Fig. 1. Integrated EP sensor. (a) Schematic of the single nanoantenna EP combined with the modified four-port integrated MZI. (b) Energy diagram of the coupled nanobars, illustrating the newly defined parameter space to balance the eigenmodes and achieve EP. (c) Microfluidic encapsulation of the detection zone for low-volume, simple sample handling in a compact biosensing setup. (d) Sensing mechanism of the integrated EP biosensor. Coupled nanobars are functionalized with specific probes to capture target biomarkers (e.g., proteins and nucleic acids) passing through the microchannel. The specifically captured biomarkers perturb the system, leading to complex eigenvalue splitting.

Fig. 2. Parameter sweeping. (a) Cross-section of the coupled nanobars placed inside the junction-waveguide. The in-plane (x-axis) component of the guided mode electric field at the crossline cutting through the middle of the gap shows the junction-waveguide enables the inserted nanoantenna to interact with the peak of the guided mode. Insets show guided mode electric field distribution, before and inside the gap, respectively. (b), (c) Resonant frequency and loss rate of the coupled nanobars system eigenvalues versus ΔX and ΔL, varying from 0 to 100 nm and from 10 to 40 nm, respectively. The crossing between two eigenmode planes determines the EP at ΔX = 62  nm and ΔL = 26  nm, as depicted by the star. The length of the larger nanobar is 220 nm, the width and height of both are 40 nm, and the gap size is set to 30 nm. It should be noted that the provided approach is called Reimann analysis, enabling EP characterization by monitoring the residuals of the complex extracted eigenvalues through the S-matrix fitting method [26,50,51]. (d), (f) Resonant frequency crossing, loss rate crossing, and the norm of the y-direction magnetic field, |Hy|, distribution at ΔL = 20  nm cross-section, showing eigenvalue real parts crossing. (g)–(i) Resonant frequency crossing, loss rate crossing, and the norm of the y-direction magnetic field, |Hy|, distribution at ΔL = 30  nm cross-section, showing eigenvalue imaginary parts crossing. Note that the eigenmode field distributions switch for the loss rate crossing case; however, they remain the same after the crossing point for the resonant frequency crossing case. All numerical modeling has been done using COMSOL Multiphysics (version 6.1).

Fig. 3. PIC design and analysis. (a) The modified four-port integrated MZI combined with single particle EP (i). Detailed demonstration of field splitting at each node of the circuit (ii). (b) The norm of electric field distribution of the TE mode, shown at the Si3N4 waveguide (1000 nm width and 220 nm thickness) cross-section (i), along the edge coupler (start-width of 200 nm, end-width of 1000 nm, and 1000 μm tapering region length) (ii), S-bending (Bezier curve with 46 μm length and 3 μm vertical shift) (iii), Y-splitter (combination of two Bezier S-bendings with length of 6 μm and vertical shift of 3 μm) (iv), and the junction-waveguide (minimum width of 300 nm and gap of 200 nm, with 400 nm tapered length) in the reference (v) and the sensing arm (vi). All distributions are derived at a 1400 nm wavelength, and the corresponding scaling ratio is shown at the bottom of each distribution. Note, the curvature of the bends for rerouting the outgoing ports is set to 40 μm. (c), (d) Measured intensities at reference (ref.), signal, and phase ports, for without and with coupled nanobars in the sensing arm cases. (e), (f) Extracted transmittance and phase of the coupled nanobar using the measured intensities at three output ports of the circuit, showing excellent agreement with the individually modeled coupled nanobars in the junction-waveguide. Note that we assumed uniform buffer concentration across the microchannel covering the sensing and reference arms. The PIC components design and the network analysis have been done using Ansys Lumerical (version 2022 R1.4).

Fig. 4. Sensing performance. (a), (b) Resonant frequency and loss rate splitting of EP and DP systems under bulk refractive index perturbation. (c), (d) Resonant frequency and loss rate splitting of EP and DP systems under local refractive index perturbation. Note that EP complex splitting follows the square root response while DP behaves linearly. Under the perturbed region, EP has a superior response, especially its loss rate splitting under local perturbation, showing about five times enhancement compared to the DP case. Note that the DP system shows negative loss rate splitting. The perturbed area is depicted in red. Note that we assumed the buffer solution has the same concentration across the sensing and reference arms.

Fig. 5. Single particle sensing. (a)–(d) Resonant frequency and loss rate splitting for up to 10 randomly captured nanoparticles with an average size of 10 nm (refractive index ∼1.43 [25]) (a), (b) and 100 nm (c), (d). Note that the measurements have been repeated 10 times. The recorded splitting shows single molecule sensitivity of the integrated EP biosensor. (e), (f) Standard deviations of the measured complex splitting for 10 and 100 nm particles, clustering captured targets with more than seven particles for 10 nm size and more than one particle for 100 nm size. Note that the resonant frequency and loss rate splitting were not calibrated, causing to have non-zero values at zero perturbation. These residual values exist due to the fact that reaching an exact EP is not possible and we only are in the vicinity of the EP. It should be noted that we considered that all flouting particles were washed out after the standard washing step and only the specifically attached particles remained on the gold nanobars.

Fig. 6. Effect of bulk refractive index on the local sensing response. (a), (b) Resonant frequency and loss rate splitting of the integrated EP sensor designed at 1.33 buffer solution refractive index with applied local refractive index perturbations. The study shows that the bulk refractive index variation slightly decreases the sensitivity of the EP sensor, especially at higher local perturbation values. Note, 0.01 increase in the buffer refractive index is equivalent to 5% increase in the NaCl concentration [24], so 0.1 can be considered as 50% increase in NaCl, already existing in the buffer solution, which may not be achievable, revealing the negligible effect of buffer content variations on the sensing performance of the EP sensor. Fitted curves are all square root functions.

Fig. 7. Effect of fabrication errors. (a), (b) Resonant frequency and loss rate splitting of the integrated EP sensor with maximum of ±5  nm variations in ΔX and ΔL relative to the EP case. Results show the asymmetric effect of ΔL variations on the system performance with over 12 THz and 15 THz maximum splitting in the resonant frequency and loss rate, respectively. However, ΔX deviation has more symmetrical effect with smaller maximum splitting compared to ΔL deviation. (c), (d) Resonant frequency and loss rate splitting of the EP sensor and −5  nm deviated cases (ΔX and ΔL) versus locally perturbed refractive index. In general, the deviated system shows smaller sensitivities compared to the EP case. Especially, loss rate splitting of the ΔL deviated case remains unaffected under local refractive index variations. Note, in all cases, the splitting is calibrated with respect to the value at Δn=0. Fitted curves are all square root functions.