure 4 shows the group index ($c/v_g$) of Mode I as a function of the frequency of the incident wave at a given grating depth and as a function of the grating depth at a given frequency of the incident wave. The group velocity $v_g$ is obtained by the slope of the dispersion curves in Fig. 3. The groupindex at the asymptotic cut-off frequency is significantly enhanced with increasing grating depth. The group velocity $v_g$ of Mode I at 200THz (1.5μm) could slow down by a factor of 800 if $h=419\text{nm}$, which is much lower than for a metal film with dielectric grating [14] and comparable with the metal grating [10].

 

Fig. 3 Dispersion relations of y-polarized (a) and x- polarized (b) SPP modes for different grating depths ($h=250\text{nm},300\text{nm},\text{ and }500\text{nm}$). Light line is dispersion relation of the fundamental mode in the microfiber without the metal grating ($h=0$). HOSM denotes the lowest higher order SPP mode.

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Fig. 4 Group index of Mode I as a function of the frequency of the incident wave (a) and the grating depth at 200THz (b).

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3.2 Rainbow-trapping of y- polarized mode

Gratings with a constant depth $h$can only slow down SPPs within a very narrow frequency range near the cut-off frequency. Graded plasmonic microfiber gratings, as shown in Fig. 1, can be used to enlarge the bandwidth of the slow SPP mode because the dispersion relation of the graded grating can change gradually along the microfiber as the depth of the grating elements is gradually increased. The group velocity of incident light of a certain frequency can be greatly reduced and finally approaches a minimum value at a specific location, where the local cut-off frequency is the same as the frequency of the incident light. Thus, SPP waves can be trapped at the corresponding positions along the propagation direction. If an appropriate grating depth is chosen, the incident light waves with different frequencies will be stopped at the corresponding grating elements with different depths in the microfiber.

To verify the prediction above, the structure shown in Fig. 1 was simulated using 3D FEM model. The grating depth $h$ is linearly changed from the left hand side $h=75\text{nm}$ to the right-hand side $h=470\text{nm}$ along the z direction. The total length of this plasmonic grating is only 15.34μm (60 periods), which meets the adiabatic condition ($\frac{\partial k^{-1}}{\partial z}\approx (\frac{1}{k_{z1}}-\frac{1}{k_{z2}})/\Lambda <<1$, where $k$ is the wave vector of the local grating) and ensures that the stop band edge of the graded grating changes slowly with the position along the microfiber as the depths increase [12]. The first element of the grating is shifted by 1.5μm in the z direction and the other parameters are the same as in Fig. 3. Figure 5(a) illustrates the normalized electric fields in they-z plane (x = 0) at three different wavelengths, i.e. 1.35μm, 1.50μm and 1.65μm. Due to the symmetry of the microfiber, a half cylinder was simulated in this study. A y-polarized core mode is launched from the left input port using boundary mode analysis and is used to excite the SPP modes in the microfiber. Figure 5(c) displays the electric field distribution of the input core mode and the arrows represent the direction of the electric field vector. The chosen boundary conditions are perfect magnetic conductor (y-polarized) at the symmetry plane and scattering boundary condition at the cylinder outer surfaces. The rainbow-trapping-like effect can be obtained and the SPP modes excited by shorter wavelengths will be localized at the positions closer to the input port of the microfiber with shallower $h$, while those excited by longer wavelengths will be localized at deeper $h$, further away from the microfiber input. The three different wavelengths are trapped at the positions close to 10.7μm, 12.6μm and 14.7μm, which correspond to grating depths of about 312nm, 361nm and 413nm, respectively. For comparison, normalized 3D electric field distributions of the graded microfiber grating with gold and without gold (i.e., gold is replaced by air) for 1.50μm wavelength are shown in Fig. 5(d) and 5(e). Light can propagate without loss along the uncoated microfiber, there is little deflection associated with the graded air grating. A pronounced rainbow trapping effect is observed for excited SPP modes in the proposed plasmonic grating. Figure 5(b) shows the calculated energy density in the x-y plane as a function of position along the microfiber for three wavelengths. It should be noted that the energy density gradually increases along the microfiber and reaches a peak when the propagation of incident light approaches the corresponding cut-off depth, and then reduces rapidly to zero.

 

Fig. 5 Electric field distributions for different wavelengths (a) and energy density as a function of position along the microfiber for three wavelengths (b). Electric field in input port (the white arrows represent the direction of electric field) (c), and 3D electric field of the grade microfiber grating with gold (d) and without gold (e) at 1.50μm.

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3.3 Rainbow-trapping of x- polarized mode

In a microfiber, it is important to investigate the polarization dependence of the rainbow trapping effect in the plasmonic graded grating. Consequently, we can realize trapped rainbow storage of light for two orthogonal polarized states. Similar to the y-polarized SPP mode, the x-polarized SPP mode can be trapped in a microfiber by gradually changing the width of the grating element, as shown in the inset of Fig. 6(a). The dispersion relations of SPP modes for different grating widths are shown in Fig. 6(a), where the grating depth is constant ($h=150\text{nm}$). As previously described, the x-polarized SPP Mode II should become the fundamental SPP mode if the depth $h$ of the grating is shallow. The cut-off frequency of Mode II can be adjusted by a change of the grating width. When $200\text{nm}<d<500\text{nm}$, thereis only one x-polarized SPP mode and no y-polarized SPP mode can be excited in the frequency range lower than 320THz. The rainbow trapping effect can be achieved using the structure illustrated in the inset of Fig. 6(a) and the normalized electric fields in the x-z plane (y = 900nm) are shown in Fig. 6(b). Here, the grating width $d$ is linearly changed from the left hand side $d=50\text{nm}$ to the right-hand side $d=550\text{nm}$ along the z direction and the whole system includes 55 grating elements. The boundary condition is perfect electric conductor (x-polarized) at the symmetry plane. An x-polarized core mode is launched from the left input and other parameters are the same as those of Fig. 5. Figure 6 shows that the x-polarized light with three different wavelengths, i.e. 1.45μm, 1.55μm and 1.65μm, can be trapped at different positions close to 9.5μm, 10.2μm and 11.5μm, which correspond to grating widths of about 335nm, 360nm and 406nm, respectively. In the microfiber, the plasmonic grating with a graded width enables the rainbow trapping effect of the x-polarized light as well as of the y-polarized light.

 

Fig. 6 Dispersion relations of x-polarized SPP modes for different grating widths ($\Lambda =260\text{nm}$, $w=130\text{nm}$, $h=150\text{nm}$, $d=200\text{nm},300\text{nm},400\text{nm}\text{and}500\text{nm}$) (a), and electric field distributions in a microfiber with gradual grating thicknesses for different wavelengths in the x-z plane (y = 900nm) (b).

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From Fig. 3 and Fig. 6(a) we can predict that the microfiber with graded depth and width can trap y- and x- polarized SPP modes at different spatial positions simultaneously and polarization splitting can be achieved. However, it is difficult to trap y- and x- polarized SPP modes with the same wavelength at the same space position, because it needs exact coincidence between the cut-off frequencies of y- and x- polarized SPP modes for the same grating elements.

3.4 Dependence of the dispersion on the grating period

The dispersion of SPP modes determines the trapping wavelengths and the positions where different light wavelengths are trapped. The grating period also strongly affects the dispersion of the SPP modes and the associated cut-off frequency. Rainbow trapping can also be obtained when the grating period is graded. The dispersion of Mode I and Mode II for different grating periods is illustrated in Fig. 7. Here the ratio between the grating thickness and the period is constant (w/Λ = 0.5). The cut-off frequencies of Mode I and Mode II shift to lower frequency with increasing the grating period. Unlike the previous cases of graded depth and width, the period affects simultaneously the dispersion of both y- and x- polarized modes, which is much more complex than the 2D planar systems that are only suitable for TM SPP mode. Meanwhile, although the graded-period grating is easier in practical fabrication, it may increase the complexity of the device design.

 

Fig. 7 Dispersion relations of y- and x- polarized SPP modes (Mode I and Mode II) for different grating periods ($h=400\text{nm}$,$d=200\text{nm}$,$\Lambda =200\text{nm},2\text{6}0\text{nm},\text{and}400\text{nm}$).

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4. Conclusion

In conclusion, a plasmonic microfiber grating is proposed to achieve in-line rainbow trapping for y-polarized and x-polarized modes for near-infrared wavelengths. The dispersion properties are engineered by simply designing the depth and width of the plasmonic graded grating. The plasmonic grating meets the adiabatic condition and in addition ensures that the cut-off frequency of the graded grating changes slowly with the position along the microfiber. The in-line rainbow trapping effect allows near-infrared light to stop and be stored at different spatial locations for two orthogonal polarized states. Different from previously reported structures that trap SPP waves in 2D planar systems, the proposed structure can trap SPP waves along a microfiber cylindrical waveguide surface, which provides an in-line way to slow or trap light signals. The in-line rainbow trapping in the microfiber offers potential applications in constructing spectrometers, optical switches, slow-light devices, and nanoscale buffers, and can also enhance light-matter interactions in fiber integrated devices. Moreover, in-line fiber rainbow trapping can also be used for polarization splitting.