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Abstract
In this paper, continuous position control of plasmonic phase singularities on a metal-air interface is achieved based on the misaligned coupling between the optical axis of vortex beam and nano ring plasmonic lens. The formula of surface plasmon polaritons field distribution in this case is derived. The offset distance and direction between the optical axis of the vortex beam and the center of the nano ring is used to control the distance and the angular distribution of the phase singularities in nanoscale, respectively. This can promote the accurate positioning of phase singularities in practical applications and provide a deeper understanding of the misaligned coupling between vortex beams and nano ring plasmonic lens.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Singular optics is an important branch of modern optics and phase singularity is a kind of important optical singularities [1,2]. In the phase singularity, the phase is indeterminate, and the light intensity is zero. A light beam with phase singularity is called optical vortex (OV). It is well known that OV has spiral wave front and carries orbital angular momentum (OAM) [3]. The unique properties of OVs have triggered numerous important applications in both classical physics and quantum science, such as OAM multiplexing communication [4], optical manipulation [5], remote detection of rotating bodies [6], quantum entanglement [7], quantum information [8] and holography [9]. Laguerre–Gaussian beam [3], perfect vortex beam [10], circular Airy vortex beam [11], etc. are typical instances of OVs. In these OVs, single phase singularity is embedded. In the more general case, an individual beam can possess multiple singularities. Recently, the investigation of multi-singularity beams is a research hotspot [12–14] since they are found to be vital to some special applications such as high-dimensional multi-partite classically entangled light creation [15], multiple particle manipulation [16], and optical communication [17]. Meanwhile, manipulation of phase singularities based on surface plasmon polariton (SPP) has also been broadly investigated since SPPs can highly confine light fields into micro-nano scale, which can enhance light-matter interaction [18] and significantly promote photonic device integration [19].
Therefore, recent years, the investigation of plasmonic vortex with multiple phase singularities has attracted wide attention from researchers. For example, Yu Wang et. al. [20] investigated the sculpturing of plasmonic vortices via Archimedean spiral slit, Yuanjie Yang et. al. [21] investigated the generation of the deuterogenic plasmonic vortices on the plasmonic vortex lenses consisting of a set of Archimedean spiral slits, Changda Zhou et. al. [22] investigated the compound plasmonic vortex generation based on spiral nanoslits, Chun-Fu Kuo et. al. [23] proposed a scheme for exciting high-density plasmonic vortex array, and Eva Prinz et. al. [24] investigated the generation and control of compound plasmonic vortex via functional meta lenses. Most of these works investigated only the generation of special multi-singularity plasmonic vortex and the control of vortex orders. In [25], however, it was pointed out that the optimal light–matter interaction requires precise control of the singularity properties, especially its location. Therefore, a method for realizing spatial position control over optical singularities on a metal–air interface by varying the polarization state of the light was reported in [25]. Despite this, the precise position control of the phase singularities is still less studied, and there still remains challenges to find new ways for achieving highly flexible, continuous and high-precision position control of the phase singularities in nanoscale.
In addition, in singular optics, it is well-known that the higher-order OVs are not stable and have tendency to split into vortices with lower order [26]. It offers a potential way to control the position of phase singularities if an actively controlled perturbation is introduced in a high order OV. Therefore, in this paper, we investigate the misaligned coupling of vortex beams and nano ring plasmonic lens. The offset distance and the orientation angle between the optical axis of the vortex beam and the center of the nano ring are introduced as the actively controlled perturbations to regulate the split of the high order plasmonic vortex exited on metal–air interface, and control the distance and the angular distribution of the phase singularities in nanoscale. Our study provides a new method for achieving continuous and high-precision position control of plasmonic phase singularities. It can be used together with other phase-singularity control methods [25], hence can greatly improve the flexibility of the accurate positioning of phase singularities in practical applications. Moreover, it can also provide a deeper understanding of the misaligned coupling between vortex beams and nano ring plasmonic lens, which may promote the design of on-chip plasmonic vortex devices.
2. Theoretical model
The schematic of our misaligned coupling scheme is shown in Fig. 1(a). The nano ring plasmonic lens consists of an annular groove etched in the gold film deposited on the silicon dioxide substrate. In the figure, ${O_1}$ is the center point of the nano ring, and (x, y, z) is the global coordinate system. The input vortex beam illuminates the structure normally from the substrate side and ${O_2}$ is the center point of the beam. There exists an offset S between the optical axis of the vortex beam and the center point of the nano ring. β is the offset orientation angle. The inserted figure of Fig. 1(a) is the intensity distribution of the SPPs on the surface of the gold film excited by the first order Bessel vortex beam with an offset along x-axis. It can be seen that the annular intensity distribution of vortex beam is broken, and there generates two phase singularities near the center of the nano ring. The distance between the two singularities is expressed by D.
Figure 1(b) shows the coordinate system, where the black solid ring and the blue dashed ring represent respectively the nano ring and circular distribution of the incident vortex beam, $(r,\theta )$, $(\rho ,\varphi )$ and $(R,\gamma )$ are respective the polar coordinates of the nano ring, incident vortex beam and the SPPs observation point near the center of the nano ring, ${r_1}$ is the radius of the nano ring, $({r_1},\theta )$ is an arbitrary point on the nano ring, ${r_2}$ is the distance between ${O_2}$ and point $({r_1},\theta )$, and L is the distance between point $({r_1},\theta )$ and the SPPs observation point $(R,\gamma )$.
Considering the incident vortex beam possessing both spin angular momentum (SAM) with spin quantum number σ and OAM with topological charge l, it can be expressed in cylindrical coordinates as
Therefore, the total electric field at the observation point is
In Eq. (4), ${r_2}$ and $\varphi $ can be represented by ${r_1},\theta ,S$ and $\beta$, hence the field distribution of the SPPs near the center of the nano ring can be calculated easily by using Eq. (4). Because of the circular symmetry of the nano ring plasmonic lens, the expressions of ${r_2}$ and $\varphi $ can be simplified by choosing $\beta \textrm{ = }0$ without loss of generality, which can be obtained according to
Therefore, we can now investigate the influences of S on the SPPs distribution and the phase singularities by using Eqs. (4) and (5).
3. Results and discussion
In this section, taking two common types of vortex beams, i.e., Bessel vortex beam and Laguerre-Gaussian (LG) beam as the input beams, we numerically investigate the position control of plasmonic phase singularities via our misaligned coupling scheme by using both the above theoretical derivation and FDTD simulation.
3.1 Incidence of Bessel vortex beam
When the input beam is l order Bessel vortex beam, i.e., ${\vec{E}_{in}}(\rho ,\varphi ,z = 0) = {J_l}(a\rho ){e^{i(l + \sigma )\varphi }}({\vec{e}_\rho } + i\sigma {\vec{e}_\varphi }),$ we have
It can be seen clearly that if the input vortex beam and the nano ring plasmonic lens is aligned, we have ${r_2} = {r_1},\varphi = \theta$, and then the excited SPPs field is an $l + \sigma$ order plasmonic vortex. Hence, when the input light is the first order Bessel vortex beam (l=1) with left circular polarization ($\sigma \textrm{ = }1$), it will excite second order SPPs vortex. In this case, if the input light and the nano ring plasmonic lens is misaligned with an offset S, it can be expected that the excited second order SPPs vortex will split into two first order vortices, and the position distribution of the two first order vortices can be controlled by S.
To investigate numerically, we should first define the FDTD simulation parameters of the nano-ring plasmonic lens. For this, we regard the micro length unit element on the narrow annular groove as a rectangle nano slit, then numerically calculate the responses of light polarization and wavelength of a rectangle nano slit with 300 nm length, 100 nm width and 100 nm depth by FDTD simulation. The calculated results of SPPs intensity are shown in the Fig. 2. Figure 2(a) are the absolute values of y- and x-polarized input light excited SPP electric field at the central point of the long edge of the nano slit and their contrast ratio versus the incident wavelength when the incident amplitude is 1 V/m, and Fig. 3(b) and (c) are the simulated distributions of the absolute values of the SPPs electric field on the top surface of the gold film under excited by 1000 nm wavelength y- and x-polarized light, respectively. It can be seen that the most efficient excitation wavelength is about 1000 nm, and in the wavelength range of 950–1050 nm, the contrast ratio is larger than 40, which means the SPPs excited by x-polarized input light is significantly less than the SPPs excited by y-polarized input light, i.e., only the input component polarized perpendicularly to the slit can couple to SPPs effectively. Therefore, in the following FDTD simulation, we choose 1000 nm incident wavelength, and annular groove with 100 nm width and 100 nm depth to obtain high excitation efficiency as well as meet the excitation condition that only the incident component polarized perpendicularly to the annular groove can couple to the SPPs effectively.
Fig. 2. (a) The absolute values of y- and x-polarized input light excited SPP electric field at the central point of the long edge of the nano slit and their contrast ratio versus the incident wavelength. (b) and (c) are the simulated distributions of the absolute values of the SPPs electric field on the top surface of the gold film under excited by 1000 nm wavelength y- and x-polarized light, respectively.
Fig. 3. The normalized absolute value and phase distributions of the SPPs field calculated by (a) the analytic solution and (b) FDTD simulation. The first and third rows are the normalized absolute value distributions of the electric field, and the second and fourth rows are phase distributions, where the phase singularities are marked by white circles.
With these knowledges, the numerically investigation can be now performed. Firstly, according to Eq. (6), the analytic results of the normalized absolute value and phase distributions of the SPPs field excited by the 1000 nm wavelength first order Bessel vortex beam with left circular polarization (l=1 and $\sigma \textrm{ = }1$) under $\beta$=0 and different offset S are calculated and shown in Fig. 3(a). In the calculation, ${r_1}$ is set to be 5µm, and $a\textrm{ = }0.638$ is used to make the nano ring match the half peak intensity position of the first intensity ring of the incident beam, since the intensity of the incident beam changes most rapidly with its radial coordinate $\rho$ at this position, which can enhance the sensitivity of the plasmonic vortex splitting to the offset S.
In the figure, the distributions of the normalized absolute value of electric field intensity and the phase under three different S are displayed, and the position of the phase singularities are marked with white circles. It can be seen clearly that by introducing a nonzero offset S, the annular intensity distribution of the SPPs is broken, the second order plasmonic vortex is split into two phase singularities near the center of the nano ring, and the distances D between the two separate phase singularities increases as S increases. The distances D are 157.2 nm, 230.3 nm and 296.5 nm when S=600 nm, 800 nm and 1000 nm, respectively. This means that we can precisely and continuously control the distance between the two separate phase singularities in nanoscale by changing the offset S. Moreover, it can also be found that in the case of β=0, i.e., the offset is along x-axis, the center of mass of the SPP intensity distribution moves along y-axis and the orientation of the two phase singularities is along y-axis, i.e., both the moving direction of the mass center of the SPP intensity distribution and the orientation direction of the two phase singularities are perpendicular to the offset direction. Here it should be noted that in the analytic calculation, the wavelength of the SPPs wave is chosen to be ${\lambda _{spp}} = {\lambda _0}{{[({\varepsilon _d} + {{\varepsilon ^{\prime}}_\textrm{m}})} / {{{\varepsilon ^{\prime}}_\textrm{m}}{\varepsilon _d}{]^{0.5}}}} = 987.9\textrm{nm}$ when the incident wavelength is ${\lambda _0} = 1000\textrm{nm}$, where ${\varepsilon ^{\prime}_m}$ is the real part of the relative permittivity of the gold and ${\varepsilon _d}$ is the relative permittivity air. The FDTD simulation results under same conditions are shown in Fig. 3(b). In the FDTD simulation, according to the results of Fig. 2, both the thickness of the gold film and the width of the annular groove are chosen to be 100 nm. It can be seen that in the FDTD simulation, the distances D are 151.7 nm, 232.8 nm and 303.0 nm at S=600 nm, 800 nm and 1000 nm, respectively. The tendency of the distances D changing with the offset S simulated via FDTD are consistent with the results of the analytic calculation except for slight difference of D, which is less than 7 nm. And there exists a small angle between the orientation of the two phase singularities and y-axis in the FDTD simulation results. We think these minor differences should be attributed to the non-circularly symmetric discrete mesh of the FDTD simulation model. It is an additional symmetry breaking which can lead to an additional splitting of the phase singularities.
Further, via a simple wave vector analysis, a physical mechanism can be given to explain the phenomenon that the moving direction of the mass center of the SPP intensity distribution is perpendicular to the offset direction. For this, we first consider the second order spiral phase of the radial component of the incident light with $l + \sigma \textrm{ = }2$, which is shown in Fig. 4(a). This spiral phase leads to an in-plane angular wave vector ${k_{OAM}} = {2 / \rho }$, which is inversely proportional to the radius $\rho $ of the incident light. The white line in Fig. 4(a) represents the schematic diagram of this angular wave vector and the arrows denotes its direction. Then we give the wave vector analysis for the cases of S=0 and S>0 according to Fig. 4(b). When S=0, the vectors ${k_{OAM}}$ on the left and right sides of the nano ring have equal amplitude and opposite direction. In this case, the directions of the vector ${k_{SPP}}$ of the SPPs excited on the two sides point to the origin point and the obtained SPPs intensity at the region near the center point of the nano ring exhibits ring distribution with circular symmetry. While when S>0, the amplitude of ${k_{OAM}}$ on the right side of the nano ring increase and the amplitude of ${k_{OAM}}$ on the left side of the nano ring decrease. It is because that a nonzero S along positive x-axis shortens the distance between the right side of the nano ring and the optical axis of the input beam while increases the distance between the left side of the nano ring and the optical axis. In this case, an additional phase tilted in the positive y-axis direction is introduced, which leads to the directions of the vector ${k_{SPP}}$ tilt in the positive y-axis direction compared to the case of S=0, as shown by the blue arrows in Fig. 4(b). Therefore, the center of mass of the SPP intensity distribution moves along positive y-axis when the offset is along positive x-axis.
Fig. 4. (a) The second order spiral phase of the radial component of the incident light. (b) the schematic of wave vector analysis.
For further comparing the results of the analytical calculation and FDTD simulation, the distribution of the SPPs normalized absolute value on y-axis and the distance D under different S are calculated via both analytical solution and FDTD simulation, the results are shown in Figs. 5(a) and 5(b), respectively. The curves T1, T2, T3 and S1, S2, S3 in Fig. 5(a) are respective the SPPs normalized absolute values of the analytical calculation and FDTD simulation results under S=600 nm, 800 nm and 1000 nm. And the blue dotted line and the red triangle in Fig. 5(b) are the distance D of the analytical calculation and FDTD simulation, respectively. It can be seen that the analytic and FDTD simulation results have high consistency. In addition, the variation of the distance between the phase singularities is significantly shorter than the variation of S, which means a large displacement of S can cause only a small displacement of the phase singularities. It can reduce the difficulty and improve the accuracy of controlling the phase singularities in nanoscale.
Fig. 5. (a) The distribution of the SPPs normalized absolute value on y-axis and (b) the distance D under different S calculated via both analytical solution and FDTD simulation.
Then for deeper understanding the splitting process of the phase singularities with the increase of S, we collect the phase distribution on the ring with 20 nm radius and centered by the singularities under different S. The results are shown in Fig. 6, where (a1, a2), (b1, b2) and (c1, c2) correspond to the phase distribution surrounding the two singularities when S=200 nm, 600 nm and 1000 nm, respectively. It can be seen that when S is small, although the second order plasmonic vortex is split into two phase singularities and the phase changes by 2π in one cycle around each phase singularity, the phase change curves are not linear. We think that this is caused by the field interference of the SPPs surround the two singularities when they are close to each other. Therefore, it can be predicted that the phase change curves will become more linear with the increase of S. This is consistent with the results of Figs. 6(b1)-(c2).
Fig. 6. The phase distribution on the ring with 20 nm radius and centered by the singularities under different S. The horizontal axis is the azimuthal angle in polar coordinates with the phase singularity as the origin point.
We then further investigate the control ability of $\beta$ on the angular position of the phase singularities. Under the same conditions of Fig. 3, the FDTD simulated normalized absolute value and phase distributions of the SPPs field under different $\beta$ are shown in Fig. 7. It can be seen that when S is 1000 nm, the angles between the orientation of the phase singularities and the x-axis are −60.4deg, −31.2deg and −2.3deg under $\beta$=30deg, 60deg, and 90deg, respectively. It demonstrates that the offset direction and the orientation of the phase singularities are always perpendicular to each other.
Fig. 7. The FDTD simulated normalized absolute value and phase distributions of the SPPs field under different $\beta$
The above calculations are all based on ${r_1}$=5µm. Therefore, it is necessary to study the influence of ${r_1}$ on the distance. The trends of distance D changing with ${r_1}$ under three different S are shown in Fig. 8. It can be seen that under a fixed S, the distance D will decrease with the increase of ${r_1}$. It is mainly due to that when S increases, to ensure the nano ring can match the half peak intensity position of the first intensity ring of the incident beam, the size of the beam should be increased, which will reduce the sensitivities of the change of the incident field intensity and the in-plane angular wave vector ${k_{OAM}}$ at the nano ring with S.
Until now, we have investigated the continuous position and orientation control of plasmonic phase singularities by the misaligned coupling between the optical axis of vortex beam and nano ring plasmonic lens. However, one can found that the intensity distributions of the excited SPPs are asymmetric and irregular, which may have trouble in practical applications, such as optical manipulation. Therefore, it is important to construct a symmetric and regular SPPs field when control the phase singularities so that it can be better applied for practical applications. We find that a symmetric SPPs field can be constructed by exciting the nano ring plasmonic lens via the superimposed beam of two vortex beams with opposite offset. The FDTD simulation results of the normalized absolute values and phase distributions of the SPPs field excited by the superimposed beam of two 1th order Bessel vortex beams with opposite offset are shown in Fig. 9. It can be seen that when the offset S is 1400 nm, i.e., the offset of one input vortex is 1400 nm and along the positive direction of x-axis and the offset of the other input vortex is 1400 nm and along the negative direction of x-axis, the distance between the two singularities D is 276.0 nm. And when the offset S is 1800nm, the distance D is 374.4 nm. Hence, we can precisely and continuously control the distance between the two separate phase singularities in nanoscale by changing the offset S as well as ensure the symmetric of the intensity distribution. Similarly, the control ability of $\beta$ on the angular position of the phase singularities is also investigated. When S is 2000nm, it can be seen that the angles between the orientation of the phase singularities and the x-axis are −59.5deg and −32.1deg under $\beta$=30deg and 60deg, respectively.
Fig. 9. The distributions of the normalized absolute value and phase of the SPPs field from FDTD simulation under different S and $\beta$. (a) is the result under S=1400 nm and $\beta$=0deg, (b) is the result under S=1800nm and $\beta$=0deg, (c) is the result under S=2000nm and $\beta$=30deg, and (d) is the result under S=2000nm and $\beta$=60deg.
3.2 Incidence of LG beam
In the above section, the case of the incidence of Bessel vortex beam is investigated. In this section, taking LG beam as an example, we prove that our misaligned coupling scheme can still control the position of phase singularities effectively when the input beam is another kind of vortex mode. The expression of the LG beam is given by
Fig. 10. The normalized absolute value and phase distributions of the SPPs field calculated by (a) the analytic solution and (b) FDTD simulation when the incident light is LG beam.
Similarly, the angular position of the phase singularities can be controlled by changing $\beta$. This is demonstrated by the FDTD simulation results of the normalized absolute value and phase distributions of the SPPs field with different $\beta$ shown in Fig. 11, which are simulated under same conditions of Fig. 10. As the control of the phase singularities are feasible in both Bessel vortex beam incidence and LG beam incidence, it proves the universality of controlling the position of phase singularities based on misaligned coupling of different kind of vortex beam and the nano ring plasmonic lens.
Fig. 11. The FDTD simulation results of the normalized absolute value and phase distributions of the SPPs field under different offset angles when the incident light is LG beam.
4. Conclusion
In conclusion, the analytical formula of SPPs vortex field excited in the case that optical axis of the incident vortex beam is not aligned with the nano ring plasmonic lens is derived. The electric field and phase distributions of the SPPs are calculated by using the obtained analytical formula and verified by FDTD simulation. It is found that by changing the distance and direction of the incident optical axis from the center of the nano ring plasmonic lens, nanoscale continuous control of the position of the phase singularities can be achieved. Besides, a large displacement of S can cause only a small displacement of phase singularities, which can reduce the difficulty and improve the accuracy of adjusting phase singularities. This phase singularity control method can be used in parallel with other phase-singularity control methods in practical applications. We believe that it can greatly improve the flexibility of the accurate positioning of phase singularities. Moreover, it can also provide a deeper understanding of the misaligned coupling between vortex beams and nano ring plasmonic lens, which may promote the design of on-chip plasmonic vortex devices.
Funding
National Key Research and Development Program of China (2019YFB2203901); National Natural Science Foundation of China (62065006, 61827819, 61975038); Natural Science Foundation of Guangxi Province (2019GXNSFAA245024, 2020GXNSFBA159059); Science and Technology Project of Guangxi (AD19245064); Middle-aged and Young Teachers' Basic Ability Promotion Project of Guangxi (2018KY0200); Guangxi Key Laboratory of Optoelectroric Information Processing (GD20103); Guangxi Key Laboratory Project of Automatic Detection Technology and Instrument (YQ20103); Innovation Project of GUET Graduate Education (2020YCXS094).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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