Controllable multiple plasmonic bending beams via polarization...

Controllable multiple plasmonic bending beams via polarization of incident waves HUI LI,1 2 YU QU,1 HAMAD ULLAH,1 BIN ZHANG,2 AND ZHONGYUE ZHANG1,* 1School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China 2Division of Public Infrastructure, De-hong Vocational College, Mang-shi 678400, China *[email protected]

Abstract: Plasmonic bending beams, which preserve their spatial shapes while propagating along curved trajectories in metal-dielectric interface, offer important applications in the fields of fiber sensor, optical trapping, and micro-nano manipulation. In this work, circular hole array, as a local point-like sources of surface plasmon polaritons, is designed on the metal film to generate multiple plasmonic bending beams. The electric field intensity of multiple plasmonic bending beams is controlled by polarization angle of input light. In addition, the electric filed intensity of multiple plasmonic bending beams relies on circle hole radius. These findings provide guidance in the design and optimization of plasmonic bending beam generators. © 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

OCIS codes: (260.2110) Electromagnetic optics; (240.6680) Surface plasmons; (060.5060) Phase modulation; (250.5403) Plasmonics; (240.5440) Polarization-selective devices.

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1. Introduction Surface plasmon polaritons (SPPs) are electromagnetic waves in two-dimensional system, that consist of collective electron oscillations in metal-dielectric interface decay, thus, their diffraction limits can be surpassed [1, 2], thereby offering important applications in the fields of photonics and electronics [3–5]. Plasmonic bending beams, as a specific SPP waves, which preserve their shapes while propagating along desired curved trajectories in metal-dielectric interface, are attracting increased attention. They offer potential applications in the fields of optical trapping [4–6] and micro-nano manipulation [7–9].

Various plasmonic bending beams, such as Airy beam [10–15], Weber beam, Mathieu beam, and arbitrary bending beams (propagating along the trajectory of xn, polynomial, and


logarithmic) [16–21], were reported in the last several years. These beams were achieved using various nanostructure arrays, such as hole array (periodic scatterers) [15–17], grains array [18], two-dimensional binary phase mask gratings [19–21], and polymethyl methacrylate microsphere (a specific refractive index structure) [22, 23]. In addition, cosine-Gauss beam, which propagates in a straight line along metallic surface, was demonstrated by a group of intersecting metal gratings [24–28]. These studies demonstrate that the phase of SPP waves was controlled to reconstruct arbitrary new SPP wavefront profiles [16, 21, 25].

Controlling plasmonic complex fields by polarization-sensitive nanostructure arrays are important for enriching the components of nanophotonic devices to analyze the polarization [29, 30]. Various polarization-sensitive nanostructure arrays, such as aperture arrays [29], F-shaped nanoslits [30], Δ-shaped nanoantennas [31], and double-lined distribution nanoslits arrays [32, 33], were designed to manipulate propagation directional of SPP waves. Multi-lined distributed nanoslits arrays were designed to achieve focused SPPs [32–34]. Cross-shaped nanoantennas were designed to generate double focused SPPs [35]. Multiple rings of nanoslits with special orientations was designed to control SPP orbitals [36]. Ring-shaped hologram consisting of slit-pair array was designed to achieve switchable SPPs [37]. All of these reports demonstrate that phase and amplitude of the generation SPP beams can be independently controlled by polarization-sensitive nanostructures. These reports also show that these tunable and compact beams have extensive potential applications [32]. However, these beams were mostly controlled by adjusting the structural orientation angles. If users want to control structural orientation angles, they need rebuild the structures, which is not convenient.

In this study, a circular hole array is used as local point-like sources of SPPs to generate SPP wavefront. The desired multiple plasmonic bending beams are generated with propagation along the desired trajectory. The intensity of the electric fields of multiple plasmonic bending beams is controlled by the polarization angle of the input light. In addition, the dependence of the intensity of the electric fields of multiple plasmonic bending beams on the radius of circular hole is studied. All these findings can provide deep applications in designing polarization-controlled plasmonic bending beam generators or polarization sensing devices.

2. Structure and calculation method All of motion of electro-magnetic can be described by the Helmholtz equation [19]. Plasmonic bending beams, the solutions of the two-dimensional (propagation in the x-y plane) Helmholtz equation, can be expressed [19, 38],

2 2( , ) ( , ) 0,sp sp spE x y k E x y∇ + = (1)

where ksp represents the wave-vector of SPPs on the metal surface. ESP(x, y) represents the distribution of SPP waves in x-y plane. Figure 1 shows the geometric diagram of curve arrangement of circular hole array. Every circular hole is regarded as an independent electric field radiation source. Both the amplitude and phase of SPP are controlled by arrangement of circular hole array. In order to achieve the distribution of plasmonic bending beams, the phase modulation method is introduced. Phase φ(x) of the required plasmonic bending beams was calculated utilizing Legendre transform method [18–20]. The geometric relationship between the phase, the angle θ, and the desired arbitrary bending trajectory f(y) is defined in Fig. 1. The phase of the required plasmonic bending beams can be obtained for the case of non-paraxial ( 2sin tan / 1 tanθ θ θ= + ) regimes [18–20, 36, 38],

20( ) tan / 1 tan ,x k dxφ θ θ= − + (2)


where tan ( )f yθ ′= , ( )f y′ is the first-order derivative of f (y). In this study, in order to deduce simply, the case of paraxial sin θ ≈tan θ was applied. The phase of the required plasmonic bending beams was obtained for paraxial regimes [20, 36, 38–40],

0( ) tan .x k dxφ θ= − (3)

The position of every radiation source was directly located by the phase ( )xφ .

Fig. 1. The geometric diagram of several radiation sources; illustration of an arbitrary bending trajectory f(y) and the tangent line, the dull-red circle points represents circle holes array (independent electric field radiation sources En).

The configuration of the Au/SiO2 structure is shown in Fig. 2(a). The thickness of Au film and SiO2 substrate are fixed at 0.1 μm and 0.3 μm, respectively. Circular hole arrays are designed on the Au film. The blue block diagram illustrates a local amplification of the structure and the definitions of structural parameters, where r represents the radius of circular hole, Dn represents the distance between adjacent radiation sources, β represents the polarization angle between the in-plane incident electric E0 and the x axis. W and Hn represent the distances in x and y direction between adjacent radiation sources, respectively.

In this study, to avoid near-field interaction between adjacent neighboring holes which can reduce the intensity of bending beams, the distance W is set as 0.6 μm. The quadratic curve (f(y) = - ay2) is chosen, where constant a is 1.13 × 10−2. According to Eq. (3), the desired phase ( 1.5( ) 1.33x kaxφ = − ) is obtained for paraxial regimes [41]. The distances

1n n nH y y+= − are calculated when the number of circular hole is equal to 20 [Fig. 2(b)]. The propagation properties of the plasmonic bending beams are simulated using finite-difference time-domain method. A plane wave with 0.633 μm wavelength is incident ( + z) into the circular holes. The relative permittivity εm of the gold film is −9.8 + 1.96i at 0.633 μm [42].


Fig. 2. (a) The configuration of the proposed structure. The blue block diagram illustrates a local amplification of the structure and the definitions of structural parameters. (b) Calculated the distances Hn of radiation source.

3. Results and discussion The electric field intensity distribution in the x–y plane is depicted in Fig. 3(a), with the polarization angle β is 0°, and the radius r is 0.2 μm. It shows that the target plasmonic bending beams (corresponding to symbol I) is generated with the propagation along the desired trajectory. The black solid curve depicts the parabolic curve f(y). Furthermore, the second (corresponding to symbol II) and third (corresponding to symbol III) plasmonic bending beams that propagate along the parabolic curve f(y) are observed. The second and third plasmonic bending beams are also satisfied with the proposed design principle. The transverse electric field intensity distributions at y = −5 μm and x = 17 μm are shown in Fig. 3(b) and 3(c), respectively. These results demonstrate that multiple plasmonic bending beams are generated by especially designed circular hole arrays.

Fig. 3. (a) Simulation electric field intensity distribution of the proposed structure with angle β = 0° and radius r = 0.2 μm. Transverse electric field intensity distributions at (b) x = 17 μm and (c) y = −5 μm..

The electric field intensity distributions with different polarization angle β in x–y plane are depicted in Fig. 4, with the radius r is 0.2 μm. It shows that, as the angle β increases from 0° to 90° [Fig. 4(a) – 4(c)], the electric field intensities of beams I and II decrease, and the electric field intensity of beam III increases. As the angle β increases from 90° to 180° [Fig.


4(c) – 4(e)], the electric field intensities of beams I and II increase, and the electric field intensity of beam III decreases.

To understand the relationship between the electric field intensity of SPP waves and the polarization angle β, the radiation properties of a circular hole array are analyzed theoretically. When every circular hole of an array is illuminated by incident light, the surface plasmons are excited and can be regarded as an electric dipole. The electric dipoles can be written as [43]:

0 .sp mnjk r

n n n mn m m mm n m n

p E A p c p eδ δ −≠ ≠

= − + (4)

The first term represents the interaction between in-plane incident electric field and the circular holes. nδ depends on the radius r of the circular hole and the permittivity of a metal εm. The second term represents the interaction among the circular holes in the array. The third term represents the field from the propagating SPPs [43]. In this study, the distance between neighboring circular holes is larger than 0.2 μm, and the interaction among other dipoles (the second and third terms) is relatively weak. Every electric dipole in the array is approximated as Pn = δnE0. Thus, the electric distribution of the n-electric dipoles can be approximatively written as

0 sin( )2sp njk r

nE E eπ β −∝ − .

The far-field distribution of SPP waves Esp(x, y) from each independent radiation source (electric dipoles) can be obtained as a superposition of the electric fields that are radiated from every source,

1

1 2

01

( , ) = ...

sin( ) (1 cos( sin )),2

sp

sp n

njk r

sp ni

E x y E E E

E e k Lπ β θ−=

+ + +

≈ − +

(5)

where n is an integer, 1 sinn n n nr r r D θ+Δ = − = ,2 2( )n nL nW h≈ + , and

1 2 ...n nh H H H= + + + . In this study, n, W, Hn, and angle θ have invariable values. Thus, the electric field intensity distribution of SPP waves have a relationship with angle β. The maximum electric field intensity occurs at β = 0°.

Electric fields at selected point A (x = 17 μm, y = −4 μm) of beam I, point B (x = 17 μm, y = 12 μm) of beam II, and point C (x = 2 μm, y = −5 μm) of beam III are studied to unveil this polarization-dependent property. The polar plots of the electric field intensity values of points A, B, and C with varied values of polarization angle β are shown in Fig. 4(f). The intensity of electric field at point B is about 0.14 times of the intensity of incident light wave (magnitude 1V/m). The equation to fit these data is given by

0sin( ).f D E ϕ ϕ= + ∗ − (6)

The solid curves in Fig. 4(f) result in the fitting of Eq. (6). The goodness for beams I, II, and III is 0.969, 0.963, and 0.977, respectively. These results demonstrate that the electric field intensity of multiple plasmonic bending beams (I, II, and III) are controlled by the polarization angle of incident light, which provides a method of dynamic controlling and arbitrarily selecting multiple plasmonic bending beams.


Fig. 4. The electric field intensity distribution the proposed structure with the different angle β, (a) β = 0°, (b) β = 45°, (c) β = 90°, (d) β = 135°, and (e) β = 180°. (f) The polar diagram for electric field intensities at Points A, B, and C with varied polarization angle β.

The effects of the radius of a circular hole on the multiple plasmonic bending beams are also investigated. Figure 5(a) – 5(c) show the electric field intensities of different circular hole radii (r = 0.1 μm, 0.15 μm, and 0.2 μm) with a polarization angle β = 0°. The transverse electric field intensity of different circular hole radii r at x = 17 μm and y = −5 μm is shown in Fig. 5(d) and 5(e), respectively. As the radius increases, the electric field intensities of the beams increase. Because as the radius of the circular hole increases, the intensity of the radiation electric field of SPPs increases, thereby resulting in the increasing intensities of incident light in the circular hole. In addition, Fig. 5(d) and 5(e) show that as the radius increases to 0.25 μm, the electric field intensities of the beams decrease. As the radius increases (the distance Dn is fixed at 0.6 μm), the interaction among dipoles (i.e., the effect of the second and third terms in Eq. (4)) is enhanced and the radiation electric field intensity of SPPs decreased. The electric dipole mode converts to quartic dipole mode, which results in the change of the orientation and intensity of radiation electric field with the increasing radii. Meanwhile, the polar plots of the electric field intensity values of point A, with varied polarization angle β values and varied circular radius r, are shown in Fig. 5(f). The solid curves represent the fitting values of Eq. (6). These results demonstrate that the value of electric field intensity can be controlled by simply selecting suitable radius circular whole arrays.


Fig. 5. Simulation electric field intensity distribution of the proposed structure with different radius r of circle hole array: (a) r = 0.1 μm, (b) r = 0.15 μm, and (c) r = 0.2 μm. Transverse electric field intensity distributions with different radius r of circle hole array at (d) x = 17 μm and (e) y = −5 μm, (f) The polar diagram for electric field intensities at Points A with varied polarization angle β values.

The third plasmonic bending beams (beam III) is smaller than the other two plasmonic bending beams because the electric field intensity contribution of the top part of beam III is smaller than the beams located far from the circular hole. In addition, the paraxial limit (i.e., θ < 20°) is considered. The results are obtained in the approximation case in this study, where sin tanθ θ≈ . Thus, the approximation conditions (i.e., θ < 20°) cannot be matched at the top part of the target curve.

4. Conclusion In conclusion, a special arrangement of circular hole array is designed on the metal film to generate multiple plasmonic bending beams. Every circular hole of the array is applied as a local point-like source of SPPs and regarded as an independent radiation source. The multiple plasmonic bending beams propagate along the desired trajectory. The electric field intensity of multiple plasmonic bending beams is controlled by the polarization angle. In addition, the radius increases with the electric field intensity. These simulated results are consistent with the theoretical calculation. All these findings offer potential applications in polarization sensors, optical tweezers, and polarization-controlled plasmonic bending beam generators.

Funding National Natural Foundation of China (61575117); Fundamental Research Funds for the Central Universities of Ministry of Education of China (GK201601008 and GK201604003).


Controllable multiple plasmonic bending beams via polarization...

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