1/f-noise-free optical sensing with an integrated heterodyne interferometer

Optical evanescent sensors can non-invasively detect unlabeled nanoscale objects in real time with unprecedented sensitivity, enabling a variety of advances in fundamental physics and biological applications. However, the intrinsic low-frequency noise therein with an approximately 1/f-shaped spectral density imposes an ultimate detection limit for monitoring many paramount processes, such as antigen-antibody reactions, cell motions and DNA hybridizations. Here, we propose and demonstrate a 1/f-noise-free optical sensor through an up-converted detection system. Experimentally, in a CMOS-compatible heterodyne interferometer, the sampling noise amplitude is suppressed by two orders of magnitude. It pushes the label-free single-nanoparticle detection limit down to the attogram level without exploiting cavity resonances, plasmonic effects, or surface charges on the analytes. Single polystyrene nanobeads and HIV-1 virus-like particles are detected as a proof-of-concept demonstration for airborne biosensing. Based on integrated waveguide arrays, our devices hold great potentials for multiplexed and rapid sensing of diverse viruses or molecules.


Supplementary Figures
Supplementary Fig. 1: Background noise. Power spectra of A I beat at low frequency region when probe light are off (gray curves) or on (light curves) at different bias frequencies. a, Conventional lock-in method (∆f = 0 Hz). b, 1/f -noise-free scheme (∆f = 30 kHz). The low-frequency noise floor is fitted by 1/f (dashed curves). Here, the signal peak in a is not presented due to the bandwidth limitation of the electrical spectrum analyzer.

Supplementary Notes
Supplementary note 1: Theoretical description of dark-field heterodyne interferometer operating at the 1/f -noise-free regime.
When a nanoparticle is deposited onto the joint sensing area in Fig. 1b, it scatters the probe light. Due to the fact that particle radius (below 100 nm) is much smaller than the optical wavelength (λ), the scattering field could be described by the dipole scattering, and its dipole scattering cross-section is where k = 2π/λ is the wave number, a is the radius of the nanosphere, ε p and ε m are the permittivity of the particle and the surrounding medium, respectively. This scattering probe light is collected by the local waveguide with the efficiency of η c ∼ f (a, r), which is a function of the radius and location of particle. This relation is influenced by the overlap of local light field and the dipole field. Therefore, the collecting efficiency can be described as Then, the collected probe light with a power of P col = E 2 probe η s interferes with the local light, the combined light field is where the subscripts j = probe or local, indicate the probe and local light with electrical amplitude E j , ω j and φ j are the angular frequency and optical phase, respectively. In our system, the frequency difference between the probe and the local light is f RF = (ω probe − ω local )/2π = 80.15 MHz. The combined light is detected by a BPD together with a reference where i BPD1 and i BPD2 represent the photocurrents of the two detectors of the BPD, R BPD is the responsivity of the commercial photodiode, ∆φ = φ probe − φ local is the phase difference between the probe and local light field, E ref is the amplitude of the reference light. Compared with the direct monitoring the transmission of scattering probe light (I probe η s ), the desired amplitude change (2 · I probe η s I local ) of the beat note has an enhancement factor of 2 · I local /I probe η s under the heterodyne concept. Here, to eliminate the laser common-mode noise [1], the power of the reference light is set with where z is the impedance of the BPD, v BPD is output voltage of signal. Since it is hard to keep a perfect balance between the two arms of BPD, a high-pass filter is introduced to remove the direct current. Such RF signal is boosted by an electrical amplifier and followed by a high-pass filter for further noise suppression. The beat signal after amplifying and filtering is v BPD = 2z · 10 (Gamp.−L filter )/10 · R BPD I probe η s I local · cos(2πf RF t + ∆φ).
This beat signal is transferred to the sampling carrier with a bias frequency ∆f = f LO −f RF = 30 kHz. which is well beneath the sampling threshold (50 kHz) of the DAQ system. Before down converting, electrical LO is divided by a 90 • power splitter with equal power but orthogonal phase: They are separately mixed with the I/Q signal, and the output beat signals are finally recorded as the original data: are the sum and difference frequency components, respectively. A E = √ 2 2 zA ELO · 10 (Gamp−L filter )/10 is the equipment-dependent scale factor. After low frequency filtering, the sum-frequency components are filtered, and only the beat amplitude changes with the bias frequency ∆f = 30 kHz are sampled as Where A s = A E I probe η s I local contains the signal information. The sampled noise amplitude of this two-way orthogonal signal is estimated in Fig. 1d and Supplementary Fig. 2.
To extract the envelope of the beat note, several digital processing procedures are applied.
First, the beat intensity I beat is recovered from the sampled A I beat and A I beat by, Here, A E remains the same in the experiments. On the contrary, I local and I probe vary from different measurements. Therefore, we track 10% of the output signal light and 1% of the input probe light simultaneously with beat signal for normalization. Finally, an average filtering method is applied to further eliminate the noise with the frequency beyond 50 Hz.

Supplementary note 2: Calculation of sampling noise amplitude
We perform a sequential measurement to reveal the influence of the bias frequency ∆f on the real-time amplitude fluctuations of the raw signal. To evaluate noise performance of this sampled raw signal A I beat , the noise amplitude defined by the standard deviation of the amplitude change δA I beat is used, where h(t) = sgn(t) and T is the duration time. According to Supplementary Equation (12), the sampled A I beat is a radio-frequency signal with the frequency of ∆f , as shown in Supplementary Fig. 2a (black curve). Therefore, the standard deviation of the amplitude change shows an oscillation characteristic over the duration T as shown (e.g., Supplementary   Fig. 2d, black curves). It is found that, the minima of the oscillating noise amplitude are free from such influence. For example, for the minimal value labelled by the pentagram in Supplementary Fig. 2d, corresponding δA I beat is shown in Supplementary Fig. 2a Figs. 2b-f). The sampling noise amplitude at T = 1 s is plotted in Fig. 1d, which illustrates an relation of 1/∆f γ (γ= 0.61) between the noise amplitude and the bias frequency ∆f .

Supplementary note 3: Nanoparticle induced chiral mode scattering
When a nanoparticle is attached to the joint sensing region, the total power of the light scattered into the local waveguide mode is where d is the induced dipole moment of a spherical scatter, α is the complex polarizability, ε exc (r) is the illuminated light field from the probe waveguide, and (r) is the electric field of the guided mode in the local waveguide. As a result, the collecting efficiency of the local waveguide is proportional to the overlap between the evanescent fields of the probe and local guided modes at the particle's position [2]. Due to the strong transverse confinement of nanowaveguide, the evanescent field has a longitudinal polarization component in the propagation direction of the light. As shown in Supplementary Fig. 3b, the evanescent fields around the probe and local waveguides in the joint sensing area are E x − iE y , −iE x − E y (the direction of S − ) and iE x − E y (the direction of S + ), respectively. The corresponding power of the probe light scattered by nanoparticle into S − and S + are Therefore, the scattering power of the probe light at different ports of the local waveguide is asymmetric and shows a position-dependent behaviour. To quantitatively calculate the scattering efficiency of a nano-object, we performed the 3D FEM simulation, and both the inhomogeneous light field and interface interactions are included ( Supplemeantary Fig. 3).
When a nano-object is deposited at the center (x = 0) of the joint sensing area, the quasilinear polarized light scattered from the the probe waveguide interacts with the interface chiral field of the local waveguide, which leads to equally scattering efficiency at the forward (S + ) and backward (S − ) direction. When a nano-object is close to left edge of probe waveguide (x ∼ 0.5 µm), the constructive chiral field interaction between the scattered light and the local waveguide mode leads to strong backward coupling (S − ). Considering the diffraction effect of probe light field, the maximum backward coupling (S − ) position is slightly deviated from the 0.5 µm. We find that the reflection effect at the joint sensing region also influences the scattering efficiency as shown in black line in Supplementary Fig.   3. To reduce the fluctuations of the probe light power, an optical isolator is required at the probe waveguide input (not drawn in Fig. 1b). Note that, during the nanotip scanning experiment, the position of the nanotip in the main text is figured out through an optical microscope. Considering radius of nanotip (∼ 200 nm) and the angle between the nanotip and the chip surface, the transmission of the scattering probe light at the S − port within the range of x ∈ [−1.75, −0.25] (µm) is used to fit the experimental results.