Interferometric position detection of levitated particles is crucial for the centre-of-mass (CM) motion cooling and manipulation of levitated particles. In combination with balanced detection and feedback cooling, this system has provided picometer scale position sensitivity, zeptonewton force detection, and sub-millikelvin CM temperatures. In this article, we develop an analytical model of this detection system and compare its performance with experimental results allowing us to explain the presence of spurious frequencies in the spectra.

In the past, interferometric position detection systems have been used in optomechanics for the detection of zeptonewton scale forces,1–3 the demonstration of sub-kelvin centre-of-mass (CM) temperatures,4–6 the measurement of Brownian motions,7 and the manipulation of levitated particles.8–12 Furthermore, this system has provided pm/Hz position sensitivity.4 In these schemes, a reference beam and the scattered light from a levitated particle interfere on a photodiode. This interference produces a signal which is directly related to the instantaneous position of the oscillator. After Fourier transformation, oscillation frequencies (ωx, ωy, and ωz) along the three axes can be retrieved from the position signals. Subsequently, these frequencies are used for parametric feedback cooling to actively control the motion of a levitated particle.1–3,5,8–14 As with other interferometric schemes, this system is well known for its high precision and resilience to noise. In optomechanical setups, this is further enhanced by a balanced detection system. A balanced detector consists of two matched photodiodes which help us to reduce common mode noise and other unwanted signals. Here, we develop a model of this interferometric scheme and present experimental evidence to justify its validity. We find that the predictions of our model match closely with the experimental results. We also show that due to the configuration of the balanced detector, it detects frequency along the desired axis as well as frequencies from the remaining two axes and the frequencies resulting from the various linear combinations of ωx, ωy, and ωz. Finally, we discuss the possible side effects of these spurious frequencies on the performance of parametric feedback cooling.

Figure 1 shows a schematic of a tweezer-based optomechanical experiment in which a high numerical aperture microscope objective forms the trap by tightly focusing a laser beam into a diffraction limited spot. The trap is normally placed inside a vacuum chamber. Once a desired particle is trapped, the chamber is evacuated and the position of the particle is monitored using the interferometric detection system. Let us assume that r=xx^+yŷ+zz^ is the instantaneous position vector of the levitated particle from the centre of the trap, where x = Ax sin(ωxt + ϕx), y = Ay sin(ωyt + ϕy), and z = Az sin(ωzt + ϕz) are the instantaneous distances of the particle along the x, y, and z axes. The angular trap frequencies are ωx, ωy, and ωz, and Ax, Ay, and Az are the respective amplitudes of oscillations along the three axes. Likewise, ϕx, ϕy, and ϕz are the phases along the three axes. In order to detect and manipulate the position of the levitated particle, balanced photo-detectors are placed along the various axes. As an example, in Fig. 1, we show one detector placed along the x-axis. This enables us to detect the trap frequency along the x-axis. From the geometry of the problem, the position vectors of the two photodiodes (D1 and D2 in Fig. 1) from the centre of the trap are r10=x0x^+y0ŷ+z0z^ and r20=(x0+Δx)x^+y0ŷ+z0z^, where x0, y0, and z0 are the distances of the two photodiodes from the centre of the trap. Δx is the position mismatch between the two photodiodes along the x-axis. This mismatch initiates an imbalance in the detector (see below for details). The distances of a levitated particle from the two photodiodes of the balanced detector can be written as r1=|r+r10|=r02+r22x0x2y0y2z0z and r2=|r+r20|=r02+r2+2x0x2y0y2z0z+Δx(Δx+2x+2x0), where r0=x02+y02+z02.

FIG. 1.

A schematic of our tweezer-based optomechanical system along with the interferometric detection system along the x-axis. The origin of the coordinate system is the centre of the trap. The inset shows the levitated particle along with the detection system in the co-ordinate system. Note that z0 actually signifies the distance between the levitated particle and the lens after it. Since the light is collimated after the lens, the distance between the lens and the diodes, shown for the sake of visualization in the main schematic, is not important. Further, different symbols correspond to L—lens, M—mirror, PBS—polarizing beam splitter, λ/2—halfwave plate, and D—diodes.

FIG. 1.

A schematic of our tweezer-based optomechanical system along with the interferometric detection system along the x-axis. The origin of the coordinate system is the centre of the trap. The inset shows the levitated particle along with the detection system in the co-ordinate system. Note that z0 actually signifies the distance between the levitated particle and the lens after it. Since the light is collimated after the lens, the distance between the lens and the diodes, shown for the sake of visualization in the main schematic, is not important. Further, different symbols correspond to L—lens, M—mirror, PBS—polarizing beam splitter, λ/2—halfwave plate, and D—diodes.

Close modal

Let us also assume that at the focus of the trap the y–polarized electric field can be expressed15,16 as El=E0w0w(z)expx2+y2w(z)2expiωtikzik(x2+y2)2R(z)+iζ(z)ŷ, where k = 2π/λ, ω = 2πc/λ, and λ and c are the trapping laser wavelength and speed in free space, respectively. w(z)=w01+z2/zr2, R(z)=z(1+zr2/z2), ζ = tan−1z/zr, and w0=λzr/π, where zr is the Rayleigh range. E0 can be expressed as 2I0/ϵ0c, where I0 is the intensity of a Gaussian trapping laser beam at the focus and ϵ0 is the dielectric constant of free space. The electric field induces a dipole moment in the trapped particle. This leads to a surface charge density if the polarization is uniform throughout the trapped bead or a volume charge density otherwise.17 

Once a charge is induced inside a particle, it starts to oscillate in the oscillating trapping field, and an oscillating charge radiates/scatters light. The scattered field from a Rayleigh spherical particle (a<<λ) that a photodiode receives can be expressed as18 

Es1Ak24πr13(x02+z02+x2+z22x0x2z0z)×E0exp  iωtkz0  1+x02+y022z02+r22x0x2y0y2z0z2z02      ŷ,
(1)
Es2Ak24πr23(x02+z02+x2+z2+2x0x+2Δxx02z0z)×E0exp  iωtkz0  1+x02+y022z02+r2+2x0x2y0y2z0z2z02+Δxx0z02    ŷ,

where A = 4πa3ϵ0(ϵ − 1)/(ϵ + 2), ϵ, and a are the polarizablity, dielectric constant, and radius of the levitated particle, respectively. We have also assumed that the electric field (E0) remains constant over the distance a levitated particle traverses inside the trap. This is valid when the amplitude of oscillation of a levitated particle is small compared to the beam waist w0 = λ/(πNA), where NA is the numerical aperture of the trapping lens and λ is the trapping laser wavelength.

In addition to the scattered light from the levitated particle, each photodiode also receives directly transmitted laser light from the trapping beam. In the far-field where z0>>zr and (x02+y02)<<z02, the directly transmitted beam unperturbed by the levitated particle can be expressed as15 [see Eqs. (A1) and (A2) of the  Appendix]

ET1zrE0z0exp[iωtikz0(1+x02+y022z02)+π/2]ŷ,ET2zrE0z0exp[iωtikz0(1+x02+y022z02)+π/2]×exp(iΔxx0kz0)ŷ,
(2)

where π/2 is the Gouy phase shift.

Considering the scattered field, and the field due to the directly transmitted light together, the difference in intensity ΔI=ID2ID1 that a balanced detector produces [see Eqs. (A6)–(A9) of the  Appendix for derivations] can be written as

ΔIx0xA2k42π2z06(x02+z02+x2+z22z0z)I0ScatteringAk2πz03(x02+z02+x2+z22z0z)cos(kr22y0y2z0z2z0)sin(x0xkz0)zrz0I0Interference2x0xAk2πz03sin(kr22y0y2z0z2z0)cos(x0xkz0)zrϵ0cE022z0InterferenceΔxx0Ak32πz04(x02+z02+x2+z22z0z+2x0x)cos(k(r22y0y2z0z2z0+x0xz0))zrz0I0Imbalance,
(3)

where we have assumed I0=ϵ0cE02/2, α = Δxx0k/z0, (x0, y0, x, y, z) <<z0, 1/r1 ≈ 1/z0, and 1/r2 ≈ 1/z0. Further, we have assumed that the depolarization of the scattered light is negligible. This is true when the levitated particle is small (a<<λ) compared to the trapping laser’s wavelength. It can be seen that there are three main terms in the signal that a balanced detector produces. These are an interference term consisting of the scattered and unscattered light, a term due to the scattered light alone, and a term owing to the imbalance (α > 0) between the two arms of a balanced detector. In the ideal scenario, where the two arms of a balanced photodetector are perfectly balanced ΔIImb = 0 (α = 0). Further, the contribution of the scattering term in the overall signal is much smaller than the interference term. As a result, below we only analyze the interference term and the term due to the imbalance and show their importance in the context of balanced detection.

Expanding cos(kr22y0y2z0z2z0), sin(kr22y0y2z0z2z0), cos(x0xkz0), and sin(x0xkz0) into their respective Taylor series and keeping only lower order terms, and substituting x = Axsin ωxt, y = Ay sin ωyt, and z = Az sin ωzt, the interference term can be written as [see Eq. (A10) of the  Appendix for details]

ΔIInter=x0zrAk3πz05I0(x02+z02)Axsinωxt2z0AxAz×cos(ωxωz)t+2z0AxAzcos(ωx+ωz)t+ f(ωx,ωy,ωz)I0,
(4)

where we have assumed ϕx = ϕy = ϕz = 0 for simplicity.

From Eq. (4), one can find that even though the balanced detector in the configuration shown in Fig. 1 is meant to detect the oscillation frequency along the x-axis, our model predicts the detection of many other frequencies f(ωx, ωy, ωz) in addition to ωx. To justify the validity of Eq. (4), Fig. 2(b) shows a Fourier transform of the measured time domain signal obtained using a balanced photodiode (PDB210C/M—Large-area balanced photodetector, Thorlabs, Ltd.) from our levitated experiment. In this particular case, a 50 nm silica particle was levitated using a dipole trap and data were collected at 3 mbar. Immediately, one can recognize the desired frequency along the x-axis, ωx/2π. One can also find two shoulders at ωxωz and ωx + ωz as predicted in Eq. (4). These frequencies are much weaker than ωx as understandable from Eq. (4). Elaborately, from our experiment, we have AxAyAz/2 ≈ 100 nm, x0 = y0 = 1 mm, and r0z0 = 10 mm. On substitution of these values in Eq. (4), one finds the ratio of the amplitudes of ωxωz or ωx + ωz, and ωx is 2Az/z0 ≈ 4 × 10−5, which is small and only in qualitative agreement with our experimental data [see Fig. 2(b)]. Mismatch between the ratios of the amplitudes of the experimental data and the theoretical model can be attributed to the different approximations and assumptions we have made in deriving the theoretical model. Other frequencies as appeared in Eq. (4) are about two orders of magnitude weaker than ωxωz or ωx + ωz. This is good for parametric feedback cooling where frequencies other than the desired frequency are problematic. A consequence of the unwanted frequencies is that they impart amplitude modulation to the intensity of the desired signal as can be seen in Fig. 2(a). This has been observed in earlier experiments1,5 as well.

FIG. 2.

A levitated silica nanoparticle (50 nm) at 3 mbars of pressure—(a) time trace as it oscillates inside the trap and (b) power spectral density. Red vertical lines in (b) represent some of the frequencies (except ωz and its harmonics) predicted by Eq. (4). Data were collected at 3 mbars of pressure.

FIG. 2.

A levitated silica nanoparticle (50 nm) at 3 mbars of pressure—(a) time trace as it oscillates inside the trap and (b) power spectral density. Red vertical lines in (b) represent some of the frequencies (except ωz and its harmonics) predicted by Eq. (4). Data were collected at 3 mbars of pressure.

Close modal

The appearance of ωz and its harmonics in Fig. 2 are not expected according to Eq. (4). Nevertheless, it can be explained by analyzing the impact of the imbalance between the two arms of a balanced detector. Specifically, in theory, α = 0 is achievable, but in a realistic laboratory environment, a minor imbalance between the two detectors is unavoidable. The consequence of this unwanted imbalance can be quite significant. For example, the ratio between the dominant imbalance [ωz, see Eq. (A11) of the  Appendix] and interference [ωx, 1st term in Eq. (4)] terms is ≈2Δx/z0. If one considers Δx = 0.01z0, then the ratio of these two terms is ≈0.02. This is equivalent to 2% of the intensity along the x-axis and is non-trivial.

For larger particles, spurious frequencies become even more pronounced as we show in Fig. 3. In this example, data were collected using a 380 nm silica particle and the detector was set to detect the frequency along the x-axis. One can see that the intensities of ωz and its harmonics as well as other frequencies are comparable to the intensity of ωx. From our laboratory experience, this happens with the majority of the larger nanoparticles that we levitate using our dipole trap. A similar phenomenon has also been detected by other groups.4 We believe that for the larger particles it is relatively easy to move outside the linear region of the trap to the non-linear part. This introduces coupling between the different modes of oscillations8,9 and hence the appearances of frequencies other than the desired one. It is also plausible that strong scattering from large particles and the ensuing interference around the trapping region alters the trapping potential profile which introduces coupling between different axes that is otherwise assumed decoupled. We also believe that larger2b3tparticles modify the propagation path of the trapping light due to refraction more strongly than their smaller counterparts. This creates severe dynamic imbalance between the two arms of a detector as the particles oscillate inside the trap and leads to the appearance of unwanted frequencies. Further, as the trapped particle becomes large, the scattered light from the particle gets depolarized.18 As a result, the interference between the scattered and the trapping light diminishes. Further, the assumption that the electric field remains constant over the distance the particle traverses breaks down.

FIG. 3.

Power spectral density (PSD) of a relatively large (380 nm) silica nanoparticle at 0.50 mbar in a dipole trap—(a) shows most of the dominant frequencies visible in the PSD while (b) is the zoomed view of (a).

FIG. 3.

Power spectral density (PSD) of a relatively large (380 nm) silica nanoparticle at 0.50 mbar in a dipole trap—(a) shows most of the dominant frequencies visible in the PSD while (b) is the zoomed view of (a).

Close modal

In the extreme case of imbalance where α>> 0, the balanced detector shown in Fig. 1 turns into an oscillation detector along the z-axis. Specifically, in the balanced detection of frequency along the z-axis, one arm of the balanced detector is fed with a fixed laser light which does not go through the trap, while the other arm of the detector is illuminated with the scattered plus the directly transmitted light that goes through the trap.5 The role of the constant laser power in the first arm is to cancel the dc term that arises in the second photodiode. The overall model is shown in the following equation [see Eq. (A12) of the  Appendix for derivation]:

ΔIzzr(x02+z02)k3A4πz05Ax2sin2ωxt+Ay2sin2ωyt+Az2sin2ωzt2x0Axsinωxt2y0Aysinωyt2z0AzsinωztI0.
(5)

One can see that the detector for the z-axis detects the frequency along the desired axis as well as frequencies along the remaining two axes. Our model of the detector also predicts the detection of harmonics of the fundamental modes albeit very weakly. Experimental data from the z-axis detector in our levitated setup are shown in Fig. 4. In agreement with the model, in our experiment, we detect all three frequencies along the three laboratory axes. Our model also agrees with the experimental power spectral density data presented by Li et al. in Ref. 4 where frequencies along all three axes are visible.

FIG. 4.

Power spectral density from a balanced detector along the z-axis.

FIG. 4.

Power spectral density from a balanced detector along the z-axis.

Close modal

Finally, it is instructive to consider the impact of the unwanted frequencies in parametric feedback cooling—particularly in experiments where large particles are levitated. It is well known that as the particle size increases the separation in frequency among the different oscillation axes diminishes. These waning gaps in frequency require a proportional reduction in the bandwidths (BWs) of the filters used in parametric feedback cooling. At some point, a further reduction of the BW becomes un-viable and filters become ineffective in suppressing unwanted frequencies. Observing this phenomenon in our levitated experiments, we wanted to derive an analytical formulation that can predict the achievable CM temperature under certain frequency noises. However, we find that an analytical model of this situation can only be derived if the spurious frequencies are the harmonics of the fundamental modes (ωx, ωy, and ωz) that one wants to cool. But this is not the case for these experiments. As a result, we are unable to provide an analytical solution of this situation. Nevertheless, we believe that an electrodynamic numerical simulation, which is not considered here, can provide quantitative answers of the impact of the unwanted frequencies in parametric feedback cooling.

We have developed a model which represents the combined interferometric and balanced detection schemes used in levitated optomechanics. According to our model, frequencies such as ωx and its harmonics as well as the sum and differences of ωx with the frequencies of oscillation along the remaining two axes and their harmonics are naturally expected from a balanced detector along the x-axis. However, the appearances of ωy and ωz and their harmonics in the detector along the x-axis can be attributed to the imbalance present in the detection system. An effect of these unwanted frequencies is the reduction of the signal to noise ratio, which might limit the ultimate temperature achievable in parametric feedback cooling. This is particularly true for systems involving large levitated particles. According to our model, designing the experiment to make x0k/z0 smaller will tend to reduce problems with imbalance as α = Δxx0k/z0. In particular, making k smaller by using a longer wavelength laser for trapping seems appropriate. Reducing x0 would also help but will reduce the amount of light that a detector receives.

This work was supported by the United Kingdom EPSRC (Nos. EP/M013243/1 and EP/J014664/1). G.W.M. is supported by the Royal Society. Raw data is available at http://wrap.warwick.ac.uk/98256.

Exploiting the impulse response of free space propagation,15 the directly transmitted light received by the photodiodes can be expressed as

ET1(x0,y0,z0)=iexp[iωtikz0]λz0E0exp[x2+y2w02]exp[iπ(x0x)2+(y0y)2λz0]dxdy=iE0exp[iωtikz0]λz0exp[iπx02+y02λz0]exp[(1w02+iπλz0)x2+i2πx0λz0x]×exp[(1w02+iπλz0)y2+i2πy0λz0y]dxdy=iE0exp[iωtikz0]λz0exp[iπx02+y02λz0]πλz0w02λz0+iπw02exp[πw02(x02+y02)λz0(λz0+iπw02)]=iπw02λz0+iπw02E0exp[iωtikz0]exp[iπx02+y02λz0]exp[πw02(x02+y02)λ2z02]=E01iz0zrexp[iωtikz0(1+x02+y022z02)]exp[πw02(x02+y02)λ2z02]E01iz0zrexp[iωtikz0(1+x02+y022z02)]zrE0z0exp[iωtikz0(1+x02+y022z02)+arctan(z0zr)]zrE0z0exp[iωtikz0(1+x02+y022z02)+π/2],RET1=zrz0E0sin(ωtkz0(1+x02+y022z02))ŷ,
(A1)

where we have used exp[ax2+ibx]dx=π/aexp[b2/(4a)], w02λr0<<1, and exp[πw02(x02+y02)λ2r02]1. Similarly, ET2 can be expressed as

ET2(x0Δx,y0,z0)zrE0z0exp[iωtikz0(1+x02+y022z02)+π/2]exp(iΔxx0kz0).
(A2)

The scattered field received by the two photodiodes can be expressed as18 

Es1=Ak2E04πr1exp{i(ωtkr1)}[(x0x)(y0y)r12x^(x0x)2+(z0z)2r12ŷ+(z0z)(y0y)r12z^]Ak2[(x0x)2+(z0z)2]4πr13E0exp{i(ωtkr1)}ŷ=[A(x02+z02)k24πr13+A(x2+z22x0x2z0z)k24πr13]E0exp{i(ωtkz0(1+x02+y022z02+r22z02x0x+y0y+z0zz02))}ŷ,R{Es1}=[A(x02+z02)k24πr13+A(x2+z22x0x2z0z)k24πr13]E0cos{(ωtkr1)}ŷ[A(x02+z02)k24πr03A(x0x+z0z)k22πr03]E0cos{(ωtkr1)}ŷ[A(x02+z02)k24πr03z0zAk22πr03x0xAk22πr03]E0cos{(ωtkz0(1+x02+y022z02+r22z02x0x+y0y+z0zz02))}ŷ
(A3)

and

Es2=Ak2E04πr2exp{i(ωtkr2)}[(x0+Δx+x)(y0y)r22x^(x0+Δx+x)2+(z0z)2r22ŷ+(z0z)(y0y)r22z^]Ak2E04πr2exp{i(ωtkr2)}[(x0+Δx+x)2+(z0z)2r22]ŷ=[A(x02+z02)k24πr23+A(x2+z2+2x0x2z0z)k24πr23+A(Δx2+2x0Δx+2xΔx)k24πr23]E0exp{i(ωtkr2)}ŷ[A(x02+z02)k24πr23+A(x2+z2+2x0x2z0z)k24πr23+Δxx0Ak22πr03]E0exp{i(ωtkr2)}ŷ=[A(x02+z02)k24πr23+A(x2+z22z0z)k24πr23+x0xAk22πr23+Δxx0Ak22πr23]E0×exp{i(ωtkz0(1+x02+y022z02+r22z02+x0xy0yz0zz02+Δxx0z02))}ŷ,R{Es2}=[A(x02+z02)k24πr03z0zAk22πr23+x0xAk22πr23+Δxx0Ak22πr23]E0cos{(ωtkz0(1+x02+y022z02+r22z02+x0xy0yz0zz02+Δxx0z02))}ŷ,
(A4)

where r1z0+x02+y022z0+r22z0x0x+y0y+z0zz0 and r2z0+x02+y022z0+r22z0+x0xy0yz0zz0+Δxx0z0.

Considering the scattered field, and the field due to the directly transmitted light (ET1 or ET2) together, the overall field at the two photodiodes can be written as

ED1zrE0z0exp[iωtikz0(1+x02+y022z02)+π/2][A(x02+z02)k24πr23+A(x2+z22z0z)k24πr23x0xAk22πr23]×E0exp{i(ωtkz0(1+x02+y022z02+r22z02x0x+y0y+z0zz02))}ŷ,ED2zrE0z0exp[iωtikz0(1+x02+y022z02)+π/2]exp(iΔxx0kz0)[A(x02+z02)k24πr23+A(x2+z22z0z)k24πr23+x0xAk22πr23+Δxx0Ak22πr23]×E0exp{i(ωtkz0(1+x02+y022z02+r22z02+x0xy0yz0zz02+Δxx0z02))}ŷ,
(A5)

where we have assumed x0, y0, x, y, z<<z0. Further, we have assumed that the depolarization of the scattered light is negligible. This is true when the levitated particle is small (a<<λ) compared to the trapping laser’s wavelength. The respective intensities can be expressed as

ID1=ϵ0cED1ED1*2=ϵ0czr2E022z02+[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]2ϵ0cE0222zrϵ0cE022z0[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]sin(k(r22z0x0x+y0y+z0zz0))ϵ0czr2E022z02+[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]2ϵ0cE0222zrϵ0cE022z0[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]sin(k(r22y0y2z0z2z0x0xz0))=ϵ0czr2E022z02+[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]2ϵ0cE0222[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03]sin(k(r22y0y2z0z2z0x0xz0))zrϵ0cE022z0+2x0xAk22πz03sin(k(r22y0y2z0z2z0x0xz0))zrϵ0cE022z0
(A6)

and

ID2=ϵ0cED2ED2*2=ϵ0czr2E022z02+(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)2ϵ0cE0222(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)zrϵ0cE022z0sin(k(r22z0+x0xy0yz0zz0+Δxx0z0))=ϵ0czr2E022z02+(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)2ϵ0cE0222(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)zrϵ0cE022z0sin(k(r22y0y2z0z2z0+x0xz0+Δxx0z0))=ϵ0czr2E022z02+(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)2ϵ0cE0222(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)zrϵ0cE022z0sin(k(r22y0y2z0z2z0+x0xz0))cos(Δxx0kz0)+cos(k(r22y0y2z0z2z0+x0xz0))sin(Δxx0kz0)ϵ0czr2E022z02+(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)2ϵ0cE0222(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)zrϵ0cE022z0sin(k(r22y0y2z0z2z0+x0xz0))+Δxx0kz0cos(k(r22y0y2z0z2z0+x0xz0))=ϵ0czr2E022z02+(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)2ϵ0cE0222(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)sin(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z02Δxx0kz0(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)cos(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z0=ϵ0czr2E022z02+(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)2ϵ0cE0222(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03)sin(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z02Δxx0kz0(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)cos(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z0=ϵ0czr2E022z02+(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)2ϵ0cE0222(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03)sin(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z02x0xAk22πz03sin(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z02Δxx0kz0(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)cos(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z0.
(A7)
ΔI=ID2ID1=ϵ0cED2ED2*2ϵ0cED1ED1*2.
(A8)

The difference between ID2 and ID1 is

ΔI=[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03]2ϵ0cE022[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]2ϵ0cE0222[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03]sin(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z0+2[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03]sin(k(r22y0y2z0z2z0x0xz0))zrϵ0cE022z02x0xAk22πz03sin(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z02x0xAk22πz03sin(k(r22y0y2z0z2z0x0xz0))zrϵ0cE022z02Δxx0kz0(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)cos(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z0=[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03]2ϵ0cE022[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]2ϵ0cE0224(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03)cos(kr22y0y2z0z2z0)sin(x0xkz0)zrϵ0cE022z04x0xAk22πz03sin(kr22y0y2z0z2z0)cos(x0xkz0)zrϵ0cE022z02Δxx0kz0(A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03+x0xAk22πz03+Δxx0Ak22πz03)cos(k(r22y0y2z0z2z0+x0xz0))zrϵ0cE022z0x0xA2k42π2z06(x02+z02+x2+z22z0z)I0ScatteringAk2πz03(x02+z02+x2+z22z0z)cos(kr22y0y2z0z2z0)sin(x0xkz0)zrz0I0Interference4x0xAk22πz03sin(kr22y0y2z0z2z0)cos(x0xkz0)zrϵ0cE022z0InterferenceΔxx0Ak32πz04(x02+z02+x2+z22z0z+2x0x)cos(k(r22y0y2z0z2z0+x0xz0))zrz0I0ImbalanceΔxx0A2k44π2z06(x02+z02+x2+z22z0z)I0Imbalance=x0xA2k42π2z06(x02+z02+x2+z22z0z)I0ScatteringzrAk2πz04(x02+z02+x2+z22z0z)(1k2(r22y0y2z0z)28z02)x0xkz0I0Interference4x0xAk22πz03(kr22y0y2z0z2z0k3(r22y0y2z0z)348z03)(1x02x2k22z02)zrϵ0cE022z0InterferenceΔxx0Ak32πz04(x02+z02+x2+z22z0z+2x0x)cos(k(r22y0y2z0z2z0+x0xz0))zrz0I0Imbalance+Δxx0A2k44π2z06(x02+z02+x2+z22z0z)I0Imbalancex0xA2k42π2z06(x02+z02+x2+z22z0z)I0Scatteringx0zrAk3πz05x(x02+z02+x2+z22z0z)(1k2(r22y0y2z0z)28z02)I0Interference4x0xAk22πz03(kr22y0y2z0z2z0k3(r22y0y2z0z)348z03)(1x02x2k22z02)zrϵ0cE022z0InterferenceΔxx0zrAk32πz05(x02+z02+x2+z22z0z+2x0x)cos(k(r22y0y2z0z2z0+x0xz0))I0Imbalance,
(A9)

where I0=ϵ0cE02/2. In order to compare the validity of Eq. (A9), Fig. 5 shows a comparison between Eqs. (A8) and (A9) where various approximations have been made. It can be seen that they match quite well.

FIG. 5.

Comparison between the exact difference signal Eq. (A8) (blue solid line) and approximate ΔI [Eq. (A9), red broken line] assuming α = 0, I0 = 2.5 × 1011 W/m2, x0 = y0 = 1 mm, z0 = 10 mm, Ax = Ay = 100 nm, Az = 200 nm, ωx = 143 kHz, ωy = 130 kHz, ωz = 33 kHz, and ϕx = ϕy = ϕz = 0.

FIG. 5.

Comparison between the exact difference signal Eq. (A8) (blue solid line) and approximate ΔI [Eq. (A9), red broken line] assuming α = 0, I0 = 2.5 × 1011 W/m2, x0 = y0 = 1 mm, z0 = 10 mm, Ax = Ay = 100 nm, Az = 200 nm, ωx = 143 kHz, ωy = 130 kHz, ωz = 33 kHz, and ϕx = ϕy = ϕz = 0.

Close modal

From Eq. (A9), the interference term is

ΔIInterx0zrAk3πz05x(x02+z02+x2+z22z0z)(1k2(r22y0y2z0z)28z02)I02x0zrxAk2πz04(kr22y0y2z0z2z0k3(r22y0y2z0z)348z03)(1x02x2k22z02)I0x0zrAk3πz05x[x02+z02+x2+z22z0zk2(x02+z02+x2+z22z0z)(r22y0y2z0z)28z02]I02x0zrxAk2πz04kr22y0y2z0z2z0k3(r22y0y2z0z)348z03I0x0zrAk3πz05x[x02+z02+x2+z22z0zk2(x02+z02)(r22y0y2z0z)28z02]I0x0zrAk3πz05x(r22y0y2z0z)I0=x0zrAk3πz05[(x02+z02)x+x3+xz22z0xz+x(r22y0y2z0z)+xk2(x02+z02)(r22y0y2z0z)28z02]I0=x0zrAk3πz05[(x02+z02)x+x3+xz22z0xz+x3+xy2+xz22y0xy2z0xzxk2(x02+z02)(r22y0y2z0z)28z02]I0=x0zrAk3πz05[(x02+z02)x4z0xz2y0xy+2x3+xy2+2xz2xk2(x02+z02)(r22y0y2z0z)28z02]I0=x0zrAk3πz05(x02+z02)Axsinωxt4z0AxAzsinωxtsinωzt2y0AxAysinωxtsinωyt+2x3+xy2+2xz2  xk2(x02+z02)(r22y0y2z0z)28z02I0=x0zrAk3πz05(x02+z02)Axsinωxt4z0AxAzsinωxtsinωzt2y0xy+2x3+xy2+2xz2  xk2(x02+z02)(r22y0y2z0z)28z02I0=x0zrAk3πz05(x02+z02)Axsinωxt2z0AxAzcos(ωxωz)t+2z0AxAzcos(ωx+ωz)t2y0xy+2x3+xy2+2xz2xk2(x02+z02)(r22y0y2z0z)28z02I0=x0zrAk3πz05[(x02+z02)Axsinωxt2z0AxAzcos(ωxωz)t+2z0AxAzcos(ωx+ωz)t+f(ωx,ωy,ωz)]I0.
(A10)

Likewise the term due to the imbalance can be written as

ΔIImb=αzrAk22πz04(x02+z02+x2+z22z0z+2x0x)cos(k(r22y0y2z0z2z0+x0xz0))I0+αA2k34π2z05(x02+z02+x2+z22z0z)I0αzrAk22πz04(x02+z02+x2+z22z0z+2x0x)(1k2(r22y0y2z0z+2x0x)28z02)I0=αzrAk22πz04[x02+z02+x2+z22z0z+2x0xk2(x02+z02+x2+z22z0z+2x0x)(r22y0y2z0z+2x0x)28z02]I0=αzrAk22πz04[x02+z022z0z+2x0x+x2+z2k2(x02+z02+x2+z22z0z+2x0x)(r22y0y2z0z+2x0x)28z02]I0.
(A11)

Finally, the signal that a balanced detector along the z-axis, excluding the DC component (see main text for details), produces can be expressed as

ΔIz=ϵ0cED1ED1*2=ϵ0czr2E022z02+[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]2ϵ0cE022+2zrϵ0cE022z0[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]sin(k(r22z0x0x+y0y+z0zz0))ϵ0czr2E022z02+[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]2ϵ0cE022+2zrϵ0cE022z0[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]sin(k(r22y0y2z0z2z0x0xz0))=ϵ0czr2E022z02+[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03x0xAk22πz03]2ϵ0cE022+2[A(x02+z02)k24πz03+A(x2+z22z0z)k24πz03]sin(k(r22y0y2z0z2z0x0xz0))zrϵ0cE022z02x0xAk22πz03sin(k(r22y0y2z0z2z0x0xz0))zrϵ0cE022z0[A(x02+z02)k22πz03+A(x2+z22z0z)k22πz03]sin(k(r22y0y2z0z2x0x)2z0)zrz0I0[(x02+z02)(r22y0y2z0z2x0x)k3A4πz04+(x2+z22z0z)(r22y0y2z0z2x0x)k3A4πz04]zrz0I0(x02+z02)(r22y0y2z0z2x0x)zrk3A4πz05I0=zr(x02+z02)k3A4πz05(Ax2sin2ωxt+Ay2sin2ωyt+Az2sin2ωzt2x0Axsinωxt2y0Aysinωyt2z0Azsinωzt)I0.
(A12)
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