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Control of plasmon-polariton vortices on the surface of a metal layer IGOR V. DZEDOLIK*
AND
VLADISLAV PERESKOKOV
Physics and Technology Institute, V. I. Vernadsky Crimean Federal University, 4 Vernadsky Avenue, Simferopol 295007, Russia
Surface plasmon polaritons (SPPs) can be excited at the interface of a metal layer and a dielectric layer by various methods. If an inhomogeneity of permittivity with a curvilinear boundary is created in the metal layer by the external electric field, the incident SPPs and the reflected ones from this inhomogeneity interfere with one another. A nonrectangular vortex lattice appears when such SPP interference occurs, and the lattice configuration can be controlled by varying the external electric field. Based on control of the SPP vortex lattice and reading out of the vortex localization, the plasmon logic gates “AND,” “OR,” and “NOT” can be realized. These logic gates represent a functionally complete basis for logical operations in processors operating at optical frequencies. © 2018 Optical Society of America
1. INTRODUCTION The design and creation of devices for plasmonics have recently attracted the particular attention of researchers in connection with applications of plasmonic devices such as signal processing at optical frequencies, surface spectroscopy, sensors of physical quantities, etc. [1–8]. The future prospects of creating optical computers working at optical frequencies have increased interest in the design and implementation of logic gates for optical processors. The operating frequencies of optical processors are higher by several orders compared with the operating frequencies of electronic processors. However, the elements of optical processors are larger in size [9] than the elements of electronic devices. Therefore, plasmonic elements operating at optical frequencies are two or three times smaller than optical elements [10], and that is promising for computer equipment applications. Surface plasmon polaritons (SPPs) arise because of 3D electromagnetic wave scattering by inhomogeneities on a metal surface and the hybridization of the evanescent electromagnetic waves, waves of medium polarization, and waves of electron gas in a metal [11]. The SPPs can be excited at optical frequencies at the interface of metal with a negative real part of the permittivity Re εM < 0 and a dielectric medium with positive permittivity Re ε0 > 0. When the real part of the metal permittivity is negative (Re εM < 0), the SPPs propagate on the surface of the metal, but at a positive value (Re εM > 0) the SPP cannot propagate because the boundary conditions εM α0 −ε0 αM are not satisfied. Here α0 > 0 and αM > 0 are the decrements along the normal axis x to the metal surface of the SPP field components ∼ exp−αx iβz − ωt, i.e., the SPPs are the
evanescent waves along the x axis, and they propagate along the z axis. Scattering of the SPPs on the inhomogeneities of various configurations at the interface of the dielectric and metal layers leads to enrichment of the mode composition of the SPPs, and it also leads to interconnection of the modes in microwave guides and micro-cavities and to radiation of bulk electromagnetic waves from the interface of the metal and dielectric layers. The propagation direction of the SPP is changed at reflection from the curvilinear boundary of the inhomogeneity in the metal layer, and the curvature of the wavefront of the surface waves is also changed. Such inhomogeneities can be realized by means of slots, grooves, or asperities on the layer, as well as by changing the permittivity at the local area of the interface [12–21]. Another way to change the permittivity of the metal layer is to effect on it by intensive an external electric field [6]. In this case, the permittivity of the metal layer can be controlled by changing the external electric field. When we have SPP interference with wavefronts of different configurations formed at the reflection, refraction, or diffraction of the surface waves, the appearance of vortices is possible [21–29]. The maxima and minima of the Poynting vector appear at the interference of incident and reflected SPPs at the area of their existence. In the interference field of the incident SPPs and the reflected ones from the inhomogeneity with a curvilinear boundary, singular points with phase dislocation arise on the surface of the metal layer. The plasmon-polariton vortices are formed in these points, where the interference fringes of the SPPs have edge dislocations [27,28] on the surface of the metal layer. SPPs that propagate forward along the z
Fig. 1. Model of generation, reflection, and reading out of SPPs. The nano-probes with negative potentials E 01 and E 02 are located above the metal layer with permittivity εM . The laser ray (red) excites the SPPs by the scattering on the diffraction grating in the form of grooves on the surface of the metal layer. The SPPs propagate along the z axis, and they are reflected from the local inhomogeneity with the curvilinear boundary of the metal permittivity. The microsensors MS01 and MS02 read out the normal component of the Poynting vector of the SPP vortices.
axis form surface waves with rectilinear wavefronts, and SPPs that propagate backward along the z axis form surface waves with curvilinear wavefronts on the surface of the metal layer (Fig. 1). These surface waves interfere, and their 2D interference pattern acquires maxima and minima. The SPP interference pattern on the metal surface x 0 depends on the form of the inhomogeneity boundary in the metal layer [24–26]. The SPP vortices do not arise if the inhomogeneity boundary is rectilinear [25]. The inhomogeneity with a curvilinear boundary (in particular, in the form of a “dovetail”) can be formed in the metal layer under the action of the electric field of negative charges localized at the nano-probes above the metal layer (Fig. 1). The negative potential of the nano-probes can change the permittivity of the metal in such a way that it acquires positive values in the optical frequency range. Then, in the area of action of the electric field, the boundary conditions of the SPP propagation are violated, and the SPPs are scattered at the boundary of the artificially created inhomogeneity of the metal permittivity (Fig. 2). We assume that the SPPs interfere in the area with length L and width 2b, (b ≤ r 0 ), and that they decay out of this area.
Fig. 2. Inhomogeneity on the surface of the metal layer with the curvilinear boundary in the form of a dovetail. The nano-probes with negative potentials create the positive “mirror” charges with radii r 0 on the surface of the metal layer. The excited SPPs with rectilinear wavefronts propagate forward along the z axis, and the reflected SPPs with a curvilinear wavefront propagate backward. The SPP angle of incidence to the inhomogeneity boundary is φ.
There is a curvilinear border in the form of a dovetail on the right side of the zone, from which the SPPs are reflected. Then the maximum value of the coordinate z for SPP propagation in the allocated zone depends on the coordinate y; z L z max . One can find the added coordinate z max y from the equation for a circle with the center at the point y b, z r 0 : jyj − b2 z − r 0 2 r 20 , from which we obtain z max r 0 − r 20 − jyj − b2 1∕2 at jyj ≤ b. One can change the form of the permittivity inhomogeneity in the metal layer by changing the values of the negative charges on the control nano-probes (Fig. 1). These changes of the inhomogeneity boundary of the area in the metal layer lead to changes in the SPP interference pattern. The positions of the SPP vortices shift by varying the intensity of the electric field of the nano-probes, due to changes in the circles’ radius r 0 of inhomogeneity, whose boundary has the shape of a dovetail (Fig. 2). One of the most promising options for exciting SPPs in plasmon processors for computer technology is the usage of spasers built into the chip [4,30–33]. The configuration of the SPP interference pattern can be read out by various ways, in particular, by the nano-antennas as nanowires or nanoparticles [6,8,34–36]. Currently, various devices based on control of the SPPs to realize logic operations have been actively designed and implemented [37–39]. However, the proposed devices do not realize the complete set for logical operations. The purposes of our work are (1) to find the conditions for the appearance of the SPP vortex lattice at the interface of the dielectric and metal layers at the reflection of SPPs from the inhomogeneity in the form of a dovetail; (2) to investigate the possibility of control of the vortex lattice by the external electric field; and (3) to propose plasmon logic gates on the basis of the vortex lattice control. The implementation of plasmon logic gates “AND” and “NOT” based on the control and readout of the SPP vortex, and “OR” based on coupled plasmon waveguides, can provide a functionally complete basis for logical operations in processors operating at optical frequencies. 2. GENERATION OF SURFACE PLASMONPOLARITON VORTICES The SPPs can be excited on the surface of a metal layer by diffraction of a bulk electromagnetic wave on the diffraction grating in the form of grooves on the surface of the metal layer [11]. Such method can be used for exciting the SPPs in microchips by injecting the radiation from an optical fiber. We assume that the external electromagnetic wave excites the SPPs with the plane wave fronts by the diffraction grating with the period Λ on the surface of the metal layer of Fig. 1. The SPPs propagate along the z axis with a propagation constant β kz m2π∕Λ, where m 1, 2, …, and their wavefronts are parallel to the y axis. An inhomogeneity of the dielectric permittivity in the form of a dovetail can be created in the metal layer by the negative charges of the nano-probes (Fig. 1). When the SPPs are reflected from the curvilinear boundary of the inhomogeneity, the scattered SPPs appear on the surface of the metal layer and propagate backwards at certain angles 2φ to the z axis, depending on the radius r 0 of the inhomogeneity boundary
(Fig. 2). The incident and reflected SPPs interfere in the linear regime, and the vortices appear at the singular points of the SPP interference pattern. Consider the excitation of vortices upon the reflection of SPPs from the inhomogeneity boundary in the form of a dovetail in the metal layer (Fig. 2). SPPs are generated at the interface of the metal layer and dielectric layer with ε0 at the
scattering of the laser ray at the diffraction grating in the form of grooves [11] on the surface of the metal layer (Fig. 1). The solutions of the system of Maxwell’s equations describe the incident and scattered SPPs (Appendix A). The transversemagnetic (TM) mode of the incident SPPs in nonmagnetic media (μ 1, B y H y ) has the components E 1x , E 1z , and B 1y . Such a surface wave has a plane wavefront that is
Fig. 3. Interference pattern of the normal component S x of the SPP Poynting vector at the interference of incident and reflected SPPs on the surface of the metal layer. The left column represents the inhomogeneity with one circle’s border at the radius r 0 5 μm; the right column represents the inhomogeneity with two circles’ borders at the radii r 0 5 μm. Pictures: (a) the interference fringes of the amplitude with one circle’s border; (b) the interference fringes of the amplitude with two circles’ borders; (c) the phase distribution with one circle’s border; (d) the phase distribution with two circles’ borders; (e) the SPP vortices with topological charge l 1 (red arrow, anti-clockwise) in the fenced area with one circle’s border; (f) the SPP vortices with topological charge l 1 (red arrow, anti-clockwise) in the fenced area with two circles’ borders. Units along the axes y, z are given in centimeters.
transverse-independent of the y coordinate. The field components of the incident SPPs are cβA cα A exp ϕT , E 1z −i 0 exp ϕT , E 1x ωε0 ωε0 B 1y A exp ϕT ,
(1)
where A const and ϕT −α0 x iβz. The propagation constant of the SPP TM mode at the interface between the dielectric medium with ε0 and the metal layer with εM has εM 1∕2 the value β ωc εε0 0ε . M The SPPs of the TM mode fall onto the inhomogeneity in the metal layer and are reflected from it. The wavevector of the SPP that is directed along the z axis after reflection from the boundary of the inhomogeneity turns at an angle 2φ, the projection to the z axis of the propagation constant becomes equal to β˜ −β cos 2φ, and the projection to the y axis becomes κ β sin 2φ. The scattered SPP has the field components E 2x , E 2z , E 2z , and B 2y , B 2z [24–26], cβA cα A sin 2φ E 2x exp ϕR , E 2y −i 0 exp ϕR , ωε0 ωε0 cα A cos 2φ E 2z i 0 exp ϕR , B 2y A cos 2φ exp ϕR , ωε0 B 2z A sin 2φ exp ϕR ,
(2)
where ϕR −α0 x iβy sin 2φ − z cos 2φ, but the normal component of the magnetic field remains equal to zero, B x 0. The number of reflected SPPs depends on the shape of the inhomogeneity boundary, i.e., on the angles 2φ for the reflected SPPs. The same modes are formed inside the metal layer near the surface. In Eqs. (1) and (2), for these modes the decrement must be replaced by −α0 → αM , and the permittivity is also replaced by ε0 → εM . The boundary conditions for the reflected SPPs are not violated, but the transverse component of the SPP wavevector κ is added to the dispersion equations c −2 ω2 ε0 α20 − κ2 − β˜2 0 and c −2 ω2 εM α2M − κ 2 − β˜2 0, where κ β sin 2φ, and β˜ β cos 2φ, i.e., κ 2 β˜2 β2 . The Poynting vector components are expressed by the components of the electric and magnetic vectors of the SPPs, S x c∕4πE y B z − E z B y , S y −c∕4πE x B z , and S z c∕4πE x B y , where E x E xTM E xE , E y E yE , E z E zTM E zE , B y PB yTM B yE , B z B zE , E jE P 0 E ϕ , and B n jE n j n B j 0 φn , where j x, y, z. The components of the SPP Poynting vector have the form S j S ja expiϕj , where S ja Re S j 2 Im S j 2 1∕2 is the amplitude component and ϕj arctanIm S j ∕Re S j is the phase component. The interference fringes can split and form the edge dislocations, and at these singular points the SPP vortices Re S j i Im S j l S jlj ja expilϕj appear. At the circle H dr∇ϕ 2πl around the singular points at the interference pattern, where Re S j 0 and Im S j 0, the phase ϕ of the surface wave changes to 2πl, where l 1, 2, … is the topological charge of the vortex [27,28]. The phase dislocations can be formed in the zero points of the 2D SPP interference pattern at the arrival of at least three plasmon-polariton waves: a forward wave and two backward waves reflected at different
angles 2φ from the curvilinear boundary of the inhomogeneity. The phases ϕj arctanIm S j ∕Re S j of the components of the Poynting vector S x , S y , and S z are different [25]. The normal component of the SPP Poynting vector S x is read out by the microsensors (Fig. 1). The distribution of the amplitude and phase of the normal component S x of the Poynting vector at the SPP interference on the surface of the metal layer are shown in Fig. 3. The SPP interference pattern includes many types of singularities [27] such as the right-hand and left-hand centers, saddles, and inward nodes. For our aim, the most interesting singularities are the righthand and left-hand centers at the screw dislocation of the phase ϕx arctanIm S x ∕Re S x of the normal component of the Poynting vector [40] for reading out by the microsensors. The plasmon vortices appear at the points of the fringe splitting at the interference pattern minima of the Poynting vector [Figs. 3(a) and 3(b), dark lines]. The changes of the nanoprobes’ potentials (Fig. 1) lead to decreasing or increasing of the circle radii r 0 of the metal permittivity inhomogeneity (Fig. 2). This causes the displacement of the vortices from their initial positions [Figs. 3(e) and 3(f )]. The sign of the topological charge l 1 corresponds to the phase rotation direction on the surface: “plus” is anti-clockwise (right helicity about the x axis) rotation. Thus, the vortex lattice appears at the interference of the SPPs, and its form depends on the number of circles of the surface inhomogeneity and their radii. The propagation direction of the surface waves and axes of the SPP phase vortices is perpendicular at the interface of the metal and dielectric layers. The SPP vortex lattice is similar to the optical vortex lattice, which was investigated, for example, in [41] with rectangular symmetry, [42] with nonrectangular symmetry, and [43] with Moire fringes. In the considered case, the 2D SPP vortex lattice has a nonrectangular form and a reflection symmetry. 3. PLASMON LOGIC GATES Consider the proposed experimental device for generating and controlling the SPP vortex lattice (Fig. 1). This device has the following parameters: the permittivity of the layer above the metal layer is ε0 1 (air), and the permittivity of the metal 0 00 iεM −12.64 i1.40 (polycrystalline layer is εM −εM gold layer with thickness 53 nm) [44] when excited by a laser beam with a wavelength in air of λ0 630 nm. In the considered case, the SPP propagation constant is β 1.08 × 105 cm−1 , and the wavelength of the SPP is λ 2π∕β 581 nm. The decrements of the SPP have the values α0 4.19 × 104 cm−1 in the air and αM 3.71 × 105 cm−1 in the metal layer. They correspond to the distances along the normal x axis of h0 238 nm and hM 26.9 nm, where the transverse field of the SPP attenuates. Suppose a point charge −q is located at the tip of the control nano-probe. It creates the field strength E 0 −q4πε0e h2p −1 , where ε0e 8.84 × 10−12 F∕m. The intensity of the electrostatic field decreases as E ξ ∕E 0 h2p ∕ξ2 at the boundary of the circle with the radius r 0 , i.e., at the distance ξ h2p r 20 1∕2 from the charge. The intensity of the electrostatic field decreases at the boundary of the permittivity inhomogeneity area (Fig. 2) with the radius r 0 10 μm as
E ξ ∕E 0 ≈ 10−5 , the area with the radius r 0 1 μm as E ξ ∕E 0 ≈ 10−3 , and the area with the radius r 0 100 nm as E ξ ∕E 0 ≈ 10−1 at the height hp 30 nm of the nano-probe above the metal layer surface. The voltage on the control probe located at the height hp is U E 0 hp . The field strength under the control probe is E 0 33.3 × 103 V∕m at the voltage on the control probe U 1 mV, and the value of the positive “mirror” charge is equal to q 4πε0e hp U 0.111 × 10−12 C. It is possible to increase the mirror charge at constant voltage U const, if a dielectric lump with permittivity ε0 → ε > 1 (but ε < jRe εM j) is placed between the control nano-probe and the metal layer. This will lead to the capacitance increasing in the space between the control nano-probe and the metal layer, and we get the charge value q CεU . The scattering of the SPPs is inelastic at the boundary of this inhomogeneity area. However, some SPPs are still reflected from the inhomogeneity, and then they interfere with the incident SPPs. One can change the radius r 0 of the inhomogeneity area of the metal permittivity by varying the voltage U at the control nano-probe U 2 ∕U 1 q2 ∕q1 r 202 h2p 1∕2 r 201 h2p −1∕2 at a fixed nano-probe height hp above the metal surface. Then the radius of the inhomogeneity area varies as r 02 g 2 r 201 g 2 − 1h2p 1∕2 , where g U 2 ∕U 1 , i.e., the radius r 0 varies in proportion to the voltage between the nano-probe and the metal layer. The radii of the inhomogeneity boundary r 01 and r 02 in the metal layer (Fig. 2) change when the potential of the nanoprobes varies. Thus, the localization of the SPP vortices on the surface of the metal layer can be controlled by changing the potential of the nano-probes, i.e., we can control the vortex lattice. The SPP vortex lattice is formed by applying two signals E 01 and E 02 with negative potential to the control nano-probes (Fig. 3, right column). The SPP vortex lattice is shifted when only one signal E 01 (or E 02 ) is applied to the nano-probes (Fig. 3, left column). The vortex size is determined by the SPP wavelength λSPP ∼ 0.5 μm. The position of the vortex is read out by a microsensor MS1 or MS2. The microsensors can be designed on various physical principles [8], which allow reading out of the SPP vortex locations. We suppose that the microsensors MS1 and MS2 are positioned above points of the metal layer where two SPP vortices are formed, while applying two signals E 01 and E 02 at the nano-probes. If only one signal E 01 or E 02 is applied to the appropriate nano-probes, the SPP vortices do not occur under the microsensors. Thus, the signals in the microsensors will depend on the signals at the nano-probes. It is possible to design the plasmon logic gate “AND” based on this principle. If two signals E 01 and E 02 are applied to the control nano-probes at the same time, they form the boundary of dovetail inhomogeneity (Fig. 2). The SPP vortices (Fig. 3, right column) are localized under the microsensors (Fig. 1), and the summary vortex signals in the microsensors MS1 and MS2 appear (Table 1). The plasmon logic gate “NOT” can be designed on the following principle. We assume that the external electrostatic field is applied to the control nano-probes, then dovetail inhomogeneities are formed (Fig. 2), and the SPP vortices appear. Then summary signal arises in the microsensors MS1 and
Table 1. Plasmon Logic Gate “AND” AND E 01 0 1 0 1
E 02
MS01 MS02
0 0 1 1
0 0 0 1
MS2 (Fig. 1). In the presence of one signal E 01 or E 02 at the control nano-probes (Fig. 3, left column), or two signals E 01 and E 02 at the same time (Fig. 3, right column), the SPP vortices are shifted [Figs. 3(e) and 3(f )], and the summary signal in the microsensors MS1 and MS2 disappears (Table 2). The plasmon logic gate “OR” can be realized by two coupled plasmon waveguides [11]. In this case the signals E 01 and E 02 are read out by one plasmon microsensor M S (Table 3). An alternative option for application of microsensors for reading out the SPP vortices is using the wide-range near-field sensor [8] located above the metal layer. It permits readout of the whole vortex lattice. Another option for reading the SPP vortex location is placing an array of short vertical nanowires on the surface of the metal layer. The SPP vortices at the interference field appear in the points of location of the nanowires, when the signals E 01 and E 02 are applied to the control nano-probes. In this case, the signals are radiated (or not radiated) by such nano-antennas in the form of bulk waves, which are then read out by electromagnetic field sensors. The next option for reading out the SPP near field is the excitation of signals in the nanoparticles located above the surface of the metal layer without contact with the metal. The plasmon signals are excited and transmitted for subsequent processing in such reading out of the nano-probes. Thus, using a wide-area near-field sensor or the array of nanowires and a row of control nano-probes will allow realization of the matrix calculations,
Table 2. Plasmon Logic Gate “NOT” NOT E 01 0 1 0 1
E 02
MS01 MS02
0 0 1 1
1 0 0 0
Table 3. Plasmon Logic Gate “OR” OR E 01 0 1 0 1
E 02
MS
0 0 1 1
0 1 1 1
as well as multifactorial logical operations in the plasmon logic gates. The proposed plasmon logic gates “AND,” “OR,” and “NOT” form a functionally complete basis for logical operations, necessary for creating the plasmon processor operating at optical frequencies ω ∼ 1015 s−1 in the format of a computer chip. Signal transmission between logic gates at such frequencies can be transmitted by plasmon waveguides, and the signals between plasmon chips can be transmitted by optical waveguides. Taking into account the fact that the wavelength of the SPP λSPP 2π∕β is shorter than the wavelength in the optical waveguide, the sizes of the plasmon logic gates are smaller than the sizes of the all-optical logic gates for the computer chip. The speed of response of the plasmon logic gates is determined by the time of the SPP propagation with velocity v SPP d β∕d ω−1 on the surface of the metal layer and by the speed of the response of the microsensors. The clock frequency in plasmon microchips is set by the built-in spaser. Now there are the all necessary technologies and components (surface etching, vacuum deposition, plasmon microsensors [8]) for the fabrication of the proposed logic gates for an optical chipset. 4. CONCLUSION The interference of the incident and reflected SPPs from the inhomogeneity of permittivity in the metal layer leads to the formation of vortices. The SPP vortices appear in the singular points on the surface of the metal layer where the amplitude of the SPP Poynting vector is equal zero and the phase is not defined. The distribution of the normal component of the Poynting vector on the surface of the metal layer determines the localization of the SPP vortices. One can read out the localization of the SPP vortices or the whole vortex lattice by use of the appropriate microsensors. Generation and control of the SPP vortices by means of an external electric field make it possible to design and realize the plasmon logic gates “AND,” “OR,” and “NOT.” These plasmon logic gates represent a functionally complete basis for logical operations in processors operating at optical frequencies. The control of the SPP vortex lattice by external signals can be applied to realize the matrix calculations in plasmon processors. APPENDIX A On the surface of a metal layer the field components of incident SPP do not change along the transverse axis y, i.e., ∂∕∂y → 0. From the Maxwell equations ∇ × H −ic −1 ωεE and ∇ × E ic −1 ωμH, we obtain the system of equations for the TM mode of SPPs in media with constant dielectric permittivity ε const and magnetic permeability μ const, ∂H y ωε i Ex, c ∂z (A1) ∂H y ωε ∂E ∂E ωμ H y: −i E z , − z x i c c ∂x ∂x ∂z We obtain from the system of Eqs. (A1) the second-order equa2 2 tion ddz 2 εμ ωc 2 α2 H y 0 for the magnetic-field component H y of the TM mode. Representing the solution of the
equation in the form H yj H 0j expiβz, we obtain the dispersion equations for the TM mode in the dielectric and in the metal layers, εj μj c −2 ω2 − β2 α2j 0,
(A2)
where j D, M . From the dispersion Eq. (A2) we find the decrements of the SPP αj β2 − εj μj c −2 ω2 1∕2 in the media. The tangential components of the TM mode of the SPPs at x 0 are continuous H yD H yM , E zD E zM , that is −1 ε−1 D ∂H Dy ∕∂x εM ∂H M y ∕∂x, whence we find the boundary conditions −εM αD εD αM :
(A3)
It follows from the boundary conditions in Eq. (A3) that the TM mode of the SPPs is excited at the interface between media with different signs of permittivities. Using the expressions for decrements and boundary conditions, we represent the dispersion equation for the TM mode of the SPPs in the form ω2 −
ε2D − ε2M c 2 β2 0: εD εM εD μM − εM μD
(A4)
It follows from the dispersion Eq. (A4) that the propagation constant of the SPP has the real and imaginary parts β β 0 iβ 0 0 , since the permittivities εD and εM are complex quantities. The SPP propagation constant is ω εD εM εD μM − εM μD 1∕2 : (A5) β c ε2D − ε2M Funding.
V. I. Vernadsky Crimean Federal University.
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