Evanescent single-molecule biosensing with quantum limited precision
N. P. Mauranyapin, L. S. Madsen, M. A. Taylor, M. Waleed, W. P. Bowen
EEvanescent single-molecule biosensing with quantum limited preci-sion
N. P. Mauranyapin , L. S. Madsen , , M. A. Taylor , , M. Waleed & W. P. Bowen , , ∗ School of Mathematics and Physics, the University of Queensland, Australia Centre for Engineered Quantum Systems, the University of Queensland, Australia Research Institute of Molecular Pathology, Vienna, Austria.
Sensors that are able to detect and track single unlabelled biomolecules are an important tool both to under-stand biomolecular dynamics and interactions at nanoscale, and for medical diagnostics operating at their ul-timate detection limits. Recently, exceptional sensitivity has been achieved using the strongly enhanced evanes-cent fields provided by optical microcavities and nano-sized plasmonic resonators. However, at high field in-tensities photodamage to the biological specimen becomes increasingly problematic. Here, we introduce anoptical nanofibre based evanescent biosensor that operates at the fundamental precision limit introduced byquantisation of light. This allows a four order-of-magnitude reduction in optical intensity whilst maintainingstate-of-the-art sensitivity. It enables quantum noise limited tracking of single biomolecules as small as 3.5 nm,and surface-molecule interactions to be monitored over extended periods. By achieving quantum noise limitedprecision, our approach provides a pathway towards quantum-enhanced single-molecule biosensors.Introduction
Evanescent optical biosensors that operate label-free and can resolve single molecules have applications ranging fromclinical diagnostics , to environmental monitoring and the detection and manipulation of viruses , proteins andantibodies . Further, they offer the prospect to provide new insights into motor molecule dynamics and biophysi-cally important conformational changes as they occur in the natural state, unmodified by the presence of fluorescentmarkers or nanoparticle labels . Recently, the reach of evanescent techniques has been extended to single proteinswith Stokes radii of a few nanometers by concentrating the optical field using resonant structures such as opticalmicrocavities and plasmonic resonators . These advances illustrate a near-universal feature of precision opti-1 a r X i v : . [ phy s i c s . op ti c s ] N ov al biosensors — that increased light intensities are required to detect smaller molecules or improve spatiotemporalresolution. This increases the photodamage experienced by the specimen, which can have broad consequences onviability , function , structure and growth . It is therefore desirable to develop alternative biosensing approachesthat improve sensitivity without exposing the specimen to higher optical intensities.Here we demonstrate an optical nanofibre-based approach to evanescent detection and tracking of unlabelled bio-molecules that utilises a combination of heterodyne interferometry and dark field illumination. This greatly suppressestechnical noise due to background scatter, vibrations and laser fluctuations that has limited previous experiments ,allowing operation at the quantum noise limit to sensitivity introduced by the quantisation of light. The increasedinformation that is extracted per scattered photon enables state-of-the-art sensitivity to be achieved with optical inten-sities four orders of magnitude lower than has been possible previously . Using the biosensor, we detect nanospheresand biomolecules as small as 3.5 nm in radius and track them with 5 nm resolution at 100 Hz bandwidth. Dark-field nanofibre based biosensor
The nanofibre sensor is immersed in a droplet of water containing nanoparticles or biomolecules (see Fig. 1a). Lightpassing through it induces an intense evanescent field extending a few hundred nanometers from the fibre surface.Likewise, light scattered in the nearfield of the nanofibre is collected by the guided mode of the fibre, providing ahighly localised objective. In contrast to previous approaches , we illuminate a small section of the nanofibre fromabove with a probe field. Since its propagation direction is orthogonal to the fibre axis, very little of the probe fieldis collected in the absence of nanoparticles and biomolecules. In this dark-field configuration, minimal backgroundnoise is introduced by the probe. When a nanoparticle or biomolecule diffuses into the illuminated region, it scattersprobe light by elastic dipole scattering. The nanofibre collects a significant fraction of this field, which we term the signal field henceforth. Diffusion of the particle in the vicinity of the nanofibre modulates the collection efficiency,encoding information about the motion of the particle on the signal field intensity.2igure 1:
Experimental setup. (a)
Nanofiber with dark field heterodyne illumination. Nanoparticles (grey spheres)in a droplet of ultra pure water are detected when entering the probe beam waist next to the nanofibre. (b)
Schematicof the optical setup for the dark field heterodyne nanofibre biosensor, including two optical isolators to suppress back-scatter of probe light which was found to increase the noise floor even for very low photon flux, a low noise New Focus1807 balanced photoreceiver with electronic noise well beneath the local oscillator shot noise level, and a home-builtultralow noise dual quadrature electronic lock-in amplifier (methods).3 oise performance
Quantum noise limited tracking of single unlabelled biomolecules is made difficult by the combination of very lowlevels of scattered power — in our case in the range of femtowatts — and technical noise sources such as laserintensity and frequency fluctuations, electronic noise, acoustic vibrations and background scattered light. These tech-nical noise sources are particularly problematic in the hertz to kilohertz frequency band of relevance for observationsof biomolecule dynamics, binding and trapping , and are a key limitation of previously reported evanescent sen-sors . To achieve quantum noise limited performance here, we use an optical heterodyne technique to amplify thesignal from the trapped particle above both the electronic noise of our measurement system and noise from backgroundlight, and to shift its frequency well away from low frequency laser, electronic and acoustic noise. In short, an opticallocal oscillator field frequency shifted by 72.58 MHz from the probe is injected into the nanofibre, and its beat withthe signal field is observed on a low noise photoreceiver (see methods). A photocurrent proportional to the absolutevalue of the signal field amplitude is acquired in real time by mixing the photoreceiver output down at the heterodynebeat frequency using a home-built dual-phase lock-in amplifier (see Fig. 1).We performed a sequence of experiments to characterise the noise performance of the biosensor and verify its quantum-limited performance (Fig 2). These confirm that electronic dark noise can be neglected at frequencies above a few hertz(Fig 2a), and that the local oscillator field is quantum noise limited (Fig 2b). The total noise floor of the biosensorwas measured as a static scattering center was illuminated with a range of powers, with results displayed in Fig. 2cas a function of collected signal power and equivalent particle size at fixed intensity. Quantum noise is found todominate at all measurement frequencies above 4 Hz, even for the highest signal power used (8 fW, equivalent radius of20.5 nm). This hertz-kilohertz frequency window is important for biophysical processes ranging from seconds to fewmilliseconds , and is the crucial window for measurements of the motion of trapped nanoparticles and biomoleculesas performed here.The quantum noise limit of our biosensor can be quantified by comparing the shot noise level due to the quantisationof light to the scattered intensity predicted from Rayleigh scattering theory (Supplementary Information II.C.1). Theminimum detectable particle cross section is σ min = 8 (cid:126) ω/ητ I probe , where (cid:126) is the reduced Planck constant, ω thefrequency of the probe light, η the total collection efficiency including detection inefficiencies, τ the measurement4igure 2: Quantum noise limited region . (a) The power spectrum of the electronic noise and the optical backgroundresponse of the system under normal operating conditions are represented by the black and orange curve, respectively. (b)
Averaged noise power spectrum over a 10 kHz bandwidth without probe light as function of the local oscillatorpower (blue points). The linear orange, quadratic red and constant black curves represent the quantum model, aclassical laser noise model and an electronic noise model, respectively. (c)
Clearance from quantum noise floor asfunction of signal power (left y-axis) or its equivalent nanoparticle radius (right y-axis) and as function of frequencies(x-axis), for a stationary scattering source. The signal power is estimated from the amplification of the scattered probefield by the local oscillator field (see methods). The noise floor is dominated by quantum noise in the white-yellowregions, and by technical noise in the orange-black regions.time and I probe the probe intensity. We estimate the collection efficiency to be in the range of 1–10%, with moreaccurate determination precluded by a strong dependence on both the nanofibre geometry and the particle position.For our experimental parameters of I probe = 7 × Wm − and τ = 10 ms, the smallest detectable particle crosssection is then predicted to be in the range of × − to × − nm , corresponding to a silica nanosphere of radius16–23 nm. 5 etection and trapping of single nanoparticles The biosensor was tested on solutions of silica and gold nanoparticles in ultrapure double processed deionized water(Sigma W3500). Unexpectedly, the sensor is able to resolve silica nanoparticles of radius down to 5 nm, significantlybeneath the quantum noise limit calculated in the previous section. We attribute this to an enhanced scattering cross-section due to the presence of surface charges on the nanoparticles, as later discussed. Figures 3a-c show sectionsof typical time domain traces that display nanoparticle detection events for 25 nm and 5 nm silica particles, and10 ×
45 nm gold nanorods, respectively. Calibration of the detected signal in terms of the particle position reveals thatthe 5 and 25 nm nanoparticles can be tracked with resolution of 5 and 1 nm, respectively, with a 100 Hz bandwidth(details on calibration in Supplementary Information VII).Further experiments were performed using a solution containing both 5 and 25 nm nanoparticles (Fig. 3d). Theobserved events are similar to those for individual particle solutions (Fig. 3a,b) showing that parallel detection anddiscrimination of different nanoparticle types is possible. These results compare favourably to other nanofiber sensors,with the smallest nanoparticles previously observed having a 100 nm radius . Moreover, the system is competitivewith the best field-enhanced evanescent sensors both using micro-cavities and plasmonic resonators , whileexposing the specimen to significantly lower optical intensities (see Discussion). This demonstrates the substantialperformance gains that can be achieved via dark field heterodyne detection and complements the recent demonstrationof quantum noise limited super-resolution imaging of stationary proteins .The power spectrum of the measured particle motion exhibits a Lorenzian shape characteristic of trapped Brown-ian motion (Supplementary Information V), which confirms that the particles are trapped. By calculating the meanposition probability distributions obtained from the 5 and 25 nm nanospheres events (see Fig. 3e) and using Boltz-mann statistics, we retrieve the trapping potentials shown in Fig. 3f (Supplementary Information VI). Unlike previousexperiments which used a combination of repulsive electrostatic forces and attractive optical forces to trap largerparticle near the nanofibre, we demonstrate theoretically and experimentally that optical attraction is negligible for oursmaller particles (Supplementary Information VIII).Recently, the surface charges that accumulate on surfaces in solution and their associated counter-ions have beenshown to greatly enhance the scattering cross section of micelles smaller than 200 nm trapped in an optical tweezer .6ince the surface area-to-volume ratio increases with reduced particle size, a considerably stronger effect could beanticipated in experiments with nanoparticles . Surface charges have also been shown to lead to long range attractiveforces between same-charged particles Ref. . We believe that this combination of effects, due to the surface chargespresent on both nanofibre and nanoparticles, may explain the enhanced scattering cross section and the ability to trapin our experiments. We rule out alternative explanations such as outside contaminants and aggregation based on thereproducibility of the signal amplitudes and potentials and control experiments (Supplementary Information IX). Totest the feasibility of signal enhancement via surface charges, we repeated the experiment with 5 nm particles whilevarying the salt concentration in the solution using DPBS (Dulbecco’s Phosphate Buffered Saline, Gibco 14040). Thespatial extent of the counter-ion distribution generated by surface charges varies strongly with salt concentration dueto screening effects. In ultrapure water, as in our initial experiments, it can extend over several hundreds nanometersand therefore significantly influence experiments, while it is reduced to nanometer-scale for even relatively modestsalt concentrations . As shown Fig. 3g both the amplitude of light scattered by the nanoparticles and the frequencyof events are effected by salt concentration, dropping significantly above a concentration of around 1 mM, similarlyto the observations in Ref. and consistent with a decrease in scattering cross section and attractive forces. Furtherstudies are required to provide a more definitive explanation of these effects, to test whether they might explainprevious observations of enhanced sensitivity in evanescent biosensors , and to explore their wider potential as anenhancement mechanism to improve detection limits of those sensors. Detection and trapping of single unlabelled biomolecules
With both quantum noise limited performance and the capability to detect small nanoparticles confirmed, we nowapply our biosensor to the detection of single unlabelled biomolecules, an application that has not previously beendemonstrated using a nanofibre sensor. We perform measurements on low concentration solutions of the biomoleculesbovine serum albumin (BSA) and anti-
Escherichia coli ( E. coli ) antibody, with molecular weights of 66 kDa and150 kDa, respectively. BSA, in particular, has a 3.5 nm Stokes radius, and is among the smallest biomolecules detected7igure 3:
Nano particle detection and trapping . (a) , (b) and (c) Time trace of the normalised amplitude (left y-axis) and corresponding position (right y-axis) of the detected signal in which a sudden rise (shaded regions) indicatestrapping events of 25 nm, 5 nm silica sphere and 10 by 45 nm gold nanorod. For clarity, these time traces are bandpassfiltered over the range 4 to 100 Hz. (d)
Time trace of the normalised amplitude for a solution containing both 25 nmand 5 nm silica particles. (e)
Mean probability density associated with all observed trapping events of the 25 nm (11events) and 5 nm (8 events) particles. (f)
Calculated trapping potentials derived from one trapping event for 25 nm(green) and 5 nm (blue) particles. The shaded bands represent the standard deviation of the potential for all observed5 nm and 25 nm trapping events. (g)
Two independent experiments monitoring the maximum amplitude (blue, leftaxis) of the trapping events and their frequency (red, right y-axis) as a function of salt concentration for 5 nm silicananospheres. The curves are guides-to-the-eye and the error bars represent the standard error of each measurement.The total number of detection events for salt concentration of { ∼ , . × − , . × − , . , , . × } mM,were {25, 26, 24, 27, 8, 3} and {9, 10, 9, 11, 6, 2} for the count rate and amplitude experiment respectively.8sing plasmonic and cavity-enhanced techniques . After performing a series of experiments using solutions of eachbiomolecule individually (not shown here), we demonstrate the ability to perform measurements in parallel using asolution containing both molecules. A time-trace which exhibits a sequence of trapping events of these moleculesis shown in Fig. 4a. A factor of five difference between the amplitudes of anti- E. coli antibody and BSA eventsallows straightforward discrimination between the molecules. This ability to discriminate is also seen in the positionprobability distributions of the biomolecules, shown in Fig. 4b. Using the same methods as for nanoparticles, wecalculate the trapping potential from the probability distribution as shown in Fig. 4b (inset) for an anti-
E. coli antibody.The repulsive part of the potential provides information about the interaction between the molecule and the fibresurface, and could be used to monitor surface particle interactions , cell membrane formation , molecule-moleculeinteractions and molecular motor motion with high sensitivity, in real time, and without recourse to ensembleaveraging.The ability to detect 3.5 nm biomolcules demonstrates that the performance of our biosensor is competitive in sen-sitivity with more complicated whispering gallery resonators and plasmonic evanescent sensors , though here thisis achieved with four orders of magnitude lower light intensity (Supplementary Information X) and a considerablysimpler, robust, sensing platform. Discussion
Specimen photodamage due to exposure to light is often a critical issue in biophysical experiments , resulting inphotochemical changes to biological processes , modifying structure and growth , and ultimately adverselyaffecting viability . For instance, Ref. found that the viability of E. Coli is affected by light intensities as low as . × Wm − . The experiments reported here used probe and local oscillator field intensities of × Wm − and × Wm − , below this threshold; while similarly-performing plasmonic sensors use intensities of around Wm − . More generally, the four order of magnitude reduction in intensity afforded by our biosensor shouldallow a commensurate increase in observation time for equivalent photodamage .9igure 4: Biomolecule detection and trapping . (a) Time trace of the normalised amplitude for a solution containingboth BSA (0.09 mg/mL) and anti-
E. coli antibody (0.03 mg/mL). The magenta regions represent trapping events ofsingle BSA molecules and the green region an anti-
E. coli antibody. (b)
Mean probability distribution obtained fromall observed trapping events of BSA and anti-
E. coli antibody, shown in magenta and green respectively. Distributionscalculated from a total of 11 and 13 events for anti-
E. coli antibody and BSA, respectively. The insert shows thetrapping potential derived from the anti-
E. coli antibody probability distribution, and the shaded band its standarddeviation. 10he combination of high sensitivity and bandwidth with low photodamage opens up a new path to explore singlemolecule biophysics. For example, it may allow single discreet steps in a free flagella motor to be observed at theirnatural frequencies. To date, such steps have only been observed by attaching a fluorescent label to the flagella andslowing down the rotation of the motor by inducing photodamage to reduce the sodium-motive force . The bandwidthof the sensor can in principle be extended to the limits of our optical detector and digitalisation device, with bandwidthsabove 10 MHz feasible. This could allow studies of small scale conformational changes, such as occur in protein sidechain and nucleic acid base conformation, occurring on time-scales of − s to − s , in solution, without markersand with minimal photodamage.Since our biosensor reaches the quantum noise limit, it can be combined with quantum correlated photons to achievesub-shot noise limited precision. Numerous approaches to quantum enhanced sensing have been developed over thepast few decades . However, as yet they have not been applied to single molecule sensing. Finally, at the costof increased intensity, our biosensing approach could be combined with plasmonic sensing by depositing plasmonicparticles on the nanotaper. In this way, it may be possible to achieve quantum noise limited plasmonic sensing andthereby allow the detection of even smaller molecules than is currently possible. MethodsHeterodyne concept:
The heterodyne strategy is similar to the approach developed in Refs. to evade low fre-quency noise sources in optical tweezers based biological measurements but, by using heterodyne detection, eliminatesthe need to phase stabilise the signal field. To implement the technique, an optical local oscillator field is injected intothe nanofibre, frequency shifted from the probe field by 72.58 MHz. The output of the nanofibre is then detected on alow noise balanced photoreceiver (see caption of Fig. 1). The interference between the collected signal field and thelocal oscillator generates a 72.58 MHz beat note in the photoreceiver output, shifted well away from technical noisesources and amplified compared to direct detection of the signal field by the factor (cid:112) P LO /P sig ∼ – , where P LO is the power of the local oscillator field and P sig is the signal power (see supplementary information II).11 xperimental setup: The light from a 780 nm diode laser is split into three beams (see Fig 1b): The first beam isfrequency shifted using an Acousto Optic Modulator (AOM) and used as probe field. This probe field is focused witha microscope objective onto the nanofibre waist. The second beam, the local oscillator, goes through the nanofibre.Polarisation controllers are used in both the probe and local oscillator fields to maximise their interferences. Afterpassing through the fibre the combined field is detected on a balanced detector together with the balance field comingfrom the last beam of the laser. The photocurrent output from the detector is then passed through a home-builtlow electronic noise dual-quadrature lock-in amplifier to produce two quadrature signals that are recorded on anoscilloscope. The lock-in amplifier first high-pass filters the photocurrent, then amplifies it, and finally mixes it downusing two radio-frequency mixers. The phases of the mixing processes are to generate orthogonal quadrature signalsquantifying the envelope amplitude of the sine and cosine components of the photocurrent. Anti-aliasing filters areemployed prior to the oscilloscope to prevent mixing of high frequency noise into the recorded signals. A final signalproportional to the scattered field amplitude is generated by taking the quadrature sum of the two quadrature signals.
Experimental procedure:
The nanofibre is positioned on top of a microscope cover slip. With a syringe a ∼ LOMO × (0.75 NA) probe objective is immersed into the droplet and focused on the nanofibre waist.Before adding nanoparticles to the water a set of noise calibration data is recorded. With a micro pipette, 20 µ L ofnanoparticles solution is added to the water at concentration of . × , . × and . × particles per mL for25 nm, 5 nm silica spheres and 10 by 45 nm gold nanorods respectively . The nanoparticle concentrations are chosensuch that multiparticle events are highly improbable, between . × − to . × − particles on average withinthe detection volume. Then, continuous time traces with bandwidth of 50–250 kHz are recorded on an oscilloscope(Tektronix MDO3054). Trapping events are identified manually and post processing is performed on a computer (seeSupplementary Information VI). Noise floor characterisation:
In our experimental configuration, the magnitude of the quantum noise introduced byquantisation of light scales differently with local oscillator power than that of technical noise sources . The electronicnoise of the detection apparatus is independent of local oscillator power; while the power of the quantum and technical12oise introduced by the optical field scale linearly and quadratically, respectively . This latter characteristic, whicharises due to the introduction of quantum vacuum fluctuations as an optical field is attenuated , provides a rigorousmethod to characterise whether the biosensor is quantum noise limited. To perform this characterisation, a spectrumanalyser was used to analyse the photoreceiver output at frequencies close to the heterodyne beat frequency undervarious conditions.First, we compare the noise power spectrum for the normal operating conditions of the biosensor using an ultrapurewater sample containing no nanoparticles or biomolecules, with the electronic noise measured for the same conditionsbut with both local oscillator and probe field blocked. As shown in Fig. 2a, the electronic noise power was found to bea uniform 8.1 dB below the laser noise over a broad frequency band centred on the heterodyne beat frequency. Spikesare observed in both the electronic and laser noise at the beat frequency. These arise from pick-up in the detection andoptical apparatus, respectively and are sufficiently narrow band to be negligible over the frequency band relevant toour measurements.Second, to test whether the local oscillator field is quantum noise limited, we characterise the scaling of the measure-ment noise with the local oscillator power in the absence of the probe field. Fig. 2b shows the noise power for a rangeof local oscillator powers between 0 and 1 mW, averaged over a 10 kHz frequency window centred at the heterodynebeat frequency. The observed linear dependence is consistent with a quantum noise model due to quantisation of thelocal oscillator field, and inconsistent with the power-squared dependence expected for technical noise. We thereforeconclude that the local oscillator field is quantum noise limited over the frequency range of interest.Finally, the measurement can be degraded by probe noise. This noise is introduced along with the collected lightintensity when the probe scatters from a trapped particle. To characterise it in isolation from the motion of the particle,we align the probe to a defect on the surface of the nanofibre. This introduces a stationary source of scattering. Thetotal noise power spectrum of the biosensor can then be characterised as a function of the collected signal power, andcalibrated to the quantum noise level via measurements using the local oscillator field alone, as discussed in the maintext. 13 uture improvement in precision: Several avenues exist that may allow further improvements in precision, with-out increased risk of photodamage. The noise floor could be reduced by a factor of four by replacing the balancedphotoreceiver with a single ultralow noise photodiode, and using homodyne detection and optical phase stabilisation,rather than heterodyne detection. Since, in our implementation, the probe intensity was one order of magnitude lowerthan the local oscillator intensity, further improvements could be obtained by increasing the probe intensity.
Nanoparticles and biomolecules:
The 5 nm and 25 nm radius silica particle are from Polysciences Inc. and thegold nano rods are from Sigma-Aldrich. BSA were purchase from Sigma-Aldrich and Anti-
E. coli antibodies fromAustralian Biosearch Inc.
Acknowledgements
This work was supported by the Australian Research Council Discovery Project (contract no. DP140100734)and by the Air Force Office of Scientific Research and Asian Office of Aerospace Research and Development (grant no. FA2386-14-1-4046). W.P.B. acknowledges support through the Australian Research Council Future Fellowship scheme FF140100650. M.A.T.is supported by a fellowship from the Human Frontiers Science Program. The authors would also like to thanks Bei-Bei Li foruseful discussions.
Author contributions
W.P.B. conceived and led the project. M.A.T. contributed towards the conceptual design. N.P.M. per-formed the experiments and the data analysis, with contributions from L.M.. Samples were prepared by N.P.M. and M.W. Themanuscript was written by N.P.M., W.P.B., and L.M.
Competing Interests
The authors declare that they have no competing financial interests.
Correspondence
Correspondence and requests for materials should be addressed to:Warwick Bowen (email: [email protected]). upplementary Information: Evanescent single-molecule biosensing with quan-tum limited precision N. P. Mauranyapin, L. S. Madsen, M. A. Taylor, M. Waleed & W. P. Bowen
Contents1 Introduction 162 Quantum noise limited measurement of dipole-scattered light 16 Introduction
In this supplementary information, we develop a theoretical model based on Rayleigh scattering in order to findthe quantum limit of our sensor. We describe how the experimental data is analysed to obtain the results in themain text and present additional data and simulations supporting our conclusions, including measurements of themotional power spectral density of the trapped particles, measurements of the trap potential as a function of opticallocal oscillator power, and simulations of the magnitude of the optical component to the trap potential. We discusscontrol experiments testing for the presence of contaminant particulates, and analysis that allows us to rule out withconfidence the presence of aggregation. We finish by comparing our results to the state-of-the-art and explain how wefabricate the nanofibres.
If an optical field containing n in photons is incident on a dielectric spherical particle, in the ap-proximation that the radius r of the particle is much smaller than the optical wavelength λ the optical field can –generally, and certainly for the results presented in the main article – be well approximated as spatially uniform acrossthe particle, and the scattered field can be well described by dipole scattering. The number of photons scattered by theparticle is then : n scat ≡ σ πw n in = ( kr ) π (cid:18) m − m + 2 (cid:19) (cid:18) λw (cid:19) n in (1)where w is the waist of the optical field, σ is the usual dipole scattering cross-section, k = 2 π/λ is the wave number,and m = n/n m is the ratio of the refractive indices of the particle ( n ) and the medium which surrounds it ( n m ). Wesee that, unsurprisingly, the scattered power depends strongly on the radius of the particle, the refractive index contrast,and the strength of focussing of the gaussian beam. The dimensionless ratio σ πλ = ( kr ) π (cid:18) m − m + 2 (cid:19) (2)contains all of the particle dependence in Eq. (1) and defines how strongly the particle will interact with a generalelectromagnetic field. It therefore quantifies, in an experimental apparatus independent way, the comparative easewith which particles may be detected (the larger the ratio is, the greater the scattering, and the easier the particle willbe to detect). We refer to it here as the detectability of the particle. It is natural to separate the detectability into two16arts: a geometrical size factor ( kr ) / (6 π ) , and the refractive index contrast ( m − / ( m + 2) . Quantum noise limit for direct detection of the scattered field
If this scattered field is collected with a collectionefficiency η and is directly detected, then from Eq. (1) the average photon number observed by the measurement is (cid:104) n det (cid:105) = η π ( kr ) (cid:18) m − m + 2 (cid:19) (cid:18) λw (cid:19) (cid:104) n in (cid:105) (3)To be confident of the presence of the particle, it must be statistically possible to distinguish this signal from the signalthat exists if no particle is present. Assuming that the incident field is shot noise limited, such that the photons incidenton the particle are uncorrelated with each other, and the scattering process is linear, the noise if the measurement isdictated by Poissonian statistics, with the variance V ( n det ) ≡ (cid:104) n det (cid:105) − (cid:104) n det (cid:105) of the measurement equal to the meanvalue, that is V ( n det ) = (cid:104) n det (cid:105) . In this case, the signal-to-noise ratio (SNR) of the measurement is SNR = (cid:104) n det (cid:105) V ( n det ) + V ( n det ) , (4)where the noise on the denominator has two components: V ( n det ) , the variance of the photon number in the presenceof a scattering particle, and V ( n det ) , the variance when no scattering particle is present. Here, since V ( n det ) = (cid:104) n det (cid:105) ,we see that, unsurprisingly, V ( n det ) = 0 (we will find later that this is not the case for heterodyne measurement). Thesignal-to-noise ratio is then SNR = (cid:104) n det (cid:105) = η (cid:104) n in (cid:105) π ( kr ) (cid:18) m − m + 2 (cid:19) (cid:18) λw (cid:19) (5)The quantum noise limit of the measurement is then found by setting the signal to noise SNR = 1 , resulting in aminimum detectable scattering cross-section σ min = 4 πw ηn in , (6)and a minimum detectable particle radius r min of r min = 12 (cid:18) wλ π (cid:19) / (cid:18) η (cid:104) n in (cid:105) (cid:19) / (cid:18) m + 2 m − (cid:19) / . (7)As one might expect, improved refractive index contrast (increased m ), improved collection efficiency, and increasedinput photon flux ( n in ) all improve the minimum detectable radius.Comparison of Eq. (2) with Eq. (6) shows the significance of the dimensionless ratio defined in Eq. (2). If this ratioequals one for a given particle, then, at the quantum noise limit, one incident photon would be sufficient to detect17he particle with a perfect efficiency detector if the incident field is focussed so that its waist size equals the opticalwavelength ( w = λ ). Quantum noise limit for heterodyne detection
In heterodyne detection, rather than directly detecting the scatteredfield, it is instead interfered with a bright local oscillator field, separated in frequency from the incident field by afrequency difference ∆ . In a quantum mechanical description the signal field (here, the scattered field) is treatedquantum mechanically, while the local oscillator is treated classically with its fluctuations neglected. This is validso long as the local oscillator field is much brighter than the signal field. In our case this approach is particularlyappropriate. The field scattered into the tapered optical fibre has photon flux in the range of × per second, whilethe photon flux of the local oscillator was × per second, 12 orders of magnitude greater. The combined fieldcan then be expressed in a rotating frame at the frequency of the scattered field as ˆ a = (cid:112) N LO e i ∆ t + (cid:112) (cid:104) N det (cid:105) + δ ˆ a, (8)where N LO and N det are the photon flux in the local oscillator and scattered fields reaching the detector, respectively,in units of photons per second, formally ˆ a is the annihilation operator of the combined field but informally it maybe thought of as an appropriately normalised complex phasor describing the amplitude and phase of the field inphase space, and δ ˆ a is a fluctuation operator with zero mean ( (cid:104) δ ˆ a (cid:105) = 0 ) that includes all of the fluctuations of thescattered field (the shot noise). In quantum mechanics the non-commutation of the annihilation and creation operators( [ˆ a, ˆ a † ] ≡ ˆ a ˆ a † − ˆ a † ˆ a = δ ( t ) , where δ ( t ) is the Dirac delta function) is ultimately responsible for the noise floor of themeasurement, once all technical noise sources are removed, and therefore for the shot noise.The combined field is then detected, resulting in a photocurrent proportional to the photon number in the field i = ˆ a † ˆ a (9) = N LO + (cid:104) N det (cid:105) + 2 (cid:112) N LO (cid:104) N det (cid:105) cos ∆ t + (cid:112) N LO (cid:0) δ ˆ a † e i ∆ t + δ ˆ ae − i ∆ t (cid:1) + δ ˆ a † δ ˆ a (10) ≈ N LO + (cid:104) N det (cid:105) + 2 (cid:112) N LO (cid:104) N det (cid:105) cos ∆ t + (cid:112) N LO (cid:16) ˆ X cos ∆ t + ˆ P sin ∆ t (cid:17) (11)where in the approximation we have made the usual approximation that the product of fluctuations δ ˆ a † δ ˆ a is muchsmaller than the other terms in the expression and that the scattered photon flux is much smaller than the local oscillatorflux ( N LO >> N det ) . For measurements that approach the quantum noise limit this is appropriate so long as the local18scillator photon number is much larger than one, which is clearly the case here. The quadrature operators ˆ X and ˆ P are defined as usual as ˆ X = ˆ a † + ˆ a (12) ˆ P = i (cid:0) ˆ a † − ˆ a (cid:1) , (13)and their commutation relation [ ˆ X, ˆ P ] = 2 iδ ( t ) results in the Heisenberg uncertainty principle V ( ˆ X ( t )) V ( ˆ P ( t )) ≥ ,with the shot noise (or quantum noise) limit of a measurement reached when V ( ˆ X ) = V ( ˆ P ) = 1 .It is clear from Eq. (11) that signals due to the mean scattered photon number n det are present both at zero frequencyand in a beat at frequency ∆ . Since, as discussed above for heterodyne detection, the local oscillator must be muchbrighter than the signal field, the zero frequency term is obscured by the the presence of the local oscillator. Further-more, since the beat term includes the square-root of the local oscillator photon number, its amplitude is much greaterthan that of the zero frequency term. Consequently, in heterodyne detection, and in our case to detect the presence ofa particle, the component at frequency ∆ is utilised. To extract this component in our experiments, since the phase ofthe beat is in practise unknown, we mix the photocurrent down with two electronic local oscillators at frequency ∆ but π/ out of phase, and integrate for a time τ . From these two photocurrents, it is possible to extract the magnitudeof the beat. The result is equivalent to mixing the photocurrent of Eq. (11) down in the following way: ˜ i = (cid:90) t + τt dt i × cos ∆ t (14) ≈ (cid:112) N LO (cid:20)(cid:112) (cid:104) N det (cid:105) τ + (cid:90) t + τt (cid:16) ˆ X cos ∆ t + ˆ P sin ∆ t cos ∆ t (cid:17) dt (cid:21) , (15)where we assume that τ is sufficiently long to remove components in the photocurrent that oscillate at frequencies fastcompared with the beat frequency ∆ . Assuming that the fluctuation terms ˆ X and ˆ P are Markovian white noise, as isthe case for a shot noise limited field or any field with a white power spectrum over the frequencies of interest, thefluctuation term in the above equation can be re-defined via Ito calculus as a new Markovian fluctuation operator ˜ X ≡ (cid:114) τ (cid:90) t + τt (cid:16) ˆ X cos ∆ t + ˆ P sin ∆ t cos ∆ t (cid:17) dt, (16)normalised such that the variance V ( ˜ X ) = 1 . This results because – in a classical sense – both the sign and valueof ˆ X and ˆ P are random (they are, classically, random Gaussian variables) as a function of time. The effect of the19inusoidal envelopes modulating each term is then just to modulate the total power of the noise – unlike a coherentsignal, integration in time of a random variable multiplied by a function such as sin ∆ t cos ∆ t does not average to zeroin the long time limit.We then find ˜ i = (cid:112) N LO (cid:18)(cid:112) (cid:104) N det (cid:105) τ + (cid:114) τ X (cid:19) (17) = √ n LO (cid:32)(cid:112) (cid:104) n det (cid:105) + ˜ X √ (cid:33) , (18)where the total photons numbers n LO and n det are given by n LO = N LO τ and n det = N det τ , respectively. Fromthis signal we wish to distinguish whether there is a scattering particle in the optical field or not, and can – as we didin the previous section on direct detection – use the signal-to-noise ratio in Eq. (4). Unlike direct detection, however,here we find that there is noise in the measurement even when (cid:104) n det (cid:105) = 0 . This exists due to the presence of the localoscillator, which amplifies the vacuum noise of the field. We then find that for a quantum noise limited field, with V ( ˜ X ) = 1 , SNR = (cid:104) n det (cid:105) , (19)exactly identical to the expression we obtained for direct detection in Eq. (5). We therefore find that, in principle, thequantum noise floors of heterodyne detection and direct detection are identical, and both governed by Eqs. (6) and (7). In our experiments we make one modification to the heterodyne detection scheme described above, following theapproach used in Refs. . Often, and in our experiments, classical laser intensity present in the local oscillator cancontaminate the measurements and preclude reaching the quantum noise limit. To eliminate this laser noise, we splitthe laser field used for the local oscillator into two equal power beams on a beam splitter. One of these beams wasinterfered with the scattered field, while the other bypassed the experiment. The two beams were then detected on abalanced detector, which allowed the classical amplitude noise to be subtracted.20athematically, extending Eq. (8), we can describe the two fields that arrive at the detector as ˆ a = ( (cid:112) N LO + X c / √ e i ∆ t + (cid:112) (cid:104) N det (cid:105) + δ ˆ a (20) ˆ a − = ( (cid:112) N LO + X c / √ e i ∆ t + δ ˆ a − , (21)where X c is the classical intensity noise on the laser, which is correlated between the two fields, while ˆ a − is an anni-hilation operator describing the cancellation field and δa − is the quantum noise on that field which, due to the actionof the beam splitter, is uncorrelated to the noise on the local oscillator field.Working through similar mathematics tothat above, and assuming that the two fields are perfectly balanced, and therefore that the classical noise is perfectlycancelled, we find that the single-to-noise ratio is quantum noise limited, but degraded by a factor of two comparedwith pure heterodyne detection. That is SNR = (cid:104) n det (cid:105) / . (22)In this case, the minimum detectable scattering cross-section σ min and particle radius r min are degraded to σ min = 8 πw ηn in , (23)and r min = (cid:18) wλ π (cid:19) / (cid:18) η (cid:104) n in (cid:105) (cid:19) / (cid:18) m + 2 m − (cid:19) / . (24)Knowing that P in = (cid:126) ω (cid:104) n in (cid:105) /τ with P in the input power, (cid:126) the reduced Plank constant, ω the frequency of the lightand τ the measurement time we have : σ min = 8 (cid:126) ωητ I in (25)and r min = (cid:18) λ (cid:126) ωπ ηI in τ (cid:19) / (cid:18) m + 2 m − (cid:19) / (26)with I in = P in /πw the input intensity.We note that this kind of intensity noise cancellation has proved to be a powerful technique for precision evanescentbiosensing, and have been utilised in a range of previous experiments, see for example Ref. .21 .0.2 Scaling of quantum and classical noise as a function of loss It is generally possible to confirm that an experiment is quantum noise limited by varying the efficiency of the mea-surement and measuring the effect this has on the signal-to-noise ratio. This is due to differences in the effect of losson quantum and classical noise due to the introduction of quantum vacuum noise. This vacuum noise contamination isa necessary consequence of the Heisenberg uncertainty principle. Mathematically, inefficiencies can be modelled bythe action of a beam splitter, one output of which is lost, and one input of which introduces the vacuum fluctuations .To see the effect of inefficiency on our experiment, we begin with Eq. (8), describing the annihilation operator of thefield to be detected via heterodyne detection. In that equation, the detected photon flux N det = ηN scat where N scat isthe scattered photon flux and η is the detection efficiency. δ ˆ a is the fluctuation operator that describes the noise on themeasurement after the inefficiencies in detection. In our initial treatment, we took this to be purely quantum noise. Ifwe include Markovian classical amplitude quadrature noise prior to the introduction of any losses, and vacuum noiseentering due to the presence of the loss then δ ˆ a can be expanded as δ ˆ a = √ η ( δ ˆ a prior + δa c ) + (cid:112) − ηδa v , (27)where δ ˆ a prior , δa c , and δa v are, respectively, the annihilation operators describing the quantum fluctuations on thefield prior to any losses, the classical noise, and the vacuum noise introduced by the loss. By examining the variance ofthe amplitude quadrature δX = δ ˆ a † + δ ˆ a we can gain some insight into the difference between quantum and classicalnoise: V ( δ ˆ X ) = ηV ( δ ˆ X prior ) + ηV ( δX c ) + (1 − η ) V ( δX v ) (28) = ηV ( δX c ) + 1 , (29)where we have used the property of quantum fluctuations of coherent light and vacuum fields, that V ( δ ˆ X prior ) = V ( δ ˆ X v ) = 1 . We see, therefore, that while the variance of the classical noise is attenuated with increasing loss(decreasing η ), the quantum noise is unchanged at unity by the action of attenuation. This being the case, we can makethe substitution δa q ≡ √ ηδ ˆ a prior + √ − ηδa v , treating the combined quantum noise as one single input quantumvacuum field. Going through the same calculation as in the main part of Section 2 but including the classical noise22erm by making the substitution δ ˆ a → δ ˆ a + √ ηa c we reach the photocurrent ˜ i = √ n LO (cid:32)(cid:112) (cid:104) n det (cid:105) + ˜ X + √ η ˜ X c √ (cid:33) (30) = √ η n LO ,prior (cid:32)(cid:112) (cid:104) n det (cid:105) + ˜ X + √ η ˜ X c √ (cid:33) (31)where n LO ,prior is the number of photons used in the local oscillator prior to any losses, and ˜ X c is defined, for theclassical noise, in the same way as ˜ X . The variance of the measurement is then V (˜ i ) = n LO ,prior (cid:16) η + η V ( ˜ X c ) (cid:17) . (32)We see that the variance of the quantum noise (first term in the brackets) scales linearly with efficiency η while theclassical noise scales quadratically. Therefore, by quantifying the measurement noise floor as a function of attenuation,as was performed in the main paper, it is possible to unambiguous determine in which regimes classical and quantumnoise dominate, and therefore whether the quantum noise limit has been reached. An alternative approach to quantum limited measurement is to perform homodyne detection. This is very closely re-lated to heterodyne detection but uses a local oscillator whose frequency matches the scattered (signal) field frequency(i.e. ∆ = 0 ). As can be seen from inspection of Eq. (15), this choice of frequency has the immediate advantage ofeliminating one of the two noise terms contributing to the measurement, and therefore can be used to improve thequantum noise limited measurement signal-to-noise ratio by a factor of two. However, heterodyne detection has themajor advantage that the signals of interest are shifted up to sideband frequencies near the beat frequency ∆ . In homo-dyne detection, this is not the case. This has severe consequences for biophysical applications where the signals to bemeasured typically reside in the hertz-kilohertz frequency range (in our case, typically beneath 100 Hz), a frequencyrange in which many low frequency technical noise sources reside . For our experiments, such noise sources werefound to preclude quantum noise limited operation by many orders of magnitude when using homodyne detection.Homodyne detection has the further disadvantage that it requires the local oscillator and scattered field to be phaselocked with high precision. In particle scattering experiments in liquid this is made difficult both by the weakness of23he scattered field and by motion of the particle which changes the path length of the scattered field. It was for thesereasons that heterodyne detection was chosen for the results reported here. There have been significant recent efforts to use the enhanced light-matter interactions available in optical microcavi-ties to allow precision nanoparticle and biomolecule detection (see for example Refs. ). Here, we derive the quantumnoise limit for such measurements. In the usual approach – reactive microcavity based sensing –the action of thenanoparticle or biomolecule is to change the average refractive index within the resonator and therefore shift its res-onance frequency. For a cavity that is initially driven with on-resonance light, the dynamics of the field within thecavity, in the presence of a biomolecule or nanoparticle induced frequency shift δω , is (see for example Ref. ) ˙ˆ a = − ( κ/ iδω )ˆ a + √ κ ˆ a in , (33)where κ is the cavity decay rate, ˆ a in is the annihilation operator describing the incident field, and we assume – toobtain a bound for the best possible predicted sensitivity – that there is no loss within the cavity, other than backthrough the input coupler. In the realistic regime where the particle can be treated as stationary over the characteristictimescales of the cavity dynamics, this equation can be solved by taking the steady-state solution where ˙ˆ a = 0 . Wethen find that ˆ a = √ κκ/ iδω ˆ a in . (34)The input-output relation ˆ a out = ˆ a in − √ κ ˆ a can then be used to determine the out-coupled field: ˆ a out = − (cid:18) κ/ − iδωκ/ iδω (cid:19) ˆ a in . (35)As before, the input field can be expanded as a bright coherent classical field (cid:104) a in (cid:105) = (cid:112) (cid:104) N in (cid:105) and a fluctuation term δa in with (cid:104) δa in (cid:105) = 0 , with (cid:104) N in (cid:105) being the mean photon flux incident on the cavity. We then obtain ˆ a out = − (cid:18) κ/ − iδωκ/ iδω (cid:19) (cid:16)(cid:112) (cid:104) N in (cid:105) + δ ˆ a in (cid:17) (36) = − (cid:18) κ / − iκδω − δω κ / δω (cid:19) (cid:16)(cid:112) (cid:104) N in (cid:105) + δ ˆ a in (cid:17) . (37)Assuming that the frequency shift is small compared to the cavity decay rate, and the optical fluctuations are muchsmaller than (cid:112) (cid:104) N in (cid:105) we can neglect all terms in this expression that include either δω or the product δωδa in , with24he result ˆ a out = − (cid:18) − iδωκ (cid:19) (cid:112) (cid:104) N in (cid:105) − δ ˆ a in . (38)It is apparent from this expression that the first order effect of the particle on the output optical field is to shift its phase.The phase quadrature of the output field is ˆ P out = i (cid:16) ˆ a † out − ˆ a out (cid:17) = 8 δωκ (cid:112) (cid:104) N in (cid:105) − δ ˆ P in . (39)A homodyne measurement can detect the phase quadrature, in principle, without any additional noise, resulting in aphotocurrent integrated over the time τ of i ∝ (cid:90) t + τt dt ˆ P out = 8 τ δωκ (cid:112) (cid:104) N in (cid:105) − (cid:90) t + τt dt δ ˆ P in . (40)Similar to our previous treatment of the quantum noise limit for heterodyne detection, from Ito calculus for Marko-vian fluctuations (such as the quantum noise of a coherent laser) the integral (cid:82) t + τt dt δ ˆ P in = √ τ δ ˜ P in , where δ ˜ P in isa Markovian noise process with – for a shot noise limit field – variance V ( δ ˜ P in ) = 1 . The detected photocurrent isthen i ∝ δωκ (cid:112) (cid:104) n in (cid:105) − δ ˜ P in , (41)where as before n in = τ N in is the total photon number incident on the detector during the measurement time. UsingEq. (5) and assuming the incident field is shot noise limited, the signal-to-noise ratio for discrimination of the presenceof the frequency shift due to the particle is then SNR = (8 δω/κ ) n in V ( δ ˜ P in ) = 32 (cid:18) δωκ (cid:19) n in . (42)Setting SNR = 1 , we find the minimum detectable frequency shift δω min = κ √ n in . (43)It now remains to determine the frequency shift introduced by a particle within the optical field. In first order perturba-tion theory, assuming that the particle is located at the position of peak intensity within the optical mode, the frequencyshift it induces is given by δω = − α Ω2 V , (44)25here α is the polarizability of the particle, Ω is the bare cavity frequency, and V ≡ (cid:82) (cid:15) r | E ( r ) | d V max {| E ( r ) | } (45)is the mode volume of the cavity optical eigenmode, with E ( r ) being the electric field distribution of the mode, r being a spatial co-ordinate in three dimensions, and (cid:15) r being the relative permittivity of the cavity medium.For the case of a dielectric sphere, at optical wavelengths the polarizability is α = 4 π(cid:15) m r (cid:18) m − m + 2 (cid:19) , (46)where (cid:15) m = n m is the relative permittivity (or refractive index square) of the surrounding medium. For a sphere, theparticle-induced optical frequency shift is therefore δω = − π(cid:15) m r V (cid:18) m − m + 2 (cid:19) Ω . (47)Substituting this expression into Eq. (43) for the minimum detectable frequency shift and rearranging, we finally arriveat the minimum detectable scattering cross section and radius using cavity enhanced sensing σ min = k V π(cid:15) m Q n in (48) r min = (cid:18) V π(cid:15) m Q (cid:19) / (cid:18) m + 2 m − (cid:19) / (cid:18) n in (cid:19) / , (49)where we have defined the optical quality factor Q ≡ Ω /κ . In this section we quantify the best sensitivity achievable by cavity enhanced and heterodyne detection sensors andcompare these predictions with examples of the state-of-the-art:For typical parameters used in our heterodyne nanofibre experiments, with an input probe power of P in = 2 πc (cid:126) n in /λτ =2 mW, with (cid:126) the Planck constant, c the speed of light, λ = 780 nm and τ = 0 . sec the measurement time, a probebeam waist of w = 3 µ m, n = 1 . , n m = 1 . and a collection efficiency η = 0 . , we find the theoretical minimalcross section detectable to be σ min fiber = 3 × − nm equivalent to a silica sphere radius of r min fiber = 23 nm.For the same input power and for typical microcavities with a quality factor of Q = 3 × , n m = √ (cid:15) m = 1 . ,and a mode volume V = 350 µ m , we find that the minimal cross section detectable is σ min = 1 . × − nm corresponding to a silica particle with a radius of r min = 3 . nm.26igure 5: Fourier transform of the trapping events of the figure 3a,b,c (green for 25 nm silica particle, blue for 5 nmsilica particle and orange for 5 by 45 nm gold nano rod). The black traces represent the laser noise of each experimentand the color traces are fitted with a Lorentzian shown in black line.These theoretical predictions show that quantum noise limited microcavities should be able to detect particles withcross section three orders of magnitude smaller than nanofibre sensors, corresponding to a reduction in radius for asilica particle of a factor of eight. Unfortunately to date it has proved difficult to reach this limit with such biosensorsand their sensitivity is at the same order of magnitude as heterodyne detection (see section 10). As shown in Ref. the power spectral density of a trapped particle can be approximated by a Lorentzian function andits corner frequency indicates the stiffness of the trap. Figures 5, 6,7 show the power spectral density of each trappingevent displayed in the main text, confirming that the particles are trapped by our system.27igure 6: Fourier transform of the trapping events of the figure 3d for a combined solution of 25 and 5 nm silicaparticleFigure 7: Fourier transform of the trapping events of the figure 4 for a combined solution of BSA (pink) and antibody(green). 28 Probability distribution and trapping potential calculations
When a particle is scattering probe light close to the nanofibre more light will be collected than when the particleis further from the fibre, thereby modulating the amplitude of the recorded signal field. This relation between theparticle position and the amplitude of the signal has the same shape as the optical field and decays approximatelyexponentially following the relation r = − k − log ( (cid:104) n det (cid:105) ) where r is the position of the particle relative to the fibre , k = 2 πn m /λ is the wave number, and λ = 780 nm is the wavelength of the light . The position of the nanoparticlecan then be calculated from the amplitude relative to the position with the highest amplitude. A histogram of theposition generates the position probability density p density ( r ) for each event. According to Boltzmann statistics it isrelated to the potential U ( r ) experienced by the trapped particle by U ( r ) /k B T = − log ( p density ( r )) . To calibrate our sensor we can calculate the resolution with which we can track the radial position of the particle. Fromthe section 6, we have the relation between the detected signal (cid:104) n det (cid:105) and the radial position r . The error in positionvariation is then: δr = − πn m λ log ( (cid:104) n det (cid:105) − δn det ) + 2 πn m λ log ( (cid:104) n det (cid:105) + δn det ) (50) δr = 2 πn m λ log (cid:18) (cid:104) n det (cid:105) + δn det (cid:104) n det (cid:105) − δn det (cid:19) (51)with δn det the standard deviation of the detected signal noise.From the results displayed in figure 3 of the main text, we find that for the 5 nm and 25 nm silica nanoparticle (cid:104) n det (cid:105) /δn det is equal to 37 and 138 respectively with average over 10 ms. The resolution is then 5 nm and 1 nmrespectively with a 100 Hz bandwidth. Because the power spectral density shape of the nanoparticles detection event is Lorentzian (see section 5), we havestrong evidences that they are trapped. In conventional nanofibre traps, a combination of attractive optical gradientforce and repulsive electrostatic forces are used together to trap the particle next to the fibre . Similar to opticaltweezers, particles diffusing in the evanescent field around the fibre will be polarised and then attracted toward thecentre of the fibre following the gradient of the light intensity. In our case, we theoretically (see following subsection 8)29nd experimentally (see following subsection 8) show that this optical force is not powerful enough explain the trappingof nanoparticles observed in our experiment. Alternatively, the attractive force could, in principle be introduced bygravity, as observed in some total internal reflection microscopy (TIRM) experiments . However, we found that theseforces are roughly five orders of magnitude too small to explain the attractive forces observed here (see section 8). Modelling of the optical trapping potentials
In a step index optical fibre the optical field is guided by total internalreflection caused by the refractive index difference between the core and the cladding. Optical nanofibres works in thesame way with the core of silica and the cladding made of the surrounding medium. As the light is guided by totalinternal reflection an evanescent optical field extends out of the fibre. Theoretical models for the extend and intensityof the evanescent fields are well developed in . The electric fields can be found analytically using the model fora step-index fibre, however finding the propagation constant is a numerical task. Here we follow reference closelyand find the propagation constant, get the normalisation for the electric fields, and calculate electric fields on a 500 by500 grid in a 4 µm area centered at the nanofibre. From the electric field we calculate the intensity as shown in Fig.8 for 1 mW of horizontally polarised 780 nm light. The trapping potential U ( r ) is obtained as function of the particleradius r and polarisability α ( r ) as U ( r ) = − α ( r ) | (cid:126)E | , with α ( r ) = 4 π(cid:15) n m r m − m + 2 , (52)where n m = 1 . is the refractive index of water.To trap particles, the depth of the nanofibre potential must be at least equal to the thermal energy k B T . Our modellingshows that this only occur for silica nanospheres of radius above 145 nm with our experimental parameter (1 mWoptical local oscillator power). The 25 nm and 5 nm particles we trap are predicted to have an optical potential depthof, 3 and 5 orders of magnitude smaller than k B T , respectively (see Figs. 8 and 9). Influence of local oscillator on attractive forces
To verify experimentally that the optical forces are not responsiblefor the attractive trapping forces we vary the optical local oscillator power and examine its influence on the trappingpotential. In Fig. 10 the local oscillator power is varied from 1.1 mW to 0.31 mW, which if optical forces are playing arole should modify the potential. However no significant changes in the trapping potential are observed in agreementwith the theoretical prediction that the optical attractive force is negligible (see Fig. 10).30igure 8: Evanescent field intensity map around the nanofibre. As a visual aid the intensity inside the fibre is set tozero giving the centered disk. The first colorbar shows the intensity in units of
W/m . At a distance of 100 nm fromthe fibre surface where the particle are expected to be trapped, the intensity is × W/m . The second and thirdcolorbars are the trapping potentials normalised to − k B T for the 25 nm and 5 nm particles respectively as markedwith the particles (not drawn to scale with the fibre). Note that in both cases the particles should not be trapped as thepotential is below k B T . 31igure 9: Theoretical trapping potential depth in function of the silica particle radius 100 nm away from the fibresurface. The blue and green points represent the 5 nm and 25 nm silica particles trapped in our experiment, respectively.32igure 10: Trapping potential of all observed trapping event of 25 nm silica particles for different L.O. power: red1.1 mW, yellow 0.78 mW, green 0.57 mW and blue 0.31 mW.An explanation can be that this attraction comes from long range electrostatic forces. Those forces come from adeformation of the counter-ion layer or electric double layer surrounding a charged particle in solution . Thislayer becomes larger as the ion concentration decreases and can be expected to be significant in deionized water. Inthe main text, we show the influence of the electric double layer on the biosensor by changing the ion concentrationusing salt water. Effect of the gravitational forces
As used in total reflection microscopy, the gravitational forces could explain theattractive part of our potential if the particles were trapped on top of the fibre. The gravitational potential is given by : U grav = 43 πr ∆ ρgh (53)with r the radius of the nanoparticle, ∆ ρ the density difference between the nanoparticle and its surrounding medium, g the gravity acceleration and h the hight. For our silica nanoparticles, r = 25 nm, ∆ ρ = 1650 kgm − which gives33 normalised potential of U grav /hk B T = 2 . × − nm − compare to the . × − nm − observed on the figure3 of the main text. We then conclude that the gravitational forces are five orders of magnitude smaller than what weobserve and can not be attributed to the attractive force of our potential. The quantum noise limit calculations in section 4 show that, without some form of scattering cross section enhance-ment, 5 nm silica particles should not be detectable by our sensor. Moreover, from figure 3 of the main text we observethat there is only a factor of ∼ between the signal amplitude of the 5 nm and 25 nm silica particles. According to thedipole scattering theory developed in section 2, the signal amplitude scales as particle size-cubed, therefore, the signalfrom a single 5 nm particle should be more than two orders of magnitude smaller than that from a 25 nm particle.We are confident that, this signal enhancement is not due to aggregation. The concentration of the particles waschosen to be very low in the detection volume ( . × − to . × − particles on average within the detectionvolume), and the samples were sonicated for at least 15 min before being studied. Furthermore, the measured trappingpotentials, probability densities and signal amplitudes are highly reproducible as shown in Fig. 3f of the main textwhere we display, as shaded bands, the potential standard deviation calculated from 9 and 8 events for the 25 nm and5 nm nanoparticles respectively. If aggregates were being detected, we would expect these parameters to vary withthe size of the aggregate. In addition, the signal enhancement is not due to contamination of the sample by outsideparticles as extreme care was taken when preparing and cleaning the apparatus. Moreover, control experiments withultra pure water and salt water, both without nanoparticles were conducted and did not exhibit any events for the entire45 minutes of the experiments, a duration comparable to the experiments performed in the main text.Our hypothesis to explain this enhancement is due to the formation of an electric double layer created by surfacecharge of the particle. The electric double layer is known in rheology to affect the hydrodynamic radius of particles.This phenomena has already been studied on silica nanospheres by Dynamic Light Scattering and has an importantcontribution to the hydrodynamic radius especially when using ultra pure water as the concentration of ion is very lowthe electric double layer become thicker . The effect of the double layer on the polarizability of charged particle hasalso been demonstrated in Ref. . They show that without the polarizability enhancement of the electric double layerthe potential of micelles smaller than 200 nm is not deep enough for the particles to be stably trapped by an optical34weezers. However, the contribution to the polarizability from the electric double layer is sufficient to enable trapping.Surface charge enhancement of the scattering cross section may also explain other observations in the literature. Forinstance, in Ref. a range of small biomolecules, including steptadivin, antibodies and Cy5, are detected using amicrocavity resonator. Without some enhancement mechanism, these molecules are too small be observable by theirapparatus and give a signal several orders of magnitude larger than expected. They attributed this signal enhancementto a thermo-optic effect. However, it was demonstrated in Ref that this effect is also too weak to explain the observedsignal magnitude and should produce signals three orders of magnitude smaller than were observed. The experimentsin Ref. used pure water, so that surface charges could be expected to produce a counter-ion distribution of significantspatial extent and therefore an enhanced scattering cross section.
10 Detected particle state of the art
The state-of-the-art of nanoparticle and biomolecule detection to-date with evanescent biosensors is presented in Fig.11 as function of the light intensity used to detect them. The state-of-the-art is compared to results of our experiments(red points). As can be seen, our sensor is still competitive with other sensors but uses four orders of magnitude lessintensity, reducing the induced photo damage on the sample.
11 Tapered fibre fabrication
Nanofibres are fabricated by pulling a regular 780 HP optical fibre from
Thorlabs Inc. under a 300 sccm torch ofhydrogen. Pulling is stopped when the fibre become single mode again and its diameter reaches about 560 nm . Ourtransmission is between 90 and 99%. 35igure 11: Overview of particle detection experiments with evanescent biosensors.
Comparison of state-of-the-art experiments. Normalised cross-section of the detected particles versus intensity experienced by the particles.Detection of nano particles are represented by circles and detection of bio-molecules by squares. The color indicate thetechniques used: yellow corresponds to techniques using plasmon resonances, blue to cavity enhanced measurements,dashed blue to hybrid cavity enhanced measurement, green for nanofibre techniques and red represents the particlesdetected in the main text. Following the contours means that for a given sensitivity, smaller particles can be detectedby increasing the incident intensity. Nano-particle detection from left to right: 5 nm silica (Si) [this paper] , 10 nmpolystyrene (PS) , 12.5 nm PS , 15 nm PS , 20 nm PS , 25 nm PS , 25 nm Si [this paper], 50 nm PS , and150 nm F e O . Bio-molecule detection form left to right: BSA ,BSA , BSA [this paper], anti E-coli antibody[this paper], biotin-streptadivin and influenza A virus .36 eferences
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