Back-scatter based whispering gallery mode sensing

Whispering gallery mode biosensors allow selective unlabelled detection of single proteins and, combined with quantum limited sensitivity, the possibility for noninvasive real-time observation of motor molecule motion. However, to date technical noise sources, most particularly low frequency laser noise, have constrained such applications. Here we introduce a new technique for whispering gallery mode sensing based on direct detection of back-scattered light. This experimentally straightforward technique is immune to frequency noise in principle, and further, acts to suppress thermorefractive noise. We demonstrate 27 dB of frequency noise suppression, eliminating frequency noise as a source of sensitivity degradation and allowing an absolute frequency shift sensitivity of 76 kHz. Our results open a new pathway towards single molecule biophysics experiments and ultrasensitive biosensors.

Here we derive the sensitivity of back-scatter based whispering gallery mode sensing, and compare it directly to typical dispersive sensing with an optical cavity. The quantum shot noise limit of the two kinds of sensing is found to be identical. However, in contrast to typical dispersive sensing, back-scatter sensing is found to be, in principle, immune to frequency noise. Furthermore, the thermorefractive noise is found to be exactly half that of typical sensing.

A BRIEF EXPLANATION OF THE APPROACH
In what follows we use a standard quantum optics approach to determining the sensitivity limits of WGM sensors (see Ref. [1] for an overview of the method). This approach is nice since it naturally includes, in a formal way, the shot noise due to quantisation of the optical field. However, the approach can be understood classically. Since the entire system is linear, an equivalent result would be obtained by neglecting quantum noise terms (terms like δã), thinking of α j = a j as being the appropriately scaled electric field amplitude in mode j, and adding shot noise phenomenologically at the end of the derivation.
In the derivation we make the approximation that the sideband frequency ω, total scattering rate between counter propagating modes g 0 + g sig , cavity detuning ∆, and thermorefractive frequency noise are all small compared to the optical linewidth γ. This approximation is accurate for the vast majority of WGM biosensing experiments in water since the optical linewidth is typically above 10 MHz, and each of the terms listed above are typically less that 1 MHz. The theory could, in principle, quite easily be derived without these approximations. However, the mathematics would become much more complex and the end result less enlightening.

DERIVING THE HAMILTONIAN
Consider two counter-propagating degenerate WGM resonances, a clock-wise mode (C) and anti-clockwise mode (A), with respective modeshapes U C (r) = V(r, θ)e imφ and U A (r) = V(r, θ)e −imφ , where r, θ, and φ are the usual cylindrical co-ordinates, and m is the azimuthal mode number. The bare Hamiltonian of the system is where we have neglected the constant term due to the zero-point energy of the system. The total Hamiltonian of the system contains both the bare Hamiltonian and terms V due to the energy of polarizable particles within the electric fieldÊ each mode. Defining the modeshapes to be normalised we have |U A | 2 dV = |U C | 2 dV = 1; and since |e ±imφ | 2 dφ = 2π, |V(r, θ)| 2 drdθ = 1/2π. Assuming no other resonant modes are nearby, the electric field within the WGM resonator can be approximated by a sum of the electric fields of each of these modeŝ = EV * (r, θ)e iΩt â † A e imφ +â † C e −imφ + h.c.
whereâ j is the annihilation operator of mode j normalised such that â † jâ j = n j and [â j ,â † j ] = 1, and E is the quantised field strength (i.e. the zero point electric field magnitude in each mode) where, since the two modes are symmetric apart from propagation direction, with degenerate frequency and volume, their quantised field strengths are identical.
We wish to consider microscopic fluctuations in the polarizability of the WGM structure. Therefore, we introduce the polarizability density ρ α (r) which describes the capacity of an electric field to induce a change in polarization around the point r. The polarization density is then just ρ p (r) = 0 ρ α (r)Ê(r) (5) = 0 Eρ α (r) V * (r, θ)e iΩt â † A e imφ +â † C e −imφ + V(r, θ)e −iΩt â A e −imφ +â C e imφ .
Since for a single dipole, the polarization energy is V = −p ·Ê = −pÊ, the polarization energy density is ρ V (r) = −ρ p (r)Ê (7) = − 0 E 2 ρ α (r) V * (r, θ)e iΩt â † A e imφ +â † C e −imφ + V(r, θ)e −iΩt â A e −imφ +â C e imφ 2 (8) Where we've made the usual rotating wave approximation, neglecting fast oscillating terms in time. The polarization energy is then where respectively represent the polarization induced self-energy of modes A and C, the interaction energy between them. We consider three different contributions to the polarizability density: the nanoparticle that we seek to detect with point-like polarizability α sig located at r sig , a static defect scattering center on the WGM with point-like polarizability α 0 located at r 0 , and a spatially and temporally fluctuating polarizability density ρ α,therm (r, t). The total polarizability density is then ρ α (r) = α sig δ(r sig ) + α 0 δ(r 0 ) + ρ α,therm (r, t).
(15) Substituting this into Eq. (13) above we find Similarly, substituting into Eq. (14) we find = 0 E 2 α sig |V(r sig , θ sig )| 2 e 2imφ sig + α 0 |V(r 0 , θ 0 )| 2 e 2imφ 0 + ρ α,therm (r, t)|V(r, θ)| 2 e 2imφ dV (20) = g sig e 2imφ sig + g 0 e 2imφ 0 + g cross therm (21) The modification to the Hamiltonian V due to polarizable particles is therefore with the total Hamiltonian being We see that each of the nanoparticle, defect centre, and thermorefractive noise introduce both a shift in the WGM resonance frequencies, and a coupling rate between them. We can simplify the problem, by moving into a rotating frame at frequency Ω − ∆ 0 , where ∆ 0 is the detuning of the laser field from the bare optical resonance. To do this, we apply the unitaryÛ = e −i(Ω+∆ 0 )t such thatH =Û †ĤÛ − (Ω − ∆ 0 ); where thed enotes throughout a Hamiltonian or operator in the the rotating frame. We then arrive at withã =âe −i(Ω−∆)t , and ∆ = ∆ 0 + g 0 being the modified cavity defining due to the scattering centre g 0 .

Variances of fluctuating energy terms
The energy terms due to thermorefractive noise g self therm and g cross therm are both zero-mean noise terms. Later on in this document it will be useful to know the relationship between their variances.
The variance of g self therm is where we've used the usual approximation that thermorefractive noise is delta-correlated, ρ α,therm (r, t)ρ α,therm (r , t) = ρ α,therm (r, t) 2 δ r=r , and, assuming a homogeneous material have taken the average thermorefractive noise to have no φ dependence. Unlike g self therm , g cross therm can in general be complex. We therefore break it into real and imaginary parts g cross therm = R cross therm + iI cross therm and calculate the variance of each independently.

DIRECT MEASUREMENT OF THE PHASE SHIFT ON A SINGLE MODE AS A BENCHMARK
The usual approach to dispersive measurement in an optical resonator is to excite the resonator on resonance, and directly measure the phase shift introduced on the field exiting the resonator. This approach has been shown, in principle, to be optimal, reaching the quantum limit due to shot noise. Here, prior to considering the full measurement protocol, we consider this simplified case to derive the quantum limit on measurements of the signal scattering rate g sig (or equivalently, the polarizability of the nanoparticle). The Hamiltonian is then found by dropping all of the coupling terms from Eq. (24), and considering only one of the two counter-propagating modes, dropping the redundant subscript.
The equation of motion for the operatorã may then be determined from the quantum Langevin equatioṅ where we have used the Boson commutation relation [ã,ã † ] = 1, γ(t) = γ in (t)+γ l (t) is the total decay rate of the optical resonator, γ in (t) is the input coupling rate with explicit time dependence included to allow noise in this parameter to be included in the model, γ in (t) is the loss rate of the resonator, andã in and δã l are respectively the bright field entering the resonator through in the input coupler and vacuum noise entering through loss, withñ =ã † inã in being the incident optical intensity in units of photons per second.

Input noise terms
Each of the annihilation operators representing input fields may be broken down into amplitudeX and phaseỸ quadratures as The Boson commutation relation, results in a commutation relation betweenX(ω) andỸ(ω), [X(ω),Ỹ(ω)] = 2i, such that their is an uncertainty principle relating the two quadratures ∆X∆Ỹ ≥ 1. It is this uncertainty principle which, from the perspective of quantum mechanics, results in the shot noise limit to sensing. If the input laser is coherent at frequency ω, δX in (ω) 2 = δỸ in (ω) 2 = 1. However, it is always the case at low frequencies of interest to biosensing experiments, that classical noise sources enter, both on the amplitude of the light and on the phase. The noise on these input fields can then be broken into a classical component and a quantum component with unity variance, eg. δX in (ω) = δX q (ω) + δX c (ω), where δX 2 q = 1. Taking the variance of each noise term, and using the fact that the quantum and classical noise is uncorrelated, we find The classical phase noise can be conveniently reexpressed in terms of absolute phase noise, i.e. δỸ c (ω) 2 is the variance of classical phase noise, scaled by the incident optical intensity α 2 in , since the absolute displacement in optical phase space is amplified by the coherent amplitude of the field. Similarly, the classical amplitude noise may be reexpressed as a relative amplitude noise δX c (ω) 2 = V RIN (ω)α 2 in . It is natural to do this because the variance of classical noise in both amplitude and phase scales linearly with intensity, whilst, due to the injection of vacuum noise when the laser is attenuated for any reason, the quantum noise variance remains constant. V ζ (ω) and V RIN (ω) are therefore intensity independent noise parameters. We then have Any fluctuation in optical path length, such as these due to thermal fluctuations and mechanical vibrations, will introduce phase (or equivalently frequency) noise into the sensing apparatus. By contrast, the introduction of amplitude noise requires fluctuations in optical attenuation which are much less common. Furthermore, optical intensity fluctuations are directly measured with a photodiode, and can therefore by compensated for in measured data, or suppressed through fed back/forward as performed in common "noise eater" arrangements. Consequently, phase and frequency noise tend to be both larger in magnitude and more difficult to suppress in optical sensors [3] Solving the mean field The first step to solving the problem is to calculate the mean fields in the resonator and leaving it back through the input coupler. We expand each of the time varying terms in Eq. (48) into their mean values and noise fluctuations (ã = ã +δã = α+δã, γ = γ + δγ =γ + δγ), take the expectation value, and linearize by neglecting noise product terms to geṫ where, since g self therm is zero mean thermorefractive noise g self therm = 0. Choosing the input field mean amplitude to be stationary and taking the steady state (α in (t) = α in ,α(t) = 0), we get where henceforth α is implicitly taken to be the steady state intracavity intensity. So that The limit relevant to high precision sensing experiments is the limit where both the frequency shift due to the nanoparticle g sig and the cavity detuning are small compared to the cavity linewidthγ. Therefore, it is reasonable to take the limit {g sig , ∆} γ. In this case, it is straightforward to show that Eq. (56) may be well approximated as The mean amplitude of the output field may be calculated using the input-output formalism with Since γ in varies with time but the amplitude of variation is small compared with the coupling rate itself, the square-root can be expanded as 2γ in (t) ≈ 2γ in + δγ in (t)/ 2γ in , so that This results in mean and fluctuation input output relations Using Eqs. (57) and (60) we then find where it will be convenient throughout to normalize rates in terms of the mean cavity decay rateγ. In these cases, the normalization will generally be denoted with a . Here, for example g sig = g sig /γ and g self therm = g self therm /γ. η =γ in /γ is the escape efficiency of the cavity, i.e. the probability that a photon in the cavity will leave through the input/output coupler. Since we wish to measure the magnitude of g sig and g sig γ, it is clear from Eq. (63) that we wish to measure a phase rotation on the output field of the resonator (g sig displaces the amplitude of the field a small amount in a direction in phase space orthogonal to the mean amplitude).

Solving for the fluctuating noise
To solve for the steady-state fluctuations on both intracavity and output fields we make the substitutionsã = α + δã for each annihilation operator in Eq. 48, substitute γ(t) =γ + δγ(t), and substitute 2γ in (t) = 2γ in + δγ in (t)/ 2γ in . Using Eq. (57) the mean terms cancel, neglecting second order fluctuating terms we find This can easily be solved in the frequency domain by taking the Fourier transform, which yields where we have substituted in for α from Eq. (57), neglected the term involving the product (g sig + ∆)g self therm since this is small compared withγ 2 , and taken ω γ. For compactness we've omitted the explicit frequency dependence ω of the fluctuations here and henceforth.
The output field fluctuations can then be found using Eq. (61) The output field annihilation operator is given byã out = α out + δã out . As can be seen from Eq. (63), the signal g sig is contained entirely within the imaginary part of α out . It can be shown from Eq. (77) that the noise variance is independent of the measured phase. Consequently, the optimal approach to extracting a signal fromã out is to measure the phase quadraturẽ Thus, an estimate of g sig may be obtained from a measurement ofỸ out as which in the limit that no noise was present in the measurement, would exactly retrieve g sig . It can be seen from this equation that the variance, or uncertainty, of the estimate is given by where we have used the relations δγ in (−ω) † = δγ in (ω) and g self therm (−ω) † = g self therm (ω) which arise since both the input coupling rate and thermal fluctuations are real parameters in the time domain; and in generalX out (ω) =ã † (−ω) +ã(ω), andỸ(ω) = i ã † (−ω) −ã(ω) . Given that each of the fluctuating terms in this expression are uncorrelated, the output phase quadrature variance is The laser input variances can be broken into a quantum part and a classical part, as described by Eqs. (52) and (53); whilst the fields entering through the sensor loss channel are vacuum and therefore quantum noise limited ( δX 2 l = δY 2 l = 1). Using these relations we find δỸ out (ω) 2 = 4η 2 (g sig + ∆ ) 2 Taking the reasonable limit that {g sig , ∆ , ω } 1, and that the presence of the particle to be sensed does not effect the noise (i.e. neglecting all noise terms multiplied by g sig ), we have This expression shows the effect of each of the four sources of noise on variance of the output field. Notice that since ∆ 1, the relative intensity noise V RIN is greatly suppressed compared to the laser phase noise V ζ in this measurement.
The uncertainty in our estimate of g sig is then finally where for compactness we have defined δγ in 2 = V γ and g self therm 2 = V therm . From this expression one sees that, as expected, only the contribution of shot noise towards the uncertainty (the final term) depends on laser power. Of the other terms, it should be expected that laser phase noise dominates, since this is typically large in the frequency range of interest to biosensing experiments and is not suppressed by the measurement technique in the same way that intensity noise is.

BACKSCATTER MEASUREMENT
To perform a similar calculation to that performed above for backscatter measurement, we return to the full Hamiltonian in Eq. (24). The equations of motion for the two intracavity annihilation operators are theṅ = i g sig + g self therm + ∆ â A + i g sig e 2imφ sig + g 0 e 2imφ 0 + g cross therm â C − γ(t)â A + 2γ in (t)â A,in + 2γ l δâ A,l (95) Solving the mean fields Similar to the above, we can find the steady state mean fields by taking the expectation values of Eqs. (95) and (97), setting the time derivative to zero, and solving. From Eq. (95) we find α A γ − i g sig + ∆ = i g sig e 2imφ sig + g 0 e 2imφ 0 α C + 2γ in α A,in Rearranging we find Similarly for α C we have where here, we consider a scenario where only the anti-clockwise mode is directly pumped, and therefore α C,in = 0. It is then straightforward to solve for α C as where we've used the assumption that {g sig , g 0 } γ. We can then find the intracavity field amplitude in mind A Where we have, as usual, assumed that g sig {g 0 ,γ}. Furthermore, since g 0 γ we immediately see that α A α C . One might, therefore think that a higher signal-to-noise could be obtained by detecting α A that α C . However, this is not this case. In fact, when the incident field is on resonance with the cavity resonance any signal arising from the field output from mode A is second order in g sig , whereas that arising from the field output from mode C is first order.
Since α C,out = α C,in − 2γ in α C , and α C,in = 0 we have Solving for the fluctuating noise Taking the fluctuating parts of Eqs. (95) and (97) we have Taking the Fourier transform and rearranging we find Substituting the first of these equations into the second, we can find an expression for δâ C in terms of the input noise sources.
Substituting g cross therm = R cross therm + iI cross therm , we have where we have neglected product terms of g 0 and α C since g 0 γ and α C α A . The output field fluctuations δã C,out can be related to the input through δâ C,out = δã C,in − 2γ in δã C − α C / 2γ in δγ in . Defining δã C,out = Ag self therm + BR cross therm + CI cross therm + Dδγ in + Eδã A,in + Fδã A,l + Gδã C,in + Hδã C,l , we then have where in the expressions for E-H we have moved into dimensionless units, and made the approximation that {ω, g sig } γ. In E we retain the ω dependence since, latter in the calculation the other terms will cancel, leaving it as the dominant term. To simplify A-D we must substitute in for α A and α C .
The output field can now be determined as a C,out = α C,out + δã C,out (141) Let us determine the signal that is received on direct detection of this field, i.e.
neglecting the noise product term δã † C,out (−ω)δã C,out (ω), and, since the signal is small the product terms of signal g sig and noise δã C,out , and the signal product term g sig 2 . The mean measured signal is then Rearranging this expression in terms of g sig we have such that, based on the measurement i an estimate of g sig may be formed as The uncertainty in this estimate is Now we must find |δi| 2 where for conciseness we have combined the phase shift due to detuning with that due to g 0 , making the definitions (1+∆ 2 )/(1+ i∆ ) 2 = e 2imφ ∆ , andφ = φ 0 + φ ∆ . In general the expectation value may be expanded in terms of a series of noise operators δã j each with coefficients A j (in our case these coefficients are A through H, and the noise operators are those associated to each of A through H).
where we have assumed, as is the relevant case here, that all noise sources are uncorrelated. It therefore only remains to calculate the contributions to the variance from each noise source.
where we've used the fact that e 2imφ e −2imφ 0 = e 2imφ ∆ = (1 + ∆ 2 )/(1 + i∆ ) 2 . From this, we see that, in contrast to sensing using a transmitted phase shift, when using backscattered light, the self-thermorefractive noise is greatly suppressed. If the cavity is put exactly on resonance ∆ = 0, then self-thermorefractive noise is exactly eliminated.
where we have taken g 0 1. From this we see that the cross-thermorefractive noise does contribute to the measured noise in the back-scatter detection protocol, so it is not possible to entirely avoid thermo-refractive noise.
where we've used the fact that I cross therm 2 = R cross therm 2 (Eq. (43)). From Eqs. (169) and (167) it can be seen that the total contribution of thermorefractive noise to the measured photocurrent is This expression is identical to the noise term due to input coupling fluctuations in the direct measurement case (see Eq. 91), with the exception that here ∆ = ∆ 0 + g 0 is replaced with simply g 0 . Hence, a disadvantage of the backscatter scheme is that input coupling noise cannot be removed by simply ensuring the light in on resonance with the cavity. However, it should be noticed, that this noise source may be arbitrarily suppressed in principle by sitting close to critical coupling where η = 0.5.
where at Eq. (177) we have made the approximation ω 1, and at Eq. (179) we have made the approximation ∆ 1. Similarly, for the second term in E we find We therefore find Substituting for the amplitude and phase noise of the incident field from Eqs. (52) and (53) we have We see that there is a fundamental noise floor due to shot noise (the 1), whilst laser phase noise is suppressed relative to intensity noise by a factor given by 4ω 2 ∆ 2 .
where to get to Eq. (195) we have used Eq. (187) combined with the fact that F is identical to E except for the substitution η 2 → η(1 − η), and to get to Eq. (196) we have used the fact that the incident field entering the system through the cavity loss channels is in a vacuum state, and therefore δX 2 A,l = δỸ 2 A,l = 1. (1 − 2η) 2 + ∆ 2 1 + ∆ 2 (206) where to arrive at Eq. (202) we have used the fact that the incident field entering the system through the input channel into mode C is in a vacuum state, and therefore δX 2 C,in = δỸ 2 C,in = 1.
e 2imφ Hδã C,l − e −2imφ H * δã C,l † 2 = 1 4 e 2imφ H δX C,l + iδỸ C,l − e −2imφ H * δX C,l − iδỸ C,l 2 (208) where to arrive at Eq. (211) we have used the fact that the incident field entering the system through the cavity loss channels is in a vacuum state, and therefore δX 2 C,l = δỸ 2 C,l = 1. Putting this all together using Eqs. (153) and (155), we finally arrive at an expression for the variance of the measured photocurrent ≈ 4η 2 g 0 2 α 2 A,in 4α 2 A,in 16η 2 g 0 2 ∆ 2 g self therm 2 + 4η 2 R cross therm 2 +4g 0 2 (2η − 1) 2 δγ in 2 + η 2 g 0 2 V RIN (ω) + 4ω 2 ∆ 2 V ζ (ω) + 1 (219) of detuning ∆ in the inset of Fig. 2 of the main paper, with a fit to the detuning dependence of the co-efficient in front of V ζ in Eq. (224) yielding excellent agreement. Finally, the first term in both expressions is the relative intensity noise. Here the situation is reversed, relative intensity noise can be perfectly suppressed in the one mode measurement, whilst it cannot be in the backscatter measurements. Converting to terminology consistent with the main paper, we have δg est sig 2 back−scatter = S (ω), V RIN = S RIN , ω 2 V ζ = S ω , V γ = S γ , V therm = S T /γ, and S shot =γ 2 /16η 2 n in so that where S ω (ω) is the laser frequency noise with it's relation to laser phase noise given, for example, in Ref. [2]. Furthermore, in the experiments reported in the paper the optical coupling is set to critical coupling such that η = 0.5, consequently the contribution of input coupling fluctuations is eliminated. Taking this case, and setting γ =γ we have which is Eq. (1) of the paper.