Far-field linear optical superresolutionvia heterodyne detection in a higher-order local oscillator mode

Fan Yang1, Arina Taschilina1, E. S. Moiseev1, Christoph Simon1, A. I. Lvovsky1,2∗1Department of Physics and Astronomy, University of Calgary, Calgary, Canada, T2N 1N4 and

2Russian Quantum Center, 100 Novaya St., Skolkovo, Moscow region, 143025, Russia

The Rayleigh limit has so far applied to all microscopy techniques that rely on linear opticalinteraction and detection in the far field. Here we demonstrate that detecting the light emittedby an object in higher-order transverse electromagnetic modes (TEMs) can help achieving sub-Rayleigh precision for a variety of microscopy-related tasks. Using optical heterodyne detection inTEM01, we measure the position of coherently and incoherently emitting objects to within 0.0015and 0.012 of the Rayleigh limit, respectively, and determine the distance between two incoherentlyemitting objects positioned within 0.28 of the Rayleigh limit with a precision of 0.019 of the Rayleighlimit. Heterodyne detection in multiple higher-order TEMs enables full imaging with resolutionsignificantly below the Rayleigh limit in a way that is reminiscent of quantum tomography of opticalstates.

Introduction. Since the invention of the optical mi-croscope, there has been a quest for enhancing its resolu-tion. The Rayleigh criterion [1] establishes the minimumresolvable distance in a direct image of a pair of sources tobe limited by diffraction according to dR = 1.22λ/2NA,where λ is the wavelength and NA is the numerical aper-ture of the objective lens. In the past century, a numberof techniques for circumventing the Rayleigh limit haveemerged [2]. These methods rely, for example, on usingshorter-wavelength radiation [3], near-field probing [4],nonclassical [5] or nonlinear [6] optical properties of theobject or switching the object’s emission on and off [7–9].However, these approaches are often expensive and notuniversally applicable. Therefore finding a linear-opticalmicroscopy technique that is operational in the far-fieldregime remains an important outstanding problem.

A promising approach to addressing this problem isby detecting the light emitted by the object in higher-order transverse electromagnetic modes (TEMs). A pointsource emits primarily into the fundamental TEM00

mode. However, when the emitter is displaced from thecenter of the fundamental mode, higher-order TEMs areilluminated. In this way, null measurements of small dis-placements are possible. For example, homodyne detec-tion in TEM01 has been used to detect small displace-ments of a laser beam [10] and to tracking particles witha nanometer resolution [11]. These experiments were per-formed for single emitters only.

More recently, Tsang et al. showed theoreticallythat sub-Rayleigh distances between two identical pointsources can be estimated by measuring the photon countrate in TEM01 [12, 13]. Remarkably, for distances belowthe Rayleigh limit, the per-photon uncertainty of thismeasurement is much less than that associated with di-rect imaging. This feature is particularly valuable whenthe number of photons the object can radiate is limited,such as in the case of photobleaching.

Inspired by these developments, we overcome theRayleigh limit by means of heterodyne detection, tak-

ing advantage of the fact that the heterodyne detectoris only sensitive to the electromagnetic field in the modeof the local oscillator (LO) [14]. We apply the techniquein a variety of settings. First, we determine the posi-tions of single objects emitting coherent and incoherentlight. Second, we determine the distance between twoidentical incoherent objects separated by a distance be-low the Rayleigh limit. In both cases, we demonstrate ameasurement precision significantly below that limit.

In addition, we utilize a mathematical analogy betweendecomposing an image into TEMs and representing thequantum state of a harmonic oscillator in the Fock basisto propose a new microscopy technique. By measuringintensity and phase of the object’s emission into all TEMsone can completely reconstruct its image, in principlewith an arbitrarily high precision. In an experiment, theimaging resolution depends on the number of TEMs inwhich the detection can be realized.Concept. Consider an optical microscope objective

lens that is used to image a plane object with transversespatial field distribution E(x). The field distribution inthe image plane is then given by the convolution

E′(x′) =

+∞∫−∞

E(x)T (x′ − x)dx, (1)

where T (·) is the transfer function of the objective lensand we have assumed the magnification to be unity forconvenience (Supplementary). The transfer function canbe approximated by a Gaussian

T (x) ≈ 1

(2π)1/4√σe−x

2/4σ2

, (2)

with the width σ ≈ 0.21λ/NA [15]. The narrower theaperture in the lens, the wider the transfer function, andthe stronger the distortion of the image.

The heterodyne detector generates a current that isproportional to the overlap between the LO and the sig-

arX

iv:1

606.

0266

2v2

[ph

ysic

s.op

tics]

9 J

un 2

016

2

nal field

J =

+∞∫−∞

E′(x′)ELO(x′)dx′, (3)

where ELO(x′) is the spatial profile of the LO. The LOis prepared in the TEM01 mode such that the corre-sponding fundamental TEM00 mode is matched to theimage of a point source located at x = 0: ELO(x′) =

1(2π)1/4σ3/2x

′e−x′2/4σ2

.

For this conceptual discussion, we assume that thesource is a point located at position xp, so that E(x) =δ(x− xp), in which case we have

J(xp) =

+∞∫−∞

T (x′ − xp)ELO(x′)dx′, (4)

which for a Gaussian transfer function reduces to J(xp) =12σxpe

−x2p/8σ

2

and the corresponding electronic power

P (xp) ∝ J2(xp) =1

4σ2x2pe−x2

p/4σ2

. (5)

The signal vanishes at xp = 0, enabling sensitive nullmeasurement of the source position with respect to thecentral point [10].

Especially useful is the enhancement associated withdetermining the distance d between two incoherent pointsources. Suppose the sources are located at x = ±d/2,where d is the distance between them. The signal inTEM01 is given by P (d/2) + P (−d/2), which is propor-tional to d2 in the leading order. The intensity in thedirect image of this object, on the other hand, is givenby I(x′) = S(x′+d/2)+S(x′−d/2), where S(·) = |T (·)|2is the point spread function of the microscope. Approxi-mating S(x′± d/2) = S(x′)± d

2∂∂xS(x′) +O(d2), we find

I(x′) = 2S(x′) + O(d2). The effect on the image due tothe separation of the slits is also of the second order ind, but with a macroscopic zeroth-order background. Anynoise in this background will have a deleterious effect onthe measurement precision of the term of interest.

Experiment. The experimental setup is shown inFig. 1. Both the signal and LO are obtained from ahome-made external cavity diode laser, producing ∼ 45mW at a wavelength of λ = 780 nm. We prepare the LOin the desired TEM by transmitting it through a temper-ature stabilized monolithic cavity with a finesse of about275 [16]. The laser frequency is locked to the cavity reso-nance by means of the Pound-Drever-Hall technique [17].The signal beam passes through an acousto-optic modu-lator operating at 40 MHz in order to reduce the effectof flicker noise in balanced detection. The modulatedbeam is collimated to an about 5 mm diameter and sentto a diaphragm with four pairs of slits of 0.15 mm widthwhose centers are separated by d = 0.25, 0.50, 0.75 and

AOM

slits

cavity

iris

laser

card

local oscillator

signal

heterodyne detector

objective lens

FIG. 1. Experimental setup. AOM: acousto-optical modula-tor.

1.00 mm (3B Scientific U14101). The power of the beamtransmitted through the slits is ∼ 200 µW.For the mea-surements with incoherent light, we place a white paperlibrary card before the slits. During the data acquisition,the card is moved in the transverse plane by a motorizedtranslation stage to achieve averaging over the incoher-ent light statistics. The incoherent optical power behindthe slits is ∼ 10 µW. After the slits, the light propagatesin free space for L = 84 cm and passes through an irisdiaphragm. The diameter of the diaphragm is measuredusing an optical microscope as 0.8 ± 0.1 mm and inde-pendently estimated with a higher precision from the fitto the data as 0.87 ± 0.01 mm. This latter value corre-sponds to a numerical aperture of NA= 0.52× 10−3 andthe Rayleigh distance of dR = 0.912 mm.

The field transmitted through the diaphragm is refo-cused by an objective lens and subjected to heterodynedetection by means of a balanced detector (ThorLabsPDB150A-SP). To align the detector, we first match thesignal’s mode to the LO prepared in TEM00. Subse-quently, the monolithic cavity temperature is changed totransmit the TEM01 mode and the LO is adjusted slightlyto minimize the interference with the signal. The outputphotocurrent from that detector is observed with a spec-trum analyzer set to a zero span mode at 40 MHz witha resolution bandwidth of 1 kHz and a video bandwidthof 10 kHz. An average of 100 traces is acquired for eachmeasurement.

Results and discussion. We first measure the positionof a single light source. In this case, the beam only passesthrough a single 0.15-mm slit, whose position xp is con-trolled by a translation stage. We conduct the measure-ment for both coherent and incoherent light.

In the coherent case, our setting resembles that of Hsuet al. [10]. We acquire the data for each point in Fig. 2(a)once, except for a set of points around xp = 0 shownin the inset. For these latter points, the signal value isacquired ten times to estimate the stochastic experimen-tal error. Two types of experimental imperfections con-

3

slit position, mm

electronic signal, nW

a)

b)

electronic signal, nW

FIG. 2. The single slit positioning experiment. The depen-dence of the heterodyne detector output signal on the slitposition measured a) with coherent light, b) with incoherentlight is displayed. The theoretical predictions are obtainedtaking into account the finite width of the slit, but are largelysimilar to those of Eq. (5) (Supplementary). The theory isfit to the experimental data by varying the vertical scale anddiaphragm diameter. The insets show the areas around theorigin, approximately corresponding to the red circles in thecorresponding main plots, magnified.

tribute to that error. First, the orthogonality betweenthe LO and signal modes is imperfect and fluctuates dueto air movements and vibration of the optics. As a result,the magnitude of the signal power fluctuation at xp = 0is comparable to the signal itself. Second, the power ofthe LO fluctuates due to the instability of the lock of thelaser to the cavity resonance. These fluctuations trans-late directly into those of the detector’s output, so theerrors at high signals are proportional to the signal.

In view of this analysis, we model the error of ourmeasurement to behave as σ2 = c2 + (gP )2, where c isthe uncertainty due to the mode mismatch fluctuationsand bP is due to the fluctuations of the LO. The actualexperimental behavior of the error is consistent with thismodel.

We use this model to find the uncertainty of estimatingthe slit position xp from the signal power. To this end,we calculate the Fisher information as a function of xp

slit distance, mm

sig

na

l, n

W

a)

b)

FIG. 3. The incoherent double slit experiment. a) Images ofthe slits with the iris diaphragm fully open (top) and closedto a 0.8 mm diameter (bottom) for d = 0.25, 0.50, 0.75, 1.00mm, left to right. The slits with d < 1.00 mm are not resolvedwith the closed diaphragm setting. b) Dependence of thesignal in TEM01 on the slit distance. The error bars show thestatistical errors of 12 measurements each.

and find that this function is maximized at xp = 0.012mm. The value of the Fisher information at this pointcorresponds to the Cramer-Rao uncertainty bound [18]of δxp = 1.4 µm, almost three orders of magnitude belowthe Rayleigh limit (Supplementary).

In the experiment with incoherent light [Fig. 2(b)], thesignal’s spatial structure is a speckle pattern, changing intime as we move the library card. The intensity of thesignal field in the mode being detected is then governedby thermal statistics, so large fluctuations, whose magni-tude is on the scale of the signal, are present even for largesignals. While the effect of these fluctuations is reduceddue to averaging, it is still the dominant source of errorfor high signal powers. For low signal powers, similarly tothe coherent case, the contribution to the uncertainty as-sociated with non-constant mode matching between thesignal and LO becomes significant. The general behaviorof the experimental error is therefore still consistent withσ2 = c2 + (gP )2, albeit with different values of c and gcompared to the coherent case. Experimentally, we eval-uate these coefficients by acquiring the heterodyne signalat each xp three times. We find that the Fisher informa-tion is maximized and the uncertainty of xp is minimizedfor xp = 0.14 mm, corresponding to δxp = 11 µm.

Next, we measure the distance between two incoherent

4

light sources. Each slit pair is centered on the laser beamand the output signal is sampled twelve times to estimatethe error. The resulting data are shown in Fig. 3(b). Theerror analysis for this setting is similar to that for a singleincoherent slit and yields a minimum of δd = 17 µm ford = 0.18 mm.

It is interesting to compare the precision of our mea-surement with what can be achieved by conventionalimaging. The images of pairs of slits acquired with aconventional CCD camera are shown in Fig. 3(a), bot-tom row. For separations significantly below the Rayleighlimit, these images do not resolve the slits and cannot beused to determine the separations. Our technique, on theother hand, permits this determination with the precisioncomparable to the camera pixel size (7 µm).

At a more quantitative level, the above error analysisimplies that the Fisher information decreases to zero, andthe estimation uncertainty tends to infinity, for d → 0.This is a common feature of both our technique and con-ventional imaging, arising because of a nonvanishing un-certainty in evaluating the signal at d = 0 (Supplemen-tary). The fundamental reason for this feature in bothcases is the shot noise. In practice, the limitation in ourexperiment is the fluctuation of the mode matching be-tween the LO and the signal, which can be reduced byusing more stable optics. Furthermore, the shot noiselevel can be reduced by using squeezing [10, 11]. Theconventional imaging technique, on the other hand, is of-ten limited by technical fluctuations of the electronic sig-nal produced by individual pixels in the camera, whichcomplicates precise measurement of minute image widthvariations associated with varying d. Note that the tech-niques of Refs. [12, 13] utilize photon counting in higherorder modes and do not suffer from the fundamental pre-cision limitation associated with the shot noise. Thesemethods are however still vulnerable to practical noisesources such as detector dark counts.

Application to imaging. Performing heterodyne de-tection in higher-order Hermite-Gaussian modes TEM0n

permits reconstruction of the full image of the objectwith sub-Rayleigh resolution. To see this, we write theheterodyne detector output photocurrent (3) as

J0n =

+∞∫−∞

E(x)J(x)dx, (6)

where J(x) is the photocurrent in response to a pointsource at x given by Eq. (4). We assume, as previously,that the transfer function is given by Eq. (2) while theLO is in TEM0n of width σ:

ELO,n(x) =Hn(x/

√2σ)

(2π)1/4√

2nn!σe−x

2/4σ2

, (7)

where Hn(·) are the Hermite polynomials. Integral (4)then corresponds to a Weierstrass transform of that poly-

nomial and is given by a remarkably simple expression

J(x) =1√n!

( x2σ

)ne−x

2/8σ2

. (8)

We see that, for objects of size . σ, photocurrent J0ngives the nth moment of the field in the object plane.

The set of photocurrents acquired for multiple modescan be further utilized to find the decomposition of E(x)into the Hermite-Gaussian basis and thereby reconstructthe full image of the object with a sub-Rayleigh resolu-tion. Let αkn be the coefficients of the Hermite polyno-mial of degree k, so that Hk(x/2σ) =

∑n αkn(x/2σ)n.

Then, according to Eqs. (6) and (8), we have

βk :=∑n

√n!αknJon =

+∞∫−∞

E(x)Hk

( x2σ

)e−x

2/8σ2

.

(9)Because Hermite-Gaussian functions form an orthonor-mal basis in the Hilbert space of one-dimensional func-tions, it follows that E(x) =

∑∞k=0 βkHk(x/2σ)e−x

2/8σ2

.Knowing all βk, we can calculate E(x). This approachis reminiscent of representing a quantum state of a har-monic oscillator as a superposition of Fock states, whose

position

intensity (arb.units)

2NA

lж цґз чи ш

position 2NA

lж цґз чи ш

intensity (arb.units)

number of TEMs

reso

lva

ble

dis

tan

cein

un

its o

f R

ayl

eig

h d

ista

nce

c)

a) b)

FIG. 4. Hermite-Gaussian microscopy. a) An image of twosingle sources positioned at 1.22λ/2NA, which corresponds tothe Rayleigh limit, expected in conventional imaging. b) Animage of the same object expected in HGM with TEM0n for0 ≤ n ≤ 20 exhibits triple enhancement of resolution. c) Res-olution of HGM, in units of the Rayleigh distance 1.22λ/2NA,as a function of the number of TEMs used. The resolution isdefined as the minimum distance between the point objectssuch that the image intensity at the center does not exceed75% of the maximum intensity.

5

wavefunctions in the position basis are given by Hermite-Gaussian functions.

We name the above-described imaging techniqueHermite-Gaussian microscopy, or HGM. Acquiring het-erodyne photocurrents for a sufficiently high number ofTEMs in principle allows HGM to reconstruct the im-age with arbitrarily high resolution. The first few tens ofTEMs, which are attainable in experimental optics, per-mit significant improvement of imaging with respect tothe Rayleigh limit, as evidenced by Fig. 4.

The above conceptual description can be readily ex-tended to practically relevant cases. Two-dimensionalimaging is possible by scanning over both indices ofTEMmn and measuring the photocurrent Jmn for eachpair (m,n) up to a desired maximum. Acquiring boththe sine and cosine quadratures of the heterodyne pho-tocurrent permits phase-sensitive reconstruction of theobject’s field.

A somewhat less trivial extension is to incoherent im-ages. In this case, the output power of the heterodynedetector is given by

〈P 〉 ∝ 1

n!

+∞∫−∞

I(x)( x

2σ

)2ne−x

2/4σ2

(10)

(see Supplementary for the derivation). Similarly to thecoherent case, we obtain moments of the field distribu-tion in the object plane. However, now we obtain onlythe even coefficients of the decomposition of I(x) intothe Hermite-Gaussian basis. Therefore the informationabout only the even component of function I(x) is re-tained. An image reconstructed with these data willbe a sum of the original intensity profile I(x) with the“ghost” image I(−x). For two-dimensional microscopy,three ghost images, I(x,−y), I(−x, y) and I(−x,−y),will be added to the original image I(x, y). Their effectcan be eliminated by placing the entire object into a sin-gle quadrant of the x-y plane.

Summary. We have used heterodyne detection in theTEM01 Hermite-Gaussian mode to obtain sub-Rayleighprecision in the measurement of the position of single co-herent and incoherent sources, as well as that of the sep-aration between two incoherent sources. With numericalapertures below 10−3, our measurement precision is ona scale of a few microns. If our technique is used withstate-of-the art microscopes with NA ∼ 1, precision onnanometer scales can be expected. By utilizing higherHermite-Gaussian modes, the technique can be extendedto full imaging with sub-Rayleigh resolution.

Note added. While working on this manuscript, webecame aware of similar research being pursued by Shengat al. [19] and Tham et al. [20].

Acknowledgments This work was supported byNSERC and CIFAR. We thank P. Barclay, R. Ghobadi,J. Moncreiff, A. Steinberg and M. Tsang for illuminating

discussions, as well as B. Chawla for helping with theexperiment.

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both, S. Stephan, G. Wellenreuther, G. Falkenberg, andC. G. Schroer, Hard x-ray scanning microscopy with co-herent radiation: Beyond the resolution of conventionalx-ray microscopes, Appl. Phys. Lett. 100, 253112 (2012).

[4] U. Durig, D. W. Pohl and F. Rohner, Nearfield opti-calscanning microscopy, J. Appl. Phys. 59, 3318 (1986)

[5] Giovannetti, V., Lloyd, S. & Maccone, L. Advances inquantum metrology, Nature Photonics 5, 222–229 (2011).

[6] S. W. Hell and J. Wichmann, Breaking the diffrac-tion resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy, Opt. Lett.19, 780–782 (1994).

[7] E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lind-wasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J.Lippincott-Schwartz, H. F. Hess, Imaging IntracellularFluorescent Proteins at Nanometer Resolution, Science313, 1642–1645 (2006).

[8] M. J. Rust, M. Bates, X. Zhuang, Sub-diffraction-limitimaging by stochastic optical reconstruction microscopy(STORM), Nature Methods 3, 793–796 (2006)

[9] R. M. Dickson, A. B. Cubitt, R. Y. Tsien and W. E. Mo-erner, On/off blinking and switching behaviour of singlemolecules of green fluorescent protein, Nature 388, 355–358 (1997)

[10] M. T. L. Hsu, V. Delaubert, P. K. Lam and W. P Bowen,Optimal optical measurement of small displacements, J.Opt. B: Quantum Semiclass. Opt. 6, 495–501 (2004).

[11] M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage,H.-A. Bachor & W. P. Bowen, Biological measurementbeyond the quantum limit, Nature Photonics 7, 229–233(2013).

[12] M. Tsang, R. Nair, X.-M. Lu, Quantum theory of su-perresolution for two incoherent optical point sources,arXiv:1511.00552.

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[14] A. I. Lvovsky and M. G. Raymer, Continuous-variableoptical quantum-state tomography Rev. Mod. Phys. 81,299–332 (2009).

[15] B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, Gaussianapproximations of fluorescence microscope point-spreadfunction models, Appl. Opt. 46, 1819–1829 (2007).

[16] P. Palittapongarnpim, A. MacRae and A. I. Lvovsky,Note: A Monolithic Filter Cavity for Experiments inQuantum Optics, Rev. Sci. Instrum. 83, 066101 (2012).

[17] Drever, R. W. P., Hall, J. L., Kowalski, F. V., Hough, J.,Ford, G. M., Munley, A. J. H. Ward (1983), Laser phaseand frequency stabilization using an optical resonator,Appl Phys B 31, 97–105 (1983).

[18] A. van den Bos, Parameter Estimation for Scientists andEngineers (John Wiley & Sons, Hoboken, 2007), pp. 45–

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7

SUPPLEMENTARY INFORMATION

Theoretical prediction for the signal. Here we calcu-late the heterodyne detector signal in response to theelectromagnetic field generated by an object of a specificshape. We used this method to obtain the theoreticalcurves in Figs. 2 and 3 of the main text. The calculationis for one-dimensional objects, but is readily extended totwo dimensions.

We start by briefly reviewing Abbe’s microscope reso-lution theory.

E(k⊥) =

+∞∫−∞

E(x)eik⊥xdx, (S1)

where k⊥ is the orthogonal component of the wavevec-tor and constant normalization factors are neglectedthroughout the calculation. The objective lens is locatedin the far field at distance L from the object plane. Theposition X in the lens plane is then related to k⊥ accord-ing to

X = Lk⊥k

=Lλk⊥

2π. (S2)

The lens, in turn, generates the inverse Fourier image inits focal plane:

E′(x′) =

+∞∫−∞

E(k⊥)T (k⊥)e−ik⊥x′dk⊥ (S3)

=

+∞∫−∞

+∞∫−∞

E(x)T (k⊥)eik⊥(x−x′)dxdk⊥,

where T (k⊥) is the transmissivity of the lens as a functionof the transverse position in its plane. If this transmissiv-ity is a constant, corresponding to an infinitely wide lens,the image is identical to the object: E′(x′) = E(x). If thelens is of finite width, the image is distorted according to

E′(x′) =

+∞∫−∞

E(x)T (x′ − x)dx, (S4)

where T (x′−x) =+∞∫−∞

T (k⊥)eik⊥(x−x′)dk⊥ is the Fourier

image of the lens. In other words, the image is a convo-lution of the object with T (·).

Heterodyne detection of the image field yields the elec-tronic signal given by Eq. (3) in the main text. If the fieldE(x) is coherent, that equation is sufficient to calculatethe output signal.

If the object is incoherent, we must take the average ofthe signal over all realizations of E(x) to find the output

power of the heterodyne detector photocurrent:

〈P 〉 =

⟨ +∞∫−∞

E′(x′)ELO(x′)dx′

2⟩(S5)

=

+∞∫∫−∞

〈E′(x′)E′(x′′)〉ELO(x′)ELO(x′′)dx′dx′′

Now using Eq. (S4) we find

〈E′(x′)E′(x′′)〉 (S6)

=

+∞∫∫−∞

〈E(x1)E(x2)〉T (x′ − x1)T (x′′ − x2)dx1dx2.

For an incoherent image,

〈E(x1)E(x2)〉 = I(x1)δ(x1 − x2) (S7)

and hence

〈P 〉 =

+∞∫∫∫−∞

I(x)T (x′ − x)T (x′′ − x) (S8)

× ELO(x′)ELO(x′′)dxdx′dx′′. (S9)

Our goal is to determine the heterodyne detector sig-nal as a function of the object configuration, E(x) in thecoherent case and I(x) in the incoherent case. Thesecalculations are simplified if we take into account aspecific shape of the transmissivity function associatedwith a round diaphragm: T (k⊥) = θ(k⊥max − |k|) withk⊥max = 2πR/(Lλ) according to Eq. (S2), R = 0.4±0.05mm is the radius of the diaphragm and θ(·) is the Heavi-side step function. In the Fourier domain this translatesinto

T (x′ − x) =J1(k⊥max(x− x′))

x− x′≈ e−(x−x

′)2/4σ2

, (S10)

where J1(·) is a Bessel function. This is further approxi-

mated as T (x′−x) ≈ e−(x−x′)2/4σ2

with σ ≈ 0.21λL/R =0.31± 0.03 mm.

The LO field in the TEM00 mode is optimized to matchthe mode E(x′) in the coherent case for E(x) = δ(x), so

ELO(x′) = e−x′2/4σ2

. Subsequently it is switched to the

TEM01 mode with ELO(x′) = x′e−x′2/4σ2

. Substitutingthis mode into the expression (3) in the main text for thecoherent case, we find

P = J2 =

+∞∫−∞

xE(x)e−x2/8σ2

2

dx. (S11)

For the incoherent case, the mean signal power (S8)equals

〈P 〉 =

+∞∫−∞

x2I(x)e−x2/4σ2

. (S12)

8

While the above results are valid for coherent and in-coherent objects of any shape, a simple analysis leadingto Eq. (5) in the main text is sufficient for the purposes ofour experiment, as evidenced by Fig. S1. The only visibledifference between the two models is that the curve cal-culated for the incoherent case using the complete modeldoes not reach zero at the slit position xp = 0. Thisis because an incoherent slit of a finite width, which canbe seen as a combination of multiple mutually incoherentpoint sources positioned around xp = 0, makes a nonzerocontribution to TEM01.

slit position, mm

signal, arb.unitsa)

slit position, mm

signal, arb.unitsb)

FIG. S1. Comparison of the theoretical predictions for thesignal power taking into account the finite width of the slit(blue thin solid line) and assuming an infinitely narrow slit(yellow thick dashed line). a) Coherent case, b) incoherentcase.

Error analysis. Our experimental setup yields theelectronic signal power P (xp) with root mean square(rms) uncertainty σ(P (xp)) as a function of the slit po-sition xp. Suppose the task is to estimate xp from theobserved power. Below, we present a method for deter-mining and minimizing the error of this estimation.

We can treat the observed power as a random variablewhose probability distribution

f(P, xp) =1√2πσ

e−[P−P (xp)]2/2σ2

(S13)

is Gaussian with rms width σ centered on the mean power

P (xp). The problem then reduces to that of estimat-ing parameter xp from this random variable. The uncer-tainty of this estimation can be found using the Cramer-Rao error bound,δxp ≥

√1/F , where

F =

+∞∫−∞

[∂f(P, xp)

∂xp

]2/f(P, xp)dP

is the Fisher information. In the neighborhood of xp =0, the power P (xp) can be approximated as ax2p, whilethe uncertainty σ2 = c2 + [gP (xp)]

2 as discussed in themain text, with constants a, c and g evaluated from theexperimental data. Accordingly, we find for the Fisherinformation in the limit g � 1

F =4a2θ2

a2g2θ4 + c2.

Next, we determine the value of xp where the Fisherinformation is maximized so the measurement is the mostsensitive. Taking the derivative of F , we find that themaximum value Fmax = 2a/gc is reached at xp =

√c/ag.

It is these values that we use to evaluate the estimationuncertainties of xp in the main text.

Derivation of Eq. (10) in the main text. We useEq. (S7) to write the output power of the heterodynedetector as the average

〈P 〉 ∝⟨J20n

⟩=

+∞∫∫−∞

〈E(x1)E(x2)〉 J(x1)J(x2)dx1dx2

=

+∞∫−∞

I(x)J2(x)dx. (S14)

Substituting Eq. (8) from the main text into the aboveresult, we obtain Eq. (10).