Broadband On-Chip Directional Coupler with Oblique Nanoslits

Abstract

Directional coupling of light at the nanoscale plays a significant role in both fundamental research and practical applications, which are crucial for the development of on-chip photonic devices. In this work, we propose a broadband directional coupler for surface plasmon polaritons (SPPs) utilizing a pair of obliquely perforated nanoslits. We demonstrate that tilting the slits significantly enhances the sensitivity of plasmonic coupling phase variation to the wavelength of the incident light, enabling precise wavelength-dependent control over SPP propagation. By optimizing the width and tilting angle of each nanoslit, we achieve an extinction ratio exceeding 10 dB with a bandwidth exceeding 400 nm and a maximum unidirectional transmission of up to 30 dB. This broadband directional SPP coupler presents a promising platform for the design and fabrication of integrated plasmonic circuits and high-performance optical devices and sensors.

1. Introduction

The integration and miniaturization of on-chip photonic devices are crucial for meeting the growing demand for higher speed, increased capacity, and improved efficiency in information acquisition, processing, and storage. Many advances have been made to enable ultra-compact devices such as waveguides, photonic crystals, nano-antennas, and meta-surfaces [1,2,3,4,5,6,7,8,9,10,11,12]. Among the numerous approaches, the employment of two-dimensional surface electromagnetic waves like surface plasmon polaritons (SPPs) has been particularly intriguing and widely investigated due to their ability to confine light to subwavelength dimensions and support a wide range of optical functionalities. Traditional SPP generators, such as prism and grating, are too large for high density integration, and plasmonic nanostructures have emerged. Many typical plasmonic devices were proposed, including light emitters [13,14], SPP emitters [15], photodetectors [16], modulators [17,18,19], biosensors [20,21], lenses [22,23], and demultiplexers [24]. A directional coupler is one of the most important plasmonic devices to launch SPP fields at desired propagating direction. To achieve this, asymmetry is introduced in either incident light [25,26] or nanostructures [27,28,29,30,31,32,33,34,35,36,37], which results in direction-dependent interference of SPP. While many efforts have been devoted to realizing the highly efficient directional coupling of plasmonic field, the operating wavelength is always limited to a narrow band due to the fixed coupling phase of the resonant element. There is a balance between the extinction ratio and bandwidth.

In this work, we introduce a novel broadband directional coupler for surface plasmon polaritons (SPPs) based on a pair of obliquely oriented nanoslits. By strategically tilting the slits, we demonstrate a significant enhancement in the sensitivity of the plasmonic coupling phase to variations in incident light wavelength. This heightened sensitivity empowers precise wavelength-dependent control over the propagation direction of the excited SPPs. Through optimization of the width and tilt angle of each nanoslit, we achieve a remarkable directional coupling performance. The ratio of SPP excitation in opposite directions, quantified by the extinction ratio, surpasses 10 dB across a broad bandwidth exceeding 400 nm, encompassing both the visible and near-infrared spectral regions. This broadband directional SPP coupler presents a promising foundation for the advancement of integrated plasmonic circuits. Its unique characteristics pave the way for the development of a new generation of high-performance optical devices and sensors, including wavelength-selective filters, optical switches, and highly sensitive biosensors.

2. Theoretical Model

The schematic diagram of the broadband directional coupler is shown in Figure 1. A pair of nanoslits is obliquely perforated into a gold film deposited on a glass substrate. When a TM polarized light is incident on the structure from the glass side, each nanoslit acts as a nanoantenna to couple free space light to plasmonic modes propagating on the surface. The wavevector of the SPP field sustained at the gold/air interface can be expressed as

$$k_{\mathit{spp}}=\frac{2\pi }{\lambda }\sqrt{\frac{{\epsilon }_m}{{\epsilon }_m+1}}$$

(1)

where λ is the wavelength of incident light and ε_m is the permittivity of gold dependent on λ. The thickness of gold film is 160 nm to block the incident light onto the gold/air interface. The directional coupling of SPPs by double-slit relies on the asymmetric excitation of plasmonic fields in each individual slit. The SPP field propagating along the ±x directions can be expressed as

$$E_{+}={\eta }_1^{+}{e}^{i{\phi }_1^{+}}{e}^{i{k}_{\mathit{spp}}L}+{\eta }_2^{+}{e}^{i{\phi }_2^{+}}$$

(2)

$$E_{-}={\eta }_2^{-}{e}^{i{\phi }_2^{-}}{e}^{i{k}_{\mathit{spp}}L}+{\eta }_1^{-}{e}^{i{\phi }_1^{-}}$$

(3)

where L is the distance between the slits and ${\eta }_i^{\pm }$ and ${\phi }_i^{\pm }$ (i = 1, 2) denote the coupling efficiency and phase of the SPP wave generated by each slit propagating along ±x axis. When the plasmonic field propagating in one direction is suppressed by destructive interference, whereas constructive interference occurs on the other side, the directional coupling of SPPs is realized. Assuming the plasmonic field is propagating along the positive x axis directionally, the coupling phase difference is expressed as

$${\phi }_1^{+}-{\phi }_2^{+}+k_{\mathit{spp}}L=0$$

(4)

$${\phi }_2^{-}-{\phi }_1^{-}+k_{\mathit{spp}}L=\mathrm{\pi }$$

(5)

Equations (4) and (5) can be satisfied at a specific wavelength by adjusting the structural parameters of each slit, and there are many studies on the directional coupling of SPPs by rectangular slit pairs [29,32,33]. In that case, the coupling phase of the plasmonic field propagating along the two sides of individual slit is identical (${\phi }_1^{+}={\phi }_1^{-}={\phi }_1$, ${\phi }_2^{+}={\phi }_2^{-}={\phi }_2$) and optimal directional coupling can be achieved with φ₂ − φ₁ = k_sppL = π/2. In general, the coupling phase is dependent on the effective refractive index of the cavity mode inside the slit, and φ₂ − φ₁ barely changes with wavelength. While the wavevector of SPPs varies significantly with wavelength, this leads to a drift of phase difference as the incidence deviates from the central wavelength, weakening the destructive interference in Equation (3). Hence, the operating wavelength in previous research is always limited to a narrow band.

To compensate the additional propagating phase term Δk_sppL as the wavelength changes, oblique nanoslits are employed in Figure 1. When incident light illuminates the gold film perpendicularly along the z-axis, oblique slits scatter light asymmetrically, exciting plasmonic fields on either side of the slit with different intensities and phases. The coupling phase exhibits a dependence on both the tilting angle of the slit and the incident wavelength, primarily due to the increased optical path length within the slit. This gives rise to varying coupling phase differences φ₂–φ₁ as the wavelength changes, which can compensate the additional propagating phase.

3. Numerical Results

The interaction between incident wave and nanoslit is simulated using the two-dimensional finite-difference time-domain (FDTD) method in the commercial software (Lumerical 2023R1). The boundary conditions in the x and z directions are set as perfectly matched layers (PML) absorbing boundary conditions. To evaluate the coupling efficiency and phase, the nanoslit etched on the gold layer is illuminated at normal incidence by a linearly polarized plane wave underneath the sample. A point monitor is placed at the gold/air interface and 10 μm away from the right edge of the slit to measure the coupling efficiency and phase. A non-uniform mesh is used throughout the whole simulation domain and the minimum mesh size is 1 nm. The unit sizes and time steps ensure that the values of the electromagnetic fields and current densities in the spatial grid are converged and stable. Figure 2a illustrates the dependence of the coupling phase of individual rectangular slits on both slit width and incident wavelength. At a fixed wavelength, the coupling phase exhibits a slight dependence on slit width, enabling directional coupling of SPPs using a pair of asymmetric slits. In contrast, the coupling phase demonstrates a significant dependence on the incident wavelength, as illustrated in Figure 2b for two distinct slit widths of 150 nm and 230 nm. However, the phase difference in Figure 2b exhibits minimal variation with wavelength, particularly for wavelengths exceeding 500 nm, which corresponds to the typical plasmonic resonance regime for gold. Hence, the operating wavelength for directional coupling is limited in terms of the rectangular nanoslit pair. As the slit is tilted, the initial coupling phase is altered due to the resultant increase in optical path length, which depends on the incident wavelength. Figure 2c illustrates the coupling phase dependence on the slit tilt angle for different wavelengths. As can be observed in Figure 2c, the coupling phase exhibits an increase with both the tilt angle and the wavevector. This provides an additional degree of freedom for manipulating the plasmonic fields excited by the nanoslits. Specifically, the coupling phase difference between two distinct oblique slits increases with the wavevector, compensating for the additional propagating phase Δk_sppL as the wavelength changes. The phase difference between the plasmonic fields for destructive interference is maintained at approximately π across a wide range of wavelengths, giving rise to broadband directional coupling of SPPs.

The dependence of the coupling efficiency η and phase φ on both slit width and tilt angle for individual slits are demonstrated in Figure 3a,b at a wavelength of 750 nm, respectively. As the tilt angle of the slit increases, the coupling efficiency shown in Figure 3a decreases, which is related to the dispersion relation of SPPs; meanwhile, the coupling phase in Figure 3b increases with the angle, consistent with the explanation provided in Figure 2. Guided by this pattern, a series of slit parameters that meet the conditions can be identified. Destructive interference of the plasmonic field occurs when the coupling phase difference between individual slits satisfies φ₂ − φ₁ = (N + 1/2)π. In addition, the coupling efficiency for each slit should be equal to optimize the directional excitation of SPPs. In order to achieve this, we select the widths and tilt angle of the slits as w₁ = 90 nm, θ₁ = −63° and w₂ = 145 nm, θ₂ = 56°, which give η₁ ≈ η₂~13%, φ₂~φ₁ ≈ π/2. The interference fields on each side exhibit oscillations as the slit spacing varies. Notably, the amplitude variations on each side of the double slit closely resemble sine functions with a distinct phase shift between them, as shown in Figure 3c. Plasmonic fields propagate directionally along the −x axis at an optimal distance of L = 230 nm. Conversely, at L = 520 nm, plasmonic fields propagate directionally along the x-axis. Figure 3d,e depicts the near-field distributions of the electric field under normal plane-wave illumination for L = 230 nm and L = 520 nm, respectively, demonstrating strong directionality in both cases. This nanoscale tilted slit can be fabricated using a focused ion beam (FIB) with tilted exposure or angled ion milling. Given the desired slit angle, the tilt angle of the FIB sample stage can be calculated to ensure the ion beam precisely mills the nanoslit at the intended angle.

Oblique slits enable directional coupling of SPPs across a broad spectral range. Figure 4a illustrates the electric field intensities on both sides of the nanoantenna under broadband light source excitation ranging from 400 nm to 1600 nm for L = 230 nm, demonstrating a directional coupling in a broadband spectrum. To quantify the directional coupling performance, the extinction ratio is defined as the ratio of the intensities of SPPs propagating in opposite directions as

$$D=10{log}_{10}{\displaystyle \frac{E_r^2}{E_l^2}}$$

(6)

where E_r and E_l denote x components of electric field along positive and negative x directions at z = 0 plane, respectively. The bandwidth of directional coupling is defined as the wavelength range within which the extinction ratio, as defined in Equation (6), exceeds 10. This extinction ratio threshold is considered sufficient for achieving efficient unidirectional SPP coupling in large-scale devices [15]. Figure 4b illustrates the simulated extinction ratio as a function of wavelength. Within the wavelength range of 550–950 nm, SPPs propagate directionally along the x axis, exhibiting a maximum extinction ratio of 30 dB. The near-field distribution of the electric field under broadband plane-wave illumination within the wavelength range of 550–950 nm is depicted in Figure 4c, confirming directional coupling across this broad spectral range. For comparison, the directional coupling performance for rectangular nanoslits is also simulated, as depicted in Figure 4d–f. In Figure 4, a notable difference in the directional coupling efficiency between the inclined slit and the rectangular slit can be observed. Under excitation by a broadband light source ranging from 400 nm to 1600 nm, there is a significant disparity in the electric field intensity on both sides of the inclined slit, which enables effective directional coupling transmission within the wavelength range of 550 nm to 950 nm. In contrast, for the rectangular slit, there is no significant difference in electric field intensity on either side when stimulated by the same light source. Figure 4c,f clearly illustrates the directional transmission capabilities of the inclined and rectangular slits.

The proposed structure demonstrates robust directional coupling performance even with deviations from the optimal slit parameters. To assess the impact of fabrication errors, we analyzed the effects of varying slit width, tilt angle, and slit spacing. As shown in Figure 5, the directional coupling behavior remains unaffected despite these parameter variations. While the coupling efficiency and extinction ratio exhibit slight changes, remaining within an acceptable range, the bandwidth maintains its similarity to that achieved with the optimal parameters. This indicates a good fabrication tolerance for the directional coupler.

4. Conclusions

In conclusion, we propose a broadband directional coupler of SPPs by applying a pair of oblique nanoslits. We demonstrate that tilting the slit can significantly enhance the sensitivity of plasmonic coupling phase variation with respect to the wavelength of incident light, enabling more precise wavelength-dependent control over SPP propagation. By optimizing the width and tilting angle of each nanoslit, the ratio of SPP excitation in opposite directions exceeds 10 dB with a bandwidth exceeding 400 nm, spanning the visible and near-infrared spectral range. The results enable efficient and flexible control over light propagation at the nanoscale, paving the way for the development of novel devices with advanced functionalities, such as ultra-compact optical interconnects, biosensors, and nonlinear optical components.