Plasmonic circuit for second-order spatial differentiation at the subwavelength scale

Yongsop Hwang; Timothy J. Davis; Jiao Lin; Xiao-Cong Yuan

doi:10.1364/OE.26.007368

1. Introduction

Localized surface plasmon (LSP) resonances are the oscillations of collective electrons that can be excited by light on the surfaces of metallic nanostructures. The optical energy is confined to nanoscale volumes by the LSP resonances with potential for high-density integration of optical components [1,2]. Various nanodevices have been designed and experimentally demonstrated based on LSP resonances exploiting the strong confinement and size reduction [3–5]. In particular, the LSPs couple evanescently among one another when the metal nanoparticles are placed in proximity, which alters the resonance properties enabling intriguing optical phenomena such as plasmon induced transparency [6–8], plasmonic edge states [9], and Fano resonances [10,11]. The resonance properties depend on the geometry and composition of the metal nanoparticles hence a desired function can be realized with a proper arrangement.

Analog optical computing affords notable benefits in the investigation of non-repetitive and statistically scarce phenomena since rapid dynamics can be demonstrated which would cost vast computation resources to be done by digital simulations [12]. Accordingly, various analog optical computing systems have been suggested [13–15] and demonstrated [16, 17]. In particular, there have been substantial efforts to realize photonic computation systems with subwavelength resolution using plasmonics [18] both theoretically [19] and experimentally [17]. For fully functional optical analog computing both first-order and higher-order differentiations in spatial domain are important mathematical operations. In our previous work [20], we showed that a configuration of three metal nanorods, that mimics a Wheatstone bridge circuit in electronics [21], performs a first-order mathematical difference operation that can be used to measure the optical phase differences, or phase gradients, across the two inputs. Subsequently, we demonstrated a two-dimensional metasurface version of this plasmonic device and showed experimentally the subwavelength mathematical operation [22]. For second-order difference operations, there have been theoretical studies proposing optical computation systems for the Laplace operator based on a multilayered scheme at a reflection mode [23, 24] but there has been no investigation of nanoplasmonic devices for spatial second-order differentiation with in-plane subwavelength resolution operating in transmission mode. Exploiting the aforementioned properties of LSP resonances such as confinement and enhancement of optical energy might resolve the problems in miniaturization of photonic devices.

Here, we investigate the properties of a plasmonic circuit that measures the second derivative of the electric field across the circuit inputs. The evanescent couplings of the localized surface plasmon modes supported by the circuit consisting of gold nanorods have been rigorously studied to verify the working principle of the second-order difference operation. Thereafter, the output intensity as a function of the phase difference induced by the oblique incidence of a plane wave has been obtained. Finally, numerical simulations using the finite-difference time-domain (FDTD) method have been performed to verify the function of the circuits in a realistic condition. The performance as a photonic circuit has been discussed with proposing potential applications.

2. Electrostatic eigenmode analysis

The analysis of the plasmonic circuit is based on the configuration shown in Fig. 1(a) which consists of five metal nanorods. Three of the rods, labeled 1, 2 and 3 act as inputs to the circuit. These rods respond to the incident electric field polarized parallel to their long axes, aligned with the y-axis. The electric field E_1,2,3 incident on each rod excites LSPs with amplitudes a_1,2,3(ω) = f(ω)p · E_1,2,3 where all three nanorods are assumed to be identical, each supporting a single LSP resonance with a dipole moment p = pŷ. The resonance factor f(ω) is the polarizability per unit volume and contains information about the LSP resonance.

Fig. 1 (a) The plasmonic circuit that outputs the second derivative of the wavefield across its inputs. The three nanorods (1, 2 and 3) aligned along the y-axis are input nanorods while the other two rods (4 and 5) aligned along the x-axis are the output nanorods. The input wavefield is linearly polarized to y-direction whereas the output is polarized to x-direction. (b) A diagram illustrates the position vectors of the rods, observer, and the displacement.

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The response of the nanorods when coupled together, as in the circuit shown in Fig. 1(a), is obtained from the Electrostatic Eigenmode Method (EEM) of Davis and Gómez [25]. The EEM provides a means by which the amplitudes ã_n of each of the nanorods n in the plasmonic circuit can be written as a linear combination of the amplitudes a_m of the isolated nanorods. The expression involves the inversion of a matrix of coupling coefficients G multiplied by the resonance factors f. In our analysis, we assume all the rods are identical and the dominant coupling is between nanorods 1, 2 and 4 and between 2, 3 and 5, as shown in Fig. 1(a). The sign of G is chosen so that G > 0 for the dipole moments as indicated in the figure.

The amplitudes of the coupled nanostructures are given by

(\begin{matrix} {\tilde{a}}_{1} \\ {\tilde{a}}_{2} \\ {\tilde{a}}_{3} \\ {\tilde{a}}_{4} \\ {\tilde{a}}_{5} \end{matrix}) = {(\begin{matrix} 1 & 0 & 0 & - f G & 0 \\ 0 & 1 & 0 & f G & f G \\ 0 & 0 & 1 & 0 & - f G \\ - f G & f G & 0 & 1 & 0 \\ 0 & f G & - f G & 0 & 1 \end{matrix})}^{- 1} (\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \\ a_{5} \end{matrix})

which involves the inversion of the matrix. This can be done analytically but the result is quite complex. Here we show the LSP amplitudes ã₄ and ã₅ induced in the two ‘output’ nanorods when the incident light has no polarisation component aligned with them, so that a₄ = a₅ = 0. The results are:

\begin{array}{l} {\tilde{a}}_{4} = \frac{f G (a_{1} - a_{2}) + f^{3} G^{3} (a_{2} + a_{3} - 2 a_{1})}{(1 - f^{2} G^{2}) (1 - 3 f^{2} G^{2})} \\ {\tilde{a}}_{5} = \frac{f G (a_{3} - a_{2}) + f^{3} G^{3} (a_{1} + a_{2} - 2 a_{3})}{(1 - f^{2} G^{2}) (1 - 3 f^{2} G^{2})} . \end{array}

The excitations in the two ‘output’ nanorods depend on differences between the LSPs induced in the three ‘input’ rods.

To understand the response of a system of such circuits we need to consider the phase of the outgoing waves polarized parallel to rods 4 and 5. We assume that the LSP in each rod radiates an outgoing spherical wave with a phase e^ikR where k = 2π/λ is the free-space wavenumber. For an observer at position r and a nanorod at position r_r we have the distance $R = | r - r_{r} | = {(r^{2} + r_{r}^{2} - 2 r \cdot r_{r})}^{1 / 2}$ as shown in Fig. 1(b). For r ≫ r_r this can be rewritten as R = r(1 + (r_r/r)² − 2r̂ · r_r/r)^1/2 ≈ r (1 − r̂ · r_r/r) where r̂ = r/r is the vector in the direction of scattering. Then e^ikR ≈ e^{ikr−ik_sr_r} where k_s = kr̂ is the wavevector in the direction of scattering. The output wave from the two nanorods is proportional to ψ = ã₄e^{−ik_s · r₄} + ã₅e^{−ik_s · r₅}. For nanorod 4 located at r₄ = r₀−d/2 and nanorod 5 at r₅ = r₀+d/2 then ψ = e^{−ik_s · r₀} (ã₄e^{ik_s · d/2} + ã₅e^{−ik_s · d/2}). Substituting the expressions for ã₄ and ã₅ from (2) yields

\begin{array}{l} ψ e^{i k_{s} \cdot r_{0}} & = \frac{f G (a_{1} - 2 a_{2} + a_{3}) cos (k_{s} \cdot d / 2)}{1 - 3 f^{2} G^{2}} \\ + i \frac{f G (a_{1} + a_{3}) sin (k_{s} \cdot d / 2)}{1 - f^{2} G^{2}} . \end{array}

For scattering normal to the substrate we have k_s · d = 0 and the output wave depends on ψ = fG(a₁ − 2a₂ + a₃)/(1 − 3f²G²) = f²Gp(E₁ − 2E₂ + E₃)/(1 − 3 f²G²) where we have substituted a_1,2,3 = fpE_1,2,3 for the input LSP amplitudes. Considering just the ŷ components E_1,2,3 at each nanorod, we can expand the incident electric field in a Taylor series E (x) ≈ E₀ + (∂E/∂x)δx + (1/2)(∂²E/∂x²)δx² about the central rod. Then for nanorod 1 we have δx = −D/2, for nanorod 2 we have δx = 0 and for nanorod 3 we have δx = +D/2 (see Fig. 1(a)). The differences in the fields becomes E₁ − 2E₂ + E₃ ≈ (1/4)(∂²E/∂x²)D² which shows that the output wave amplitude depends on the second derivative of the input electric field. For scattering away from the surface normal we also need to take into account the second term in (3) which is no longer zero. This term depends on a₁ − a₃ = fp(E₁ − E₃) ≈ fp(∂E/∂x)D which is a function of the first derivative of the field. For near-normal incidence this term will be small but will become more dominant for larger scattering angles.

Note also that the two terms in (3) have different resonance frequencies that are shifted from the main plasmon resonance by an amount depending on the coupling strength G. This can be shown explicitly by writing the resonance factor in the form f(ω) = −A/(ω − ω_r + iΓ/2) where ω_r is the LSP resonance frequency and Γ is the full width at half maximum of the resonance. For simplicity we write this as f =−A/(ω − ω̃_r) where ω̃_r = ω_r − iΓ/2 includes the imaginary term. With this expression in (3) and using a_1,2,3 = fpE_1,2,3 we have

\begin{array}{l} ψ e^{i k_{s} \cdot r_{0}} & = \frac{A^{2} p G (E_{1} - 2 E_{2} + E_{3}) cos (k_{s} \cdot d / 2)}{{(ω - {\tilde{ω}}_{r})}^{2} - 3 A^{2} G^{2}} \\ + i \frac{A^{2} p G (E_{1} - E_{3}) sin (k_{s} \cdot d / 2)}{{(ω - {\tilde{ω}}_{r})}^{2} - A^{2} G^{2}}, \end{array}

which obviously shows the amplitude of the output wave is proportional to the second-order difference of the incident electric fields E₁ − 2E₂ + E₃ as noted in the first term. The resonances occur where the real parts of the denominators are zero. The first term in (4) has resonances when

ω = ω_{r} \pm \sqrt{3 A^{2} G^{2} + Γ^{2} / 4}

whereas the second term has resonances at

ω = ω_{r} \pm \sqrt{A^{2} G^{2} + Γ^{2} / 4}

. When the coupling is weak, these resonances will lie within the resonance curve of the individual nanorods, as determined by Γ. For strong coupling, we should find that the resonances of the two components are different.

The coupled electrostatic LSP modes which can be supported by the five-nanorod configuration are displayed in Fig. 2 along with the scattering cross-section spectra which are obtained using the boundary element method [26, 27]. The gold nanorod is 90 nm long and 40 nm wide with the thickness of 30 nm and the edge-to-edge separation of 10 nm both in x - and y-directions is introduced between the two perpendicularly placed adjacent rods. The optical parameters of gold are taken from the work of Johnson and Christy [28]. The surface charge density ρ profiles of the five resonance modes with the lowest energy are shown as categorized into bright modes and dark modes. The bright modes m = 1, 3, 5 can be excited by illuminating a linearly polarized plane wave whereas the dark modes m = 2, 4 are unable to be excited by the same illumination. The corresponding wavelengths of the bright modes are identified in the scattering cross-section spectra. Depending on the polarization of the incident wave, different modes are selectively excited: m = 1, 5 and m = 3 are induced by x - and y-polarized light, respectively. Note that the surface charge density of mode m = 3 in the output nanorods aligned in x-direction is nearly zero when the normal incident plane wave is applied.

Fig. 2 The spectra of the scattering cross-section σ_S depending on the polarization of the incident wave and the profiles of the surface charge density ρ of the corresponding modes. The circuit is composed of five identical gold nanorods whose physical dimension is 90 × 40 × 30 nm³. The second and the fourth modes are not shown in the spectra since they are dark modes.

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3. Oblique incidence of a plane wave

One method for creating optical phase gradients is to illuminate the structure with light at some angle θ off-normal incidence. If we let the electric fields have the same amplitude E₀ but a phase difference ϕ, then the fields can be written as E₁ = E₀e^−iϕ/2, E₂ = E₀ and E₃ = E₀e^iϕ/2, where the phase shift is given by ϕ = 2π(D/λ) sin θ. Then the field amplitude in (4) can be written as

\begin{array}{l} ψ e^{i k_{s} \cdot r_{0}} & = \frac{- 4 A^{2} p G E_{0} {sin}^{2} (ϕ / 4) cos (k_{s} \cdot d / 2)}{{(ω - {\tilde{ω}}_{r})}^{2} - 3 A^{2} G^{2}} \\ + \frac{2 A^{2} p G E_{0} sin (ϕ / 2) sin (k_{s} \cdot d / 2)}{{(ω - {\tilde{ω}}_{r})}^{2} - A^{2} G^{2}}, \end{array}

since cos (ϕ/2) − 1 = −2 sin²(ϕ/4). If the scattered light is viewed in the direction of the incident light, which is often the situation, then k_s = k_i so that k_s · d/2 = π(d/λ) sin θ. On the other hand, assuming the scattered light only to the normal direction is collected, then k_s · d/2 = 0 so that the second term in (5) vanishes. Then the output intensity is written as a function of the phase difference as

I (ϕ) = I_{0} {sin}^{4} (ϕ / 4) .

As discussed previously, the second-order derivative leads to a quadratic dependence of the wave amplitudes on the phase (where for small phase sin⁴(ϕ/4) ≈ (ϕ/4)⁴), leading to the power of four dependence for the intensity. The first-order derivative leads to a linear dependence of the amplitudes on the phase and a squared dependence I(ϕ) ≈ I₀ϕ² for the intensity that we measured previously with the difference circuit [20, 22]. In Fig. 3, the output intensity of the second derivative circuit as a function of the incidence angle θ of a plane wave is shown along with that of the difference circuit for the first order differentiation for comparison. The schematic of the oblique incidence on the second derivative circuit is shown in the inset. It is clear that the EEM based calculation shows a quartic output intensity curve as expected from the analytic derivation.

Fig. 3 Scattered intensities of the plasmonic circuits of first and second derivatives as functions of incidence angle θ of a plane wave polarized along the input nanorods. The calculation is performed in the electrostatic regime using the boundary element method. The inset indicates the oblique incidence on the second derivative circuit.

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4. Time-dependent simulation

Since the electrostatic analysis does not include effects such as retardation, the conclusions derived in the electrostatic limit requires verification with additional analysis considering retardation. Therefore, we performed FDTD simulations for the second-order differentiation circuits with two different structure sizes. The scattered intensities and the electric field profiles obtained from the FDTD simulations are shown in Fig. 4. The calculated scattered intensities are marked with points overlaid with the fit curves using (6) with an offset; I(ϕ) = I₀ sin⁴(ϕ/4) + C. Since the retardation is more significant in a structure larger than the wavelength, we investigated circuits with two different sizes which have a plasmonic resonance at a similar wavelength. According to the electrostatic derivation where the evanescent coupling among the nanorods occurs instantaneously as in (1), the scattered intensity of the x-polarized output wave at ϕ = 0 must be zero for both circuits. In the electrodynamic analysis, however, the coupling between nanorods occur with a temporal delay which causes an offset. And it is observed that the offset is greater in the case of a full-size structure (90 × 40 × 30 nm³) than the half-size circuit (50 × 20 × 20 nm³). This indicates that the effect of the retardation increases as increasing the structure size. The resonance wavelengths of the third mode shown in Fig. 2 are used for the simulations which are 715 nm and 700 nm for the large and small circuits, respectively. Given the center-to-center distances D between the rods 1 and 5 are 380 nm and 200 nm for the large and small circuits, respectively, the circuits have the subwavelength areal resolutions of 0.53 and 0.29 λ⁻¹, accordingly.

Fig. 4 FDTD simulation results for a large (90 × 40 × 30 nm³) and a small (50 × 20 × 20 nm³) circuits for second-order differentiation. (a) Calculated scattered intensities (points) as functions of the phase difference ϕ overlaid with the fit curves (lines). (b) The induced E_x-field profiles at the indicated ϕ. The scale bar is 100 nm.

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The induced electric field in the direction perpendicular to the polarization of the incident field is shown in Fig. 4(b). Since the incident light is polarized to the y-direction, the x-component of the electric field is not from direct incidence but from excitation of the plasmonic modes. The induced E_x-field on the output nanorods (4 and 5) is nearly zero for the zero phase difference (ϕ = 0) whereas a positive or negative value for the large phase differences. It is worth to note that both for the large and small circuits, the induced electric field on the two output nanorods show opposite signs which makes the second-order differentiation large enough to be detected.

We would like to note that the condition for the electrostatic limit is known as ∊_b(kD)² << 1, where ∊_b is the electric permittivity of the background material and D is the size of the plasmonic system [25]. The center-to-center distance between the rods 1 and 5 is taken as the largest physical dimension D of the circuits, which are 380 nm and 200 nm for the large and small circuits, respectively. When this criterion is applied to the proposed differentiation circuits, 16.7 and 4.8 are given for the large and small circuits, respectively, where ∊_b = 1.5. Due to the size of the circuit which is comparable to the wavelength, effects such as retardation must be considered. As compared in Fig. 4, the large circuit shows much higher offset for ϕ = 0 owing to the retardation. The small circuit with less offset, however, interacts more weakly with the incident wave than the large one, hence the output intensity is 2 orders of magnitude smaller. Although the offset is inevitable owing to the physical dimension of the considered circuits, it can be reduced by optimizing the design.

The configuration of five gold nanorods can be modified to perform the second-order differentiation in y-axis instead of x-axis for the same y-polarized incident field. Then a two-dimensional spatial second-order difference operator, Laplacian, can be realized by alternately placing the x - and y-circuits similarly to our previous work for the gradient operator [22]. Moreover, a unit cell consisting of the x - and y-circuits can be periodically located to form a functional metasurface, where further studies are required to resolve some expected issues of the inter-circuit crosstalk and the noise caused by diffraction. Yet the proposed configuration as a single circuit can be scanned on a surface of interest to obtain the areal map of the second-order derivative of the light wave.

5. Conclusion

In conclusion, we present a plasmonic nanodevice which performs a second-order differentiation on the incident light field with a subwavelength resolution in transmission mode. The suggested circuit consists of five metallic nanorods which are located to couple to the neighbor nanorods via evanescent coupling. The electrostatic analysis was performed to design the circuit and predict its operation, and the optical characteristics of the supported modes are discussed based on the calculated results with the boundary element method. The numerically obtained quartic curve of the scattered intensities as a function of the incident angle showed the designed circuit performs second-order derivative of the phase. Additionally, the effect of retardation was studied numerically by performing the FDTD simulations for the proposed circuits of different sizes. The suggested plasmonic device is expected to enable miniaturizing all-optical computation systems.

Funding

National Natural Science Foundation of China (61490712, 61427819, U1701661); National Key Basic Research Program of China (973) (2015CB352004); the Leading Talents of Guangdong Province Program (00201505); Natural Science Foundation of Guangdong Province (2016A030312010); Science and Technology Innovation Commission of Shenzhen (KQTD2015071016560101, ZDSYS201703031605029).

Acknowledgments

YH acknowledges the helpful discussion on the boundary element method with D. E. Gómez.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Plasmonic circuit for second-order spatial differentiation at the subwavelength scale

Abstract

1. Introduction

2. Electrostatic eigenmode analysis

3. Oblique incidence of a plane wave

4. Time-dependent simulation

5. Conclusion

Funding

Acknowledgments

Disclosures

References and links

Cited By

Figures (4)

Equations (6)

Optics Express