Quantum Reconstruction of the Mutual Coherence Function

Z. Hradil,1 J. Rehacek,1 and L. L. Sanchez-Soto2

1Department of Optics, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic2Departamento de Optica, Facultad de Fısica, Universidad Complutense, 28040 Madrid, Spain

(Received 11 January 2010; published 29 June 2010)

Light is a major carrier of information about the world around us, from the microcosmos to the

macrocosmos. The present methods of detection are sensitive both to robust features, such as intensity, or

polarization, and to more subtle effects, such as correlations. Here we show how wave front detection,

which allows for registering the direction of the incoming wave flux at a given position, can be used to

reconstruct the mutual coherence function when combined with some techniques previously developed for

quantum information processing.

DOI: 10.1103/PhysRevLett.105.010401 PACS numbers: 03.65.Wj, 03.65.Ta, 42.50.Tx

Introduction.—Three-dimensional objects emit charac-teristic wave fronts that are determined by different objectfeatures, such as shape, refractive index, density, or tem-perature. Wave front sensing thus constitutes an invaluabletool to acquire information about our surroundings, espe-cially because ordinary detection devices are not phasesensitive and only capture two-dimensional intensity data.

The measurement of the wave front phase distribution isa key issue for many applications, such as noncontactmetrology, adaptive optics, high-power laser systems, andophthalmology [1,2]. Since the frequency of light waves isso high, no detectors of sufficient time resolution exist andindirect methods for phase measurements have been de-veloped [3,4], each one with their own pros and cons. Inshort, practical and robust wave front sensing is still anunresolved and demanding problem.

The purpose of this Letter is to point out that thesestandard methods may be underrated and do not fullyexploit the potential of registered data. By recasting theirfunctioning principle in a quantum language, one canimmediately foresee that, by using the methods of tomo-graphic reconstruction [5], these devices allow in fact for afull evaluation of the mutual coherence function of thesignal, which conveys full information. This should becompared with the partial phase information retrieved bythe standard wave front reconstruction techniques, wherefull coherence of the detected signal is assumed and im-perfect correlations are ignored. Going beyond such astandard interpretation may constitute a substantial stepahead as it is indeed an interdisciplinary task involvingwave and statistical optics, as well as protocols of quantumstate reconstruction.

Classical theory of wave front measurements.—To bespecific, we sketch the principle of a Hartmann-Shacksensor [6], which is general enough for our purposes hereand is schematized in Fig. 1. To keep the discussion assimple as possible, we restrict ourselves to a one-dimensional model and denote by x the position in thescanning aperture. The incoming light field is divided intoa number of subapertures by a microlens array that creates

focal spots. The deviation of the spot pattern from areference measurement allows the local direction anglesto be derived, which in turn allows the reconstruction of thewave front. In addition, the light intensity distribution atthe detector plane can be obtained by integration andinterpolation between the foci.These devices provide a simultaneous detection of po-

sition x and angular spectrum p of the incident radiation,which is determined by the position of the detected signalon the screen. As we shall see below, p determines directlythe transverse impulse px of incident radiation. Let � bethe complex amplitude of the signal emitted by the source.The propagation of this signal is described by the convo-lution with the optical transfer function h [7], so that at theplane of the apertures it can be expressed as

�apðxÞ ¼Z

dx0�ðx0Þhðx0 � xÞ: (1)

The amplitude in the focal plane due to the jth microlens ofextension Aj thus reads

x p

object

scanningaperture

focalplane

FIG. 1 (color online). Simplified scheme of a general scanningsetup. Two of the microlenses of the periodic array are shown.By moving the aperture, the field is sampled at different posi-tions, whereas by moving the detector in the focal plane the fieldis sampled at different angles.

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�jðpÞ ¼ZAj

dx�apðxÞAjðxÞ exp�ik

fxp

�; (2)

where k ¼ 2�=� is the wave number, f the focal length,and the integration is performed in the transverse plane.The function Aj is the aperture (or pupil) function of the jth

microlens. Since only intensities can be detected, the cor-responding signal at the focal plane is

Sj;p � hj�jðpÞj2i¼

ZAj

dxdx0dx00dx000Qðx; x0Þhðx� x00Þh�ðx0 � x000Þ

� �j;pðx00Þ��j;pðx000Þ: (3)

Here h�i denotes the averaging performed by the detectionand Qðx; x0Þ ¼ h�ðxÞ��ðx0Þi is the mutual coherence func-tion, which fully characterizes the source properties (up tosecond order). The combined factor �j;pðxÞ ¼ AjðxÞ�expðikxp=fÞ condenses the lens focusing. Without lossof generality we can assume a smooth Gaussian ap-proximation of this aperture profile, so that AjðxÞ ¼exp½� 1

4 ðx� xjÞ2=ð�xÞ2�, where xj is the position of the

center of the jth microlens and (�x) is the Gaussian width,which can be considered identical for them all.

To show in a very simple manner the operating principleof these devices, let �ðxÞ represent the wave front underconsideration. The mutual coherence function then readsQðx; x0Þ ¼ expfik½�ðx0Þ ��ðxÞ�g. If �ðxÞ changes slowlyat each microlens, it can be approximated by a linearexpansion near the central point, so that �ðxÞ ’ �ðxjÞ þðx� xjÞ�0ðxjÞ . . . , where �0ðxjÞ is the derivative evaluatedat xj. Keeping the Gaussian approximation, the signal due

to each microlens can written, after a simple integration, as

Sj;p ’ expf�2k2ð�xÞ2½p=f��0ðxjÞ�2g: (4)

The limit of geometrical optics can be characterized by thecondition k2ð�xÞ2 ! 1. By employing the identity

lim�!1ffiffiffiffiffiffiffiffiffiffi�=�

pexpð��t2Þ ¼ �ðtÞ, we immediately get

Sj;p ’ �ðp=f��0ðxjÞÞ; (5)

whose interpretation is quite convincing: the detection canbe viewed as a local tilting of the wave front. This alsoshows that the displacement of the focal spot can be used tofind the gradient of the surface �ðxÞ, whence we cancompute the wave front from a direct integration (or otheradapted numerical schemes).

However, this is an approximation: the scheme alsodepends on the coherence of the signal. Loosely speaking,coherence expresses a mutual relation between two ormore sources. Consider, for example, the imaging of twodistant pointlike sources described, in the far field, by theplane waves expðik1 � xÞ and expðik2 � xÞ (see Fig. 2). Twolimiting cases can be distinguished: for an incoherentsuperposition, the detector will record two spots indicatingthe two respective directions, while for coherent signals,

the two plane waves will superpose resulting in a singlewave with a modulated amplitude 2 cos½ðk1 � k2Þ �x=2� exp½iðk1 þ k2Þ � x=2�. In general, coherences in thesignal will have a significant effect on the detected inten-sity profile. For example, when the two sources are sepa-rated by the Rayleigh distance the detected image consistsof a single blurred point. In this particular case, twocoherent point sources are not resolved by a single aper-ture, while incoherent sources at the same positions are justresolved. In practice, the situation is even trickier: anyrealistic signal is always generated by a superposition ofmany plane waves, which may only be partially coherent.Though the role of the coherence in image processing is

clear, standard optical arguments always assume coherentor incoherent signals [7]. Partial coherence is mostly as-sumed in the context of interferometry and manifests by areduction of the visibility. In fact, the celebratedvan Cittert–Zernike theorem states the way in which co-herence changes during the propagation.Coherence is the main obstacle for the geometrical

interpretation of the signal in a Shack-Hartmann sensor.Indeed, when registering the position and direction, there isstill the uncertainty of whether a single ray or severalinterfering rays were registered. This problem is solvableonly at the level of the mutual coherence function. In viewof these arguments, the role of coherence seems to beunderestimated in current data processing and it mightprovide new schemes. This is why we wish to bring intoprominence the idea that the performance of optical scan-ning devices can be pushed up to the ultimate limits byresorting to tomographic reconstruction methods.Quantum formulation.—We now make the important

observation that the central equation (3) can be recast as

Sj;p ¼ h�j;pjUyQUj�j;pi; (6)

where we have introduced an obvious Dirac notation. Theunitary transformation U accounts for the propagator. Forexample, employing the x representation, we have thathðx� x0Þ ¼ hxjUjx0i, which gives a direct meaning to U.In addition, Qðx; x0Þ ¼ hxjQjx0i plays the role of a densitymatrix and �j;pðxÞ ¼ hxj�j;pi. Equation (6) may appear as

incoherentsignal incoherent

Coherentsignal Coherent

image

lens image

FIG. 2 (color online). Coherent (orange) and incoherent (redand green) images of two point sources separated by theRayleigh distance. Notice how the recorded image varies withrespect to the coherence properties of the signal.

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a simple reformulation of the problem; however, it is thebasis for all our results: it can be seen as the opticalinterpretation of the quantum projection postulate. Thescanning apparatus thus provides the same kind of infor-mation about the mutual coherence function as an ordinarymeasurement does about the state of a quantum system.

Note that special cases of (6) are the position intensityscan IðxÞ ¼ hxjQjxi and the spectral (angular) intensityscan IðpÞ ¼ hpjQjpi, where jpi are the momentum (angu-lar) states obtained from jxi by Fourier transformation.Therefore, the intensity IðxÞ is a projection onto a preciselydefined position, while the spectral intensity is a projectionalong the direction of the incoming wave. These intensitiesare measured by all optical devices with imaging capabil-ities, such as telescopes and photographic cameras.

To go beyond the standard image processing, we pro-pose to reinterpret the operation of these devices as ageneralized measurement of noncommuting variables.Indeed, position and momentum in the transverse planecannot be measured simultaneously with arbitrary preci-sion. In a quantum language, this represents a projectioninto a minimum uncertainty (squeezed) state, determinedby the effective width of the �x detection delimited by themicrolens aperture. In terms of optics, the conjugate vari-able can be attributed to the impulse px ’ kp=f and fulfillsthe Heisenberg uncertainty principle �x�px � @=2.

Simultaneous detection of noncommuting variables(such as position x and momentum px) is well known inquantum optics. This issue was addressed by Arthurs andKelly [8] and can be properly formulated as a quantumestimation problem [9,10]: one looks for the registeredclassical variables, let us call them X and Px, that bringinformation about original quantum observables x and px.These new variables obey an uncertainty relation in whichthe lower bound is twice as large as the original Heisenberguncertainty �X�Px � @.

The simultaneous measurement of x and px can also bemodeled as the measurement of the non-Hermitian opera-tor A ¼ xþ ipx. Denoting the eigenstates of this operatoras j�i, � being the complex number with the real partcorresponding to the observable x and the imaginary partcorresponding to px, the probability distribution for theirsimultaneous detection is the projectionPHð�Þ ¼ h�j%j�i,also known as the Husimi function [11] of the quantumstate %. Since PH is fully equivalent to %, the projectionsj�i are informationally complete. In practice, however, PH

can be sampled only by a finite number of detections �,and the tools of quantum state estimation [12–17] areespecially germane to reconstruct % from experimentaldata.

The strategy we suggest is straightforward.(i) The signal is measured with a wave front sensor

consisting of a position sensitive detector placed in thefocal plane of an array of microlenses. Alternatively, asingle lens can be moved with respect to the signal.

(ii) To each pixel (of angular coordinate p) in the focalplane of the jth subaperture we associate a projection j�j;pi

performed on the signal beam with the expectation valueSj;p. As we have shown above, in this way the signal

reconstruction is converted to a quantum state reconstruc-tion problem.(iii) The mutual coherence function Q of the signal is

reconstructed by inverting the relation (6) with the con-straint Q � 0. For example, the maximum likely mutualcoherence matrix is obtained by iteratively solving theoperator equation RQ ¼ GQ [18], where

R ¼ Xj;p

sj;pSj;p

j�j;pih�j;pj; G ¼P

j;p sj;pPj;p Sj;p

Xj;p

j�j;pih�j;pj

(7)

are positive semidefinite matrices and sj;p are measured

(noisy) data.Once Q is known, any information of interest, such as,

e.g., position or angular intensity scans in any transverseplane, coherence properties, etc., can be obtained from Qby postprocessing. Note also that the quantum aspects ofthe problem are manifested by the positive semidefinite-ness constraint Q � 0.Discussion.—The scanning devices of the type discussed

above are not rare in nature. For example, the compoundeyes of insects [19] consist of thousands of identical units(ommatidia) and bear some similarity with a Hartmann-Shack sensor (see Fig. 3). As we have just argued, such adetector can, in principle, be used for reconstructing themutual coherence function of the observed signal. Thismay have intriguing consequences for biology. For ex-ample, it is a commonly accepted hypothesis that fliesare shortsighted because the compound eye lacks the abil-ity to accommodate and therefore to focus at differentdistances. Things may not be that simple: since the mutualcoherence contains full information, focusing (or any otheroptical transformation) can be done by postprocessing theregistered data, and sharp images can be obtained withoutany need for optical focusing.This can be clearly seen in Fig. 4, which presents an

object consisting of a pair of spatially separated brightspots. For a particular choice of the parameters, a singleblurred spot is registered by a detector placed at the imageplane of a single lens. A Hartmann-Shack device with just

FIG. 3. Schematic drawing of a cut through an ommatidium ofa fruit fly. The process of coupling the signal into the light-sensitive cell and the observed position, where the light isabsorbed, provide, in principle, information about the angularspectrum of the incident light. In this respect each ommatidiumresembles a subaperture of a Hartmann-Shack sensor.

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10 subapertures fitted with 10-pixel detectors providesenough information to get a faithful reconstruction of theoriginal object. We leave as an open question whether ornot nature actually takes advantage of the physics behindthese scanning devices.

As the second example, where the coherence could bechallenging, are recent experimental results for tempera-ture deviations of the cosmic microwave background. Theanisotropy is mapped as spots on the sphere [20], repre-senting the distribution of directions of the incoming ra-diation. To get access to the position distribution, thedetector has to be moved and, in principle, such a scanningsensor then brings information about the position anddirection simultaneously. When the aperture is moving, itscans the field repeatedly at different positions, denotedhere by the index j. Consequently, the registered signal

expressed in a fixed reference frame reads Sj;p ¼h�pjUy

j QUjj�pi.The ideal blackbody radiation in thermal equilibrium

should be homogeneous, isotropic, and incoherent, as aconsequence of Bose-Einstein statistics. However, recentmissions of the projects COBE and WMAP have demon-strated convincingly that there is an anisotropy in thedistribution of cosmic microwave background radiation.These deviations encode a wealth of information on theproperties of the Universe in its infancy. The objective ofthe Planck mission is to measure these properties withunprecedented accuracy and level of detail [21]. This couldalso be an excellent chance to investigate the subtle coher-ence properties of the relict radiation. To our best knowl-edge, this question has not yet been posed. Though thecoherence cannot be measured directly by standard inter-ferometric techniques, it could be inferred indirectly fromthe image patterns generated by partially coherent sources.The methods of quantum tomography are especially ade-quate for this task.

In summary, coherence can be used as an importantsource of information, provided that the mutual coherenceis properly sampled by the detection. This possibility is notexploited in the standard protocols of image processing.The proposed statistical inversion, according to the receiptsof quantum tomography, is the main result of this Letter.

Notice that the acquired mutual coherence function has theextraordinary property of recording all views of an object,and the lack of resolution or accommodation could be morethan compensated by the remarkable possibility of post-processing this complete information once it is accessible.This postprocessing is compatible with the laws of nature,and can surely be a source of inspiration for the technology.We acknowledge Hubert de Guise for a careful reading

of this manuscript. This work was supported by the CzechMinistry of Education, Project No. MSM6198959213, theCzech Ministry of Industry and Trade, Project No. FR-TI1/364, and the Spanish Research Directorate, GrantNo. FIS2008-04356.

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FIG. 4. Simulated reconstruction of a scanned optical signal.Panels from top to bottom: The simulated object, the blurredimage registered by a short-sighted optical system, and theintensity distribution at the object plane as reconstructed fromdata provided by a scanning device.

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