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Abstract
A device incorporating a series of periscope-like waveguides to achieve bidirectional focusing and plasmon launching is proposed. Optimizing the number, positions, and dimensions of the waveguides and tuning the waveguide optical paths both produce the required phase shifts to shape wavefronts and achieve constructive interference at the desired points. Due to the symmetry and reversibility of the structure, the lens can focus the light incident on both sides. Energy redistribution to a specific multi-focus can also be achieved by applying appropriate phase shifts. This simple and high performance structure makes the bidirectional plasmonic launcher easy to implement in various application situations.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon polaritons (SPPs) excited by light interacting with nanostructures have received widespread attention in nanophotonics [1]. The ultimate goals of nanophotonics are the nano-level manipulation of light and application of light–matter interactions [2–6]. These goals can be achieved through proper understanding of the interaction mechanisms between surface plasmons and nanostructures. The potential application of nanophotonics in optoelectronics and quantum optics has further stimulated interest [7]. This has led to the development of SPP resonance sensors [8], multi-mode resonator for compact on-chip optical circuits [9], polarization-controlled coupling [10], focusing, phase transformation and diffraction using nanoslits [11–16], plasmon launchers [17], and double-focusing waveguide structures [18,19]. A plasmon launcher is typically composed of nanoslits and incorporates a metal layer, although a technologically-advanced double-turn waveguide structure has recently been proposed as an alternative [20].
Although a range of plasmon lens designs has been proposed [21–25], many designs are limited in function; for example, they are unidirectional, or are difficult to manufacture. To overcome these challenges, we propose a periscope-like nano-waveguide structure to realize multi-focusing and plasmon launching in both forward and reverse directions. Compared with previous research, our design have fully considered the novel functionality and manufacturability. The optimal thickness of the lens can reach 270 nm, and the error rate of the focus position can be as low as 0.25%, the maximum nanoslit ratio (depth to width) can be as small as 1.43, which is more competitive than the lenses reported in the previous literature. the proposed novel periscope-like plasmonic waveguide plane lenses can achieve achieve multiple functions with only one theoretical design, is easier and more efficient to achieve bidirectional multiple focusing or splitting in this simple plane waveguide lens. Here we attempt to adopt multilayer metallic films to construct the desired lenses. The double-layer films are used to demonstrate the design concept. The desired spatial phase modulation could be achieved by adjusting the distance between the two films and their structural parameters. Therefore, we cannot only decrease the aspect ratio of nanoslits but also realize the superfocusing capability.
2. Structure and methods
The waveguide uses only two constituents: gold and a homogeneous dielectric. Its bidirectional periscope-like structure provides the desired flexibility in manipulating light wave fronts. A mirror-symmetric array of these waveguides forms a focusing lens, as illustrated schematically in Fig. 1(a). This type of lens is theoretically capable of simultaneously focusing incoherent waves of different wavelengths incident at different angles. Figure 1(b) shows the details of the mirror-symmetric central pair of waveguides, where the channel width w is constant throughout the waveguide. The fabrication of this design structure is a little complicated than straight waveguide structure. The double-layer films are used to fabricate the designed periscope-like plasmonic lenses in our processing receipt. The other fabrication processes are similar to the traditional planar waveguides. When the electromagnetic wave is incident on the metal surface, there will be a skin effect. According to formula (1), the skin depth is 28 nm. In order to avoid the coupling effect [13], the thickness of each film of metal is set to 100 nm, which is more than three times the skin depth. And the thickness of the channel connection area is determined by the thickness of the waveguide on both sides, which is d = w. The length along the z-axis is modeled as infinite. Since the incident wave is a coherent wave, as the tilt angle θ increases, the cross-sectional area of the homogeneous dielectric at the corner will decrease accordingly, interference of two waves leads to an increase of waveguide transmission loss. and the smaller the angle, the better; But at the same time, as the tilt angle θ decreases, the light intensity per unit area irradiated by the incident light will increase, increasing the damage to the device. The incident radiation excites SPPs in the lens which propagate through the plane waveguide.
Fig. 1. (a-b) Schematic of the bidirectional photonic lens composed of periscope-like waveguides. (a) The waveguide array of five mirrored pairs showing focusing. (b) Details of the central mirrored pair of waveguides. (c) The magnetic field (Hz) distribution for a pair of waveguides.
When the wavelength is longer than the waveguide width, the TM mode propagation constant β can be calculated using the following equation [20]
An incident TM plane wave propagating in the y-direction, with its electric vector oscillating in the x-direction, excites the waveguide. A typical resulting magnetic field distribution is illustrated in Fig. 1(c), obtained with waveguide width w = 70 nm, wavelength λ = 430 nm, and metal relative permittivity ɛm = −12.8915 + 1.2044i. [13]
For a given dielectric in the waveguide, the effective refractive index Neff is obtained by solving Eq. (1) numerically. Figure 2(a) shows the plot of Neff as a function of waveguide width w and dielectric refractive index n. As w decreases, Neff increases. Figure 2(b) depicts graphs of propagation constant β versus wavelength for different waveguide widths. We see that β decreases nonuniformly as wavelength λ increases. Neff is related to the incident wavelength, in whichever direction the wave impinges on the lens.
Fig. 2. (a) Effective refractive index Neff versus waveguide width w and dielectric refractive index n. (b) Dependence of β on wavelength λ and waveguide width w.
For plane wave propagation behind the lens, the phase shift φ, due to propagation in the waveguide and in air, is given by the equation [19]
where f is the focal length, x is the distance from the waveguide exit to the centerline of the structure, l is the bending distance (see Fig. 1(b)), and m = 0, 1, 2… is an integer.Based on the SPP interference principle, this periscope-like waveguide can achieve focusing behind the lens. The SPP can also be tailored to interfere constructively at the ends of the waveguide in both the left and right directions, and to launch plasmons.
φ1 and φ2 are the phases of the SPP waves excited at the exit of the waveguide. If the waveguide is symmetric, φ1 and φ2 are equal. They contribute to the launch of plasmons according to the equation [17]
3. Results and discussion
3.1 Discussion of single focus
The phase shifts for all five pairs of waveguides calculated from Eq. (2) are plotted in Fig. 3(a) for focal lengths of 2 µm, 3 µm, and 5 µm. Figure 3(b) shows the magnetic intensity distributions for three focal lengths. The simulation focal position is different from the preset position. This may be due to the difference between the phase shift equation and the numerical simulation model. In this paper, all structures are analyzed using finite element method (FEM) and simulated by COMMSOL software. In Fig. 3(c) the magnetic intensity is plotted as a function of (negative) distance along the y-axis. The peak positions indicate the focal points, verifying the focal length magnitudes of 2 µm, 3 µm, and 5 µm. The depth of focus (DOF), measured by the peak width, increases with increasing focal length; the DOF values are 0.557 µm, 0.779 µm, and 0.952 µm, respectively. When the focus position becomes farther, the angle between the wave vectors of the focus position decreases, and the interference distance in the y-direction increases, resulting in an increase in the peak width of the focus point.
Fig. 3. Control of focal point position. (a) Phase shift dependence on waveguide number for lenses with three different focal lengths. (b) Magnetic field intensity distribution for focal length 2 µm, 3 µm, 5 µm, respectively, with the waveguide width w = 70 nm. (c) Magnetic field intensity distribution along the focal axis for the same three focal lengths.
The number of waveguide pairs in the lens influences the focal length and focus intensity. Figure 4(a) shows the magnetic field intensity distribution for lenses with 3, 4, and 5 pairs of waveguides. The intensity at the focal point increases as the number of waveguide pairs increases. The focal point position and DOF are also obviously related to the number of waveguide pairs, as shown in Fig. 4(b). According to the interference theorem, the intensity and size of the focus point will be affected by light in different directions. The accuracy of the design focus point will increase with the increase in the amount of light in different directions due to the interference construct. However, due to the different direction of each wave vector, the interference effect around the focus point will be unstable, which will cause the contrast between the focus point and the surroundings to increase, closer to the design position, and the DOF that characterizes the size of the focus will also be smaller. The DOF decreases uniformly as the number of waveguide pairs increases from 3 to 8. In order to describe the relative size of the lens, we define divergence angle as the angle to the x-axis of the line connecting the focal point and the outermost waveguide exit from the centerline (this line corresponds to a yellow arrow in Fig. 1(a)). Figure 4(c) shows the divergence angle increasing as the number of waveguide pairs increases. With the rising of the number of waveguide pairs, the angle of the outmost wave vector reaching the focus position designed according to Eq. (2) will consistently increase, which is beneficial to the accuracy of focus position and focus quality.
Fig. 4. Effect of number of waveguide pairs on lens characteristics. (a) Magnetic field intensity distribution with 3, 4, 5 waveguide pairs with the waveguide width w = 70 nm. (b) Dependence of DOF and focal point position on number of waveguide pairs. (c) Divergence angle vs. number of waveguide pairs.
The waveguide width w determines the amount of light exciting the SPPs, and therefore, influences the resultant intensity at the focal point. However, with increasing w, the condition for waveguide positioning deteriorates, thus leading to defocusing. It is, therefore, important to find the optimal width with the maximum focal point intensity. Lenses with 5 waveguide pairs with widths 30 nm, 50 nm, 70 nm, and 90 nm were examined at focal length 2 µm. Figure 5 shows the results, indicating that the optimal width, at which the maximum intensity is observed, is 70 nm.
Fig. 5. Effect of waveguide width w on focal point intensity, for w = 30 nm, 50 nm, 70 nm, and 90 nm. Upper: two-dimensional magnetic field intensity distribution, lower: intensity along the line through the focal point parallel to the array.
The waveguide widths may not be uniform owing to manufacturing defects, which may influence the lens characteristics. To understand this influence, we consider the effects of the maximum deviation Δw = 10 nm, 20 nm, 30 nm, and 40 nm from the optimal width. Figure 6(a) shows the dependence of the measured focal point intensity on the number of waveguide pairs with the width deviation. The width deviation causes the focal point intensity to decrease, and the width deviation of the second and third waveguides has the great influence on the lens quality.
Fig. 6. (a) Dependence of focal point intensity on the number of waveguide pairs with width deviations. (b) Focal length sensitivity S dependence on number of waveguide pairs with width deviations.
The sensitivity of the focal length to the width deviations was assessed using the formula
where f is the focal length of the lens with waveguide width deviations, and f* is the focal length of the optimal lens. In Fig. 6(b), this sensitivity is plotted against the number of deviations waveguide pairs. Based on the construction of multiple pairs of waveguides, we can find that the focal length will almost unchange (sensitivity is less than 2%) with adjusting each of waveguide pairs.In addition, our designed periscope waveguide has many great advantages compared with previous work. The optimal thickness of the lens can reach 270 nm, and the error rate of the focus position can be as low as 0.25%. At the same time, the maximum nanoslit ratio of depth to width can be as small as 1.43, which is more advantageous than the lenses reported in the previous papers (see Table 1).
Table 1. Different lens’ focus position error rate and maximum nanoslit ratio
3.2 Discussions of plasmon launch
According to Eq. (3), this structure can act as a plasmon launcher. Assuming a plane wave incident in the positive y-direction, the simulated magnetic intensity distribution is shown in Fig. 7(a). We design each pair of waveguides to constructively interfere at the specified precise position, while the other positions interfere destructively. The phase shift profile of the lens is shown in Fig. 7(b), where the waveguide pairs are numbered ±1, ±2… with respect to the centerline, where the phase shift is assumed to be zero. The points indicate the phase shift for constructive interference of each waveguide. Figure 7(c) shows the intensity profile in the x-direction. The FWHM of each intensity peak is 0.48 µm, the distance s between the two peaks is 6.06 µm, and the peaks are closely symmetric.
Fig. 7. (a) Magnetic field intensity distribution above the lens for excitation at wavelength of 870 nm incident from below, with waveguide width w = 70 nm. (b) Phase shift dependence on waveguide number. The waveguides are numbered ±1, ±2… with respect to the centerline, where the phase shift is assumed to be zero (c) Magnetic field intensity distribution in the x-direction for constructive interference. The position of the field distribution is 0.3 µm outside the lens exit, which has been marked with a red dotted line in (a)
3.3 Discussions of bidirectional multifunction
The lens has a reciprocal design, which makes it capable of focusing light incident on either side. This is shown schematically in Fig. 8(a), where plane waves of 430 nm and 550 nm, incident on the upper and lower surfaces respectively, are focused. The design focus lengths are 2 µm. The corresponding intensity distributions are shown in Fig. 8(d). When the wavelength of the incident radiation reaches a certain value, as in Figs. 8(b) and (c), the wave energy is transferred to excitement of SPP waves along the ends of the lens surface. The corresponding intensity distributions are shown in Figs. 8(e) and (f) respectively. Because of its mirror-symmetric periscope structure, this type of lens has different focusing wavelengths and SPP excitation wavelengths in the two directions. This is shown in Figs. 8(c) and (f), where waves incident from above and below excite SPPs at 830 nm and 870 nm respectively.
Fig. 8. Upper and lower surfaces illuminated with wavelengths of (a) 430 nm and 550 nm, (b) 430 nm and 870 nm, (c) 830 nm and 870 nm, showing focusing and plasmon excitation. Resulting magnetic field intensity distributions for (d) 430 nm and 550 nm, (e) 430 nm and 870 nm, (f) 830 nm and 870 nm. They are all obtained with the waveguide width w = 70 nm.
According to Eqs. (2) and (3), we can construct a series of periscope-like lenses, assuming constant waveguide width. Since the calculations on both sides of the waveguide are similar, we only consider the positive y-direction. Table 2 shows parameters for the two-function lenses at different wavelengths. $\textrm{\; }\bar{l}\textrm{\; }$represents the average bending distance of the waveguide, and S is the distance between the two intensity peaks in the waveguide xz-plane.
Table 2. Parameters for the two functions, focus and plasmon launch, at various wavelengths
3.4 Discussions of multiple focus
For certain applications, multi-focus lenses are required to distribute the energy of the incoming plane wave between different points or regions on the focal axis or in the focal plane [26–28]. Such a lens is designed specifically for the required multi-focus pattern. We can vary our lens design to realize such requirements. For example, we can split the focus intensity equally between two locations in the focal plane, as shown in Fig. 9, or create two foci of different intensity along the focal axis, as shown in Fig. 10. With appropriate design of waveguide geometries and calculation of phase delays, even a combination of four focal points (two along the focal axis and two in the focal plane) can be realized; this is illustrated in Fig. 11.
Fig. 9. Focus splitting in the focal plane. (a) The geometry of the beam focusing. (b) Magnetic field intensity distribution (right) in two dimensions, and (inset) along the dashed line through the two focal points, with the waveguide width w = 70 nm.
Fig. 10. Focus splitting along the focal axis. (a) The geometry of the beam focusing. (b) Magnetic field intensity distribution along the axis. (c) Magnetic field intensity distribution in two dimensions, with the waveguide width w = 70 nm.
Fig. 11. Focus splitting along the focal axis and in the focal plane. (a) Magnetic field intensity distribution in two dimensions, and (insets) in the x and y directions, with waveguide width w = 70 nm. (b) Phase shift as a function of waveguide number.
4. Conclusions
We have described the design and simulation of a bidirectional plasmonic lens, which can realize single-focus, multi-focus and plasmon launch for light incident on either surface. Increasing the number of mirror-symmetric periscope-like waveguide pairs increases divergence angle and reduces the FWHM of the focus intensity maximum. Each lens design has an optimal waveguide width, although small variations will not considerably affect the performance. Lenses with different operating wavelengths can be designed using phase shift theory. The realization of multiple-focus lenses with energy redistribution presents the prospect of a wide range of applications. We have shown that this type of simulation can be used to design two- or four-focus lenses. In addition, the device can function as a bidirectional plasmon launcher, making it even more versatile.
Funding
National Natural Science Foundation of China (11811530052, 11947028); Intergovernmental Science and Technology Regular Meeting Exchange Project of Ministry of Science and Technology of China (CB02-20); China Postdoctoral Science Foundation (2017M611693, 2018T110440); National Laboratory of Solid State Microstructures, Nanjing University (M32056); National College Students Innovation and Entrepreneurship Training Program (201910295051Z, 201910295067).
Disclosures
The authors declare no conflicts of interest.
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