Vector beam generation based on the nanometer-scale rectangular holes

Qi Zhang; Peiyu Li; Yanying Li; Han Wang; Lixia Liu; Yuan He; Shuyun Teng

doi:10.1364/OE.25.033480

1. Introduction

Polarization is a significant optical property of light, even more important than the spectra and intensity distribution in many applications. Vector beam (VB) usually points to an optical field with spatially inhomogeneous polarization and cylindrically symmetric intensity distribution [1]. Among various vector beams, the radially and azimuthally polarized beams are the most common vector beams. Due to the advantages of VB in focusing [2, 3] and imaging [4], the generation and applications of VB attract much attention and the study about VB becomes a hot research spot in modern optics and photonics. Various methods to generate VB including subwavelength gratings [5, 6], optical fiber [7, 8], spatial light modulator [9], optical crystals [10, 11], and ring resonator [12, 13] have been reported. And VB has been also applied in optical trapping and rotating [14–16], optical microscopy [17–19], and quantum information processing [20, 21]. For promoting the applications of VB in microoptics and nanometer photonics, these optical elements for generating VB should be compact and easy to manufacture.

In recent years, the surface plasmon polaritons (SPPs) excited by nanometer structures have gained researchers’ attention because they can effectively manipulate light in subwavelength scale [22–24]. Various nanometer devices based on the excitation of SPPs have been continuously reported, such as plasmonic lens [25, 26], plasmonic polarization state analyzer [27–29], plasmonic wave plate [30–33], and optical vortex generator [34, 35]. These researches provide us an enlightenment to design the compact plasmonic VB generator with the nanometer units. In this work, we aim to propose a kind of plasmonic VB generator based on the space-variant rectangular holes etched in a silver film, and this kind of generator can transform the circularly polarized light into VBs including radially and azimuthally polarized light. Each rectangular hole can be taken as a quarter-wave plate and it can realize the polarization conversions of circularly and linearly polarized light. The parameters of rectangular holes are optimized by three-dimensional finite-difference time-domain (FDTD) method. Adjust the direction of these rectangular holes and arrange them along different Archimedes' spirals, the vector beams are produced under the circularly polarized light illumination. The theoretical analysis based on Jones matrix predicts the formation mechanism of VB generation. The measured intensity distributions of the proposed metallic structure in practical experiment show the properties of VB.

2. Design principle of the VB generator

To obtain the VB, we firstly optimize the size of a single rectangular hole so that it possesses the property of a quarter-wave plate, namely, it can convert the circularly polarized light into the linearly polarized light, and vice versa. The schematic diagrams for the polarization conversation of an optimized rectangular hole are shown in Fig. 1. Where the rectangular hole is etched in a silver film which is deposited on a glass substrate and the hole can be described by the length L₁, the width L₂ of the rectangular hole and the thickness h of the silver film. These three parameters are optimized according to FDTD method. In numerical simulations, the wavelength of the incident plane wave takes λ = 632.8nm, the dielectric constant for the silver film is ε = −15.92−j1.075 [36] and the corresponding SPP wavelength is λ_spp = 612.5nm. The perfect matched layer is chosen as the boundary condition and the minimum mesh step is set at 2.5nm. A linear monitor on z axis just starts from the center of the hole on the exit surface, and a two-dimensional transverse monitor with z normal is set at 3μm above the silver film.

Fig. 1 The polarization transformation of the light beam through a rectangular hole, (a) linear polarization into circular polarization, (b) circular polarization into linear polarization

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The optimized parameters for the rectangular hole are L₁ = 310nm, L₂ = 200nm and h = 200nm, and Fig. 2 gives the diffraction distributions of the optimized hole with different polarized light illumination. Figures 2(a)-2(d) show the curves for the electric components E_x (red) and E_y (green) varying with the coordinate z with 45° linear polarization, −45° linear polarization, right-handed circular polarization (RCP) and left-handed circular polarization (LCP) illumination, respectively, they are detected by the linear monitor which is set at the center point of the hole with x = 0 and y = 0. Color patterns in Figs. 2(e)-2(h) are the corresponding phase distributions of the electric field component E_z on the preset two-dimensional transverse monitor plane with z = 3μm, where the black arrows represent the polarization states of the incident light and the black squares denote the positions of the holes. From Figs. 2(a) and 2(b), we can see that the oscillations of E_x and E_y dislocate one-fourth period, and this means E_y has π/2 delay for the former and E_x has π/2 delay for the latter. Figures 2(c) and 2(d) show the oscillations of E_x and E_y are synchronous for RCP polarization illumination and reverse for LCP polarization illumination, namely in phase and out of phase, respectively. For these four cases, E_x and E_y have the same amplitude. This means that the 45° linear polarization can be transformed into LCP, and the monitored phase distribution increasing from −π to π in anticlockwise gives the verification, as shown in Fig. 2(e). The −45° linear polarization is transformed into RCP, and the phase variation in clockwise shown in Fig. 2(f) demonstrates this conclusion. Similarly, Figs. 2(g) and 2(f) verify the conversation of the circular polarizations into the 45° and −45° linear polarizations, respectively.

Fig. 2 Distribution curves of Re(E_x) and Re(E_y) of a rectangular hole illuminated by (a) 45° linear polarization light, (b) −45° linear polarization light, (c) RCP light and (d) LCP light, and their corresponding phase distributions (e-h)

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These simulated results demonstrate that the optimized rectangular hole can change the polarization state of the incident light just having the same function with a quarter-wave plate, and the fast axis of this equivalent quarter-wave plate is along the long edge of hole. The Jones matrix expression of the rectangular hole with the cross angle α between the fast axis of the hole and x axis can be written as,

T (α) = (\begin{matrix} \cos α & \sin α \\ \sin α & - \cos α \end{matrix}) (\begin{matrix} 1 & 0 \\ 0 & i \end{matrix}) (\begin{matrix} \cos α & \sin α \\ \sin α & - \cos α \end{matrix}) = (\begin{matrix} \cos^{2} α + i \sin^{2} α & \sin α \cos α (1 - i) \\ \sin α \cos α (1 - i) & \sin^{2} α + i \cos^{2} α \end{matrix})

Naturally, if many optimized holes are oriented respectively towards the different directions, the spatially homogeneous circular polarization light can be modified into the spatially inhomogeneous polarization light. Suppose a series of holes are arranged on a determined trajectory and the position coordinates of one arbitrary rectangular hole are denoted by the polar coordinates (r, θ), as shown in Fig. 3. The directional angle α of the fast axis for this hole and the azimuthal angle θ should satisfy the following relation,

Fig. 3 Schematic diagrams of VB generator (left) and the magnified part (right)

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α = θ + π / 4

For LCP and RCP incident lights, their Jones vectors can be expressed as,

E_{L C P}^{0} = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ i \end{matrix}), E_{R C P}^{0} = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ - i \end{matrix})

Under the illumination of LCP and RCP light, the transmission field of a single hole can be approximated by E¹ = T(α) × E⁰. According to Eqs. (1)-(3), the transmission fields of a single hole for two cases can be written by,

{\begin{array}{l} E_{L C P}^{1} = e^{i (θ + π / 4)} (\begin{matrix} \cos θ \\ \sin θ \end{matrix}) \\ E_{R C P}^{1} = e^{- i (θ + π / 4)} (\begin{matrix} - \sin θ \\ \cos θ \end{matrix}) \end{array}

If the rectangular holes are arranged on one spiral with the radial coordination satisfying r = r₀ + mλ_sppθ/(2π), where r₀ is the initial radius of the spiral, m is an integer representing the geometrical charge of the spiral, and θ is the azimuthal angle, as shown by the right diagram in Fig. 3, the electric field near the center of the structure can be expressed approximately by the following integral,

{\begin{cases} E_{L C P} = \int_{0}^{2 π} e^{i k_{s p p} \cdot (ρ - r)} e^{i \frac{π}{4}} e^{i θ} (\begin{matrix} \cos θ \\ \sin θ \end{matrix}) d θ \\ E_{R C P} = \int_{0}^{2 π} e^{i k_{s p p} \cdot (ρ - r)} e^{- i \frac{π}{4}} e^{- i θ} (\begin{matrix} - \sin θ \\ \cos θ \end{matrix}) d θ \end{cases}

where k_spp is the wave vector of SPP. When the observation point is close to the center of the structure, the directions of k_spp and r are almost opposite, and k_s_pp·r and k_spp·ρ can be written as k_spp·r = k_sppr and k_spp·ρ = -k_sppρcos(β-θ), respectively. Insert them into Eq. (5) and perform the integral operations, the electric field distributions near the center of the structure are,

{\begin{cases} E_{L C P} = B_{1} i^{- m} e^{i m β} (\begin{matrix} J_{- m} (- k_{s p p} ρ) - e^{i 2 β} J_{- (m + 2)} (- k_{s p p} ρ) \\ i e^{i 2 β} J_{- m} (- k_{s p p} ρ) + i J_{- (m + 2)} (- k_{s p p} ρ) \end{matrix}) \\ E_{R C P} = B_{2} i^{- m} e^{i m β} (\begin{matrix} i e^{- i 2 β} J_{- (m - 2)} (- k_{s p p} ρ) - i J_{- m} (- k_{s p p} ρ) \\ J_{- m} (- k_{s p p} ρ) + e^{- i 2 β} J_{- (m - 2)} (- k_{s p p} ρ) \end{matrix}) \end{cases}

Where B₁ and B₂ are the constant terms. If the integer m takes −1 for LCP illumination and 1 for RCP illumination, and neglect the constant phase terms, Eq. (6) can be simplified by,

{\begin{cases} E_{L C P} \propto J_{1} (- k_{s p p} ρ) (\begin{matrix} \cos β \\ \sin β \end{matrix}) \\ E_{R C P} \propto J_{1} (- k_{s p p} ρ) (\begin{matrix} - \sin β \\ \cos β \end{matrix}) \end{cases}

It is easy to find that the electric field distributions are the multiples of the first-order Bessel function and the Jones matrix of the radial light beam and the first-order Bessel function and the Jones matrix of azimuthal light beam. These equations represent a radial polarization beam forming with LCP illumination and an azimuthal polarization beam forming with RCP illumination. The theoretical analysis indicates that designed structures really generate the vector beams.

3. Simulations for VB generation

For examining the function of VB generators, the diffraction of the VB generators mentioned in section 2 are simulated according to FDTD method. The structures of the VB generators with m = 1 and m = −1 are drawn in Figs. 4(a) and 4(e), and Fig. 4(a) is for generating azimuthal polarization beam with RCP illumination, and Fig. 4(e) is for producing the radial polarization beam with LCP illumination. The rectangular holes are chosen as the optimized ones, and the initial radius of the spirals for the VB generators are r₀ = 10λ_spp. For the rotation angles of holes, they are adjusted slightly to eliminate the influence of the interference between holes. The rotation angles of holes in Fig. 4(a) increases 6°, and the rotation angles of holes in Fig. 4(b) decreases 6°. The color patterns in Figs. 4(b)-4(d) are the distributions of the electric field |E|² and the components |E_x| and |E_y| for the first structure, and those in Figs. 4(f)-4(h) are for the second structure. The phase distributions of the electric fields for two cases are inserted in Figs. 4(b) and 4(f). Where the monitor planes are set at 3μm above the silver film. The electric field intensity distributions in Figs. 4(b) and 4(f) show the annular field distributions with the central dark points, and the phases at the center are homogeneous. The symmetric bright stripes in Figs. 4(c) and 4(d) always appear on the direction perpendicular to the given polarization. This indicates that the pure azimuthal polarization field generates. However, the symmetric bright stripes in Figs. 4(g) and 4(h) always exist on the direction parallel to the given polarization direction. This represents that a radial polarization beam produces. These results verify the theoretical predictions of Eq. (7).

Fig. 4 The azimuthal polarization generator (a) and its electric field distributions (b)-(d) with RCP illumination, and the radial polarization generator (e) and its electric field distributions (f)-(h) with LCP illumination

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4. Experimental verification for the vector beam

For evaluate and verify the performance of the VB generator, we manufacture the samples of the VB generators. The silver with the thickness 200nm is firstly deposited on glass substrates by magnetron sputtering method, and then the rectangular holes are etched in the silver film by the focused ion beam etched technique. The fabricated holes have the same size of the length 300nm and the width 100nm, and they are arranged on the Archimedes' spiral of r = r₀ ± λ_sppθ/(2π) with r₀ = 6.13μm. The SEM pictures for the samples of the vector beam generators are given in Figs. 5(a) and 5(b). The former is for the azimuthal polarization beam generators and the latter is for radial polarization beam generator. Place one VB generator sample into the experimental light path like Fig. 5(c), and measure its diffraction intensity distribution. In the experiment setup, the linear polarization beam from He-Ne laser passes through a quarter wave plate (QWP) and changes into the circular polarization beam. Then LCP or RCP beam illuminates the fabricated sample (S) to excite the SPP field and generate the vector beam. A microscopy objective (MO) is set behind the sample to magnify and image the electric field intensity distribution of VB generator. Two reflectors M1 and M2 are used to adjust the direction of the light path and a polarizer (P) is applied to select the polarization component of the vector beam. The intensity distribution of VB generator is received by a charge-coupled device (CCD) with No. DU-888U3.

Fig. 5 Two samples of VB generators (a, b), the experiment setup for the verification of the vector beam(c), and the measured results (d-i)

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Figures 5(d)-5(f) give the intensity distributions of the sample in Fig. 5(a) which is illuminated by RCP light. Two latter figures correspond to the cases with the polarizer insert in the light path, respectively, where the inserted white arrows represent the polarization directions of the polarizer. Without the polarizer, the intensity distribution seems to be an annular shape with a dark center, and with the polarizer, the split stripes appear along the direction perpendicular to the polarization direction of the polarizer. This is just the property of the azimuthally polarized light. Figures 5(g)-5(i) are the intensity distributions of the sample in Fig. 5(b) when it is illuminated by LCP. The annular distribution also appears when the polarizer is not set in the light path, as shown in Fig. 5(g), and the split stripes take on when the polarizer is placed in the light path. The symmetric stripes appear along the polarization directions of the polarizer, and this is just the property of radially polarized light. All the experiment results conform very well to the numerical simulations, and they also tally with theoretical analysis. This means our designed VB generators are effective to generate the azimuthally and radially polarized light. Carefully observe the measured patterns, we also see the symmetry of the intensity distributions is not perfect. We think this is because the fabricated sample using the focused ion beam etched technique is not flawless, and maybe some rectangular holes are not etched well. The deviation of the sizes and the rotation angles of the holes from the optimized ones may influences the rotational symmetry of the intensity distributions.

5. Conclusion

In conclusion, this paper proposes a kind of metallic structure consisting of rectangular holes to generate azimuthal and radial polarization beams. Each optimized rectangular hole is taken as a quarter-wave plate to control the polarization state of light. Adjust the direction of the rectangular holes, arrange these holes in spiral trajectory and choose the chirality of circular polarization, the pure azimuthal and radial polarization beams can be generated. The detailed theoretical analysis about the diffraction of the rectangular holes arranged in Archimedes' spirals provides the foundation for the generation of the azimuthal and radial polarization beams. The directions of rotation of Archimedes' spirals correspond to the RCP and LCP illumination, respectively. The numerical simulations and experiment measurements give the strong verification for VB generation. This kind of plasmonic VB generator has the advantages of the compact structure and the ease to use and it will promote the applications of vector beams in nanometer photonics.

Funding

National Natural Science Foundation of China (NSFC) (0874105); Shandong Provincial Natural Science Foundation of China (2015ZRB01864).

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Vector beam generation based on the nanometer-scale rectangular holes

Abstract

1. Introduction

2. Design principle of the VB generator

3. Simulations for VB generation

4. Experimental verification for the vector beam

5. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (7)

Optics Express