document: pushing the resolution limits of light microscopy , and understanding optical phenomena on scales below the diffraction limit , has been the driving force of what is known today as nano - optics  @xcite . to overcome this limit , 
   most of the early work was focused on near - field optical microscopy and related techniques  @xcite . however , in recent years , new concepts in fluorescence microscopy have pushed the resolution of far - field imaging down to the nanometer range  @xcite . 
   most of these methods  @xcite rely on the accurate localization of individual fluorescent markers , that are isolated from one another on the basis of one or more distinguishing optical characteristics , or by selective or random activation of a bright and a dark state  @xcite . 
   determining the location of an isolated fluorescent marker is only limited by photon noise , and not by the diffraction barrier . 
   a key issue affecting these subwavelength imaging methods is the optical transparency of the media surrounding the light emitters . 
   taking advantage of the transparency of cells , fluorescence microscopy uniquely provides noninvasive imaging of the interior of cells and allows the detection of specific cellular constituents through fluorescence tagging 
   . however , certain biological tissues or soft - matter systems ( such as foams or colloidal suspensions ) look turbid due to intense scattering of photons traveling through them  @xcite . 
   the image formed at a given point in the observation plane consists in a superposition of multiple fields , each arising from a different scattering sequence in the medium . 
   this gives rise to a chaotic intensity distribution with numerous bright and dark spots known as a speckle pattern , producing a blurred image carrying no apparent information about the source position  @xcite . 
   techniques to measure the distance between individual nano - objects without actually imaging their position exist  @xcite , fluorescence resonance energy transfer ( fret ) being the most widespread example  @xcite . 
   it relies on the near - field energy transfer between two fluorophores ( donor and acceptor ) emitting at different wavelengths . 
   the fret signal ( _ e.g. _ the ratio between the intensities emitted by the donor and the acceptor at different wavelengths ) depends on the donor - acceptor distance in the range @xmath0  nm . as such , it is not very sensitive to scattering problems . 
   however , determining distances between two emitters in the range of 10 to 500  nm in a scattering medium still remains a challenging problem , not accessible either by fluorescence microscopy or fret techniques . 
   our main goal here is to introduce a new approach to obtain information about the relative distance between two identical incoherent point sources in a disordered environment , based on the analysis of the fluctuations of the emitted light . 
   this is an issue of much interest , for example , in the study of conformational changes in biomolecules in living tissues . sensing the distance between two incoherent sources in a complex medium 
   could also provide an alternative to green s function retrieval techniques based on the correlations of the isotropic ambient noise measured at two receivers  @xcite .    in this paper 
   , we propose a method to capture the interaction between two identical sources in a scattering environment , based only on the measurement of intensity fluctuations . 
   the principle of the method is schematically illustrated in fig .  1 , and is based on the analysis of the intensity - intensity correlation function and the intensity fluctuations in the speckle pattern formed by two identical and mutually incoherent point sources . 
   this approach permits , in principle , to monitor the relative distance between the sources in the range 10 - 500 nm , with a precision that is not limited by diffraction , but by the microstructure of the scattering medium . in application to green s function retrieval in complex media , the approach replaces the two - point field - field correlation of the background noise by a measurement at a single point of the intensity noise due to the two fluctuating sources . 
   this might simplify the technique , in particular at visible or near - ir frequencies where time - domain field - field correlations are not easy to measure . 
   the result in this paper also illustrate the fact that multiple scattering , that had long been considered as an unavoidable nuisance , can actually enhance the performance of sensing , imaging and communication techniques  @xcite , as already demonstrated in the context of spatio - temporal focusing by time reversal  @xcite , wavefront shaping of multiply scattered waves  @xcite , or improvement of information capacity of telecommunication channels  @xcite . 
   we consider two point sources of light ( electric dipoles ) located at @xmath1 and @xmath2 in a disordered medium . 
   the sources are characterized by their electric dipole moments @xmath3 and @xmath4 , that are fluctuating quantities of the form @xmath5 \exp(-i\omega t ) { { \bf u}}_k$ ] with @xmath6 a slowly varying random phase , @xmath7 a complex amplitude and @xmath8 a unit vector defining the orientation of the dipole moment . 
   this corresponds to a classical model for a quasi - monochromatic temporally incoherent source , such as a fluorescent source emitting at frequency @xmath9 . 
   we assume that the two sources are uncorrelated ( or mutually incoherent ) , so that @xmath10\exp[-i\phi_2(t)]}=0 $ ] , where the bar denotes averaging over the fluctuations of the sources . using the ( dyadic ) green function @xmath11 of the disordered medium , 
   the electric fields at any point @xmath12 can be written  : @xmath13 the intensity associated to this field is a time fluctuating and spatially varying quantity that forms a time - dependent speckle pattern . 
   let us first consider the total power @xmath14 emitted by the two sources . 
   it reads @xmath15 where @xmath16 is a sphere with radius @xmath17 that encloses the disordered medium and @xmath18 is the speed of light in vacuum . for a _ non 
   absorbing _ medium , the following relation can be derived from the vector form of green s second identity  @xcite ( the frequency dependence in the green function is dropped for simplicity ) : @xmath19 { { \bf p}}_2^ * \ . 
   \label{eq : green_identity}\end{aligned}\ ] ] from eqs . 
   ( [ eq : field_general]-[eq : green_identity ] ) , we obtain @xmath20 where the notation @xmath21 { { \bf u}}_{j^\prime}$ ] has been introduced for the sake of simplicity . 
   we first assume that a temporal averaging over the fluctuations of the sources can be performed , in one configuration of the disordered medium ( frozen disorder ) . 
   the fluctuation time scale of the emitted power can be associated to the coherence time , as usually defined for partially coherent sources  @xcite . for fluorescent sources , 
   this time is on the order of the lifetime @xmath22 of the excited state . for emission in the visible range , expected orders of magnitude are @xmath23 ns for dye molecules or quantum dots , and @xmath24s for rare - earth ions . 
   the averaged power @xmath25 is simply the sum of the averaged power emitted by each source independently , since the terms with @xmath26 in eq . 
   ( [ eq : p2 ] ) vanish upon time averaging . 
   it reads as = ( |p_1|^2 _ 11 + |p_2|^2 _ 22 ) [ eq : pmoy ] where @xmath27 \mathrm{im } g_{jj}$ ] is the electric part of the local density of states ( ldos ) at point @xmath28  @xcite . 
   however , a cross - term survives in the fluctuations of the total emitted power . indeed , calculating the variance of @xmath14 from eq . 
   ( [ eq : p2 ] ) , one obtains  @xcite @xmath29 \ . 
   \label{eq : varp_time}\ ] ]    the imaginary part of the two - point green function @xmath30 in eq . 
   is known to enter the expression of field - field spatial correlations in random fields , such as blackbody radiation or volume speckle patterns  @xcite , the description of time - reversed fields  @xcite , and is at the core of green s function retrieval techniques based on ambient noise correlations  @xcite . 
   it is proportional to the cross density of states ( cdos ) , that was introduced in a different context for the description of spatial coherence in complex systems  @xcite . 
   physically , the cdos counts the number of photonic eigenmodes connecting two points ( in our case the source points ) at a given frequency  @xcite . 
   more precisely , the cdos connecting @xmath31 to @xmath28 is given by @xmath32 \mathrm{im } g_{jk}$ ]  @xcite . using the cdos , and assuming that the two sources have the same amplitude ( @xmath33 ) , the variance of the total emitted power can be rewritten as @xmath34 this equation is the first result in this paper . 
   it provides a direct relationship between the temporal fluctuations of the total power emitted by two incoherent sources and the cdos connecting the source points in an arbitrary environment . 
   this suggests that a retrieval of the amplitude of the cdos ( or equivalently of the imaginary of the green function at two different points ) is possible in a structured medium from a measurement of temporal fluctuations of the emitted power emerging from two incoherent sources . 
   such a measurement would resemble the green function retrieval from ambient noise correlations , based on the relationship between field - field correlations and the imaginary part of the green function given by the fluctuation - dissipation theorem , initially introduced in the context of electromagnetic thermal fluctuations  @xcite . 
   the generality of this relationship , also valid for field fluctuations in speckle patterns  @xcite , has stimulated the development of green s function retrieval techniques in acoustics , seismology or with low - frequency electromagnetic waves  @xcite . in this approach , 
   the statistical isotropic ambient noise is detected by two receivers , while in the method suggested here the green function is encoded in the noise due to the fluctuations of the two sources . 
   the possibility to measure power fluctuations instead of field - field correlations might be an advantage for electromanetic green s function retrieval in the visible or near - ir frequency range . 
   equation  ( [ eq : varp_time_2 ] ) also shows that the power fluctuations encode the interdistance between the sources . at this stage , since the cdos @xmath35 is specific to the sample under study and unknown , changes in power fluctuations could reflect changes in the interdistance , but the interdistance could not be determined without solving a difficult inverse problem . 
   we will see how the problem can be simplified in the presence of multiple scattering in a disordered medium by peforming an ensemble averaging over the configurations of disorder . 
   we now assume that in addition to a temporal averaging over the fluctuations of the sources , an average over the configurations of the disordered medium can be performed . 
   a specific situation would be that of sources embedded in a dynamic medium , with configurational changes occurring on a time scale much larger than the characteristic time of the fluctuations of the sources . 
   an equivalent situation is that of sources moving inside a frozen disordered medium , also on a sufficiently large time scale , as schematically shown in fig . 
   if both the sources and the disordered medium are fixed ( i.e. the medium itself does not fluctuate ) , an artificial configurational averaging process could be induced by an external moving diffuser surrounding the medium , as shown in fig . 
   2(b ) . in all these situations both averaging processes can be performed independently and subsequently . 
   the ( temporal ) variance of the total emitted power averaged over the configurations of the disordered medium is readily obtained from eq . 
   : @xmath36 where we use the notation @xmath37 for configurational averaging .        in practice 
   , it is often convenient to work with normalized statistical quantities . for later convenience 
   , we can introduce the contrast of the power fluctuations , that we define as _ 
   [ eq : def_sigmap ] let us point out again the specific use of two non - commuting averaging processes in this definition , one over the temporal fluctuations of the sources and subsequently one over the configurations of the disordered medium . from eqs .   and 
   , the power contrast can be written in terms of the ldos and cdos , leading to _ 
   [ eq : expr_sigmap ] in the next section we will derive a simple relationship between the power contrast @xmath38 and the speckle contrast @xmath39 deduced from the radiated intensity measured in a single direction , or equivalently at a single point in the far field speckle pattern . 
   a measurement of the total emitted power @xmath14 requires a detection integrated over @xmath40 steradians . 
   measuring fluctuations of the intensity radiated in a given direction , or intensity correlations between two different directions , could be a more convenient approach in practice . 
   the time - averaged far - field intensity emerging in a given direction @xmath41 , per unit solid angle , and for a polarization state @xmath42 , is given by & = & + [ inocros ] where @xmath43 is the intensity radiated by the point source @xmath44 . 
   the latter reads as i_j(*u * ) = _ r r^2 |*e*_*g*(*r*,*r*_j ) p_j * u*_j|^2 [ ijsigma ] with @xmath45 . 
   the configurational average of the ( time averaged ) total emitted power and fluctuations can be rewritten as & = & _ 4 * u*[exact1 ] + & = & _ ^ _ 4 * u * * u*^ [ exact2 ] where @xmath46 means integration over the solid angle , @xmath47 runs over the two orthogonal polarization states in the far field , and @xmath48 . 
   we now assume that the radiated field is a random statistically isotropic field , under the only constraints given by eqs .   and . 
   in this case one has @xmath49 and = ( 1 + _ ^ _ * u**u*^ ) [ avip1ip1 ] where @xmath50 is the usual kronecker delta and @xmath51 is a kronecker delta with respect to detection angles . 
   the derivation of eq .   is given in appendix a.    at this stage , it is worth noticing that relation between the speckle intensity - intensity correlation function and the fluctuations of the total emitted power is the origin of the so - called @xmath52 correlation known for a speckle produced by a single source  @xcite . 
   if we consider the particular case of a single source at @xmath1 ( @xmath53 ) that does not fluctuate in time , the normalized intensity - intensity correlation function @xmath54 follows directly from eq . and 
   is given by c(*u*,;*u*^,^ ) & = & -1 + & = & c_0 + ( c_0 + 1)_^ _ * u**u*^ [ c1]where c_0 & = & is the normalized variance of the ldos at the position of the source , that is at the origin of the infinite - range @xmath52 contribution to the speckle correlation function  @xcite . 
   therefore , for a single source , eq .   provides another way of deriving the well - known results related to the @xmath52 correlation . 
   moreover , from eq . , the speckle contrast ( defined as the normalized variance of the intensity in a specific direction and a given polarization channel @xmath47 ) is given by = 1 + 2 c_0 a result already obtained by shapiro  @xcite , based on a microscopic diagrammatic approach for scalar waves . 
   this result leads to non - universal corrections to the rayleigh statistical distribution of the intensity in speckle patterns produced by multiple scattering  @xcite .    for two fluctuating sources , assuming again that configurational changes in the disordered medium occur on time scales larger than the characteristic time of the source fluctuations 
   , we also have [ the derivation is similar to that leading to eq . 
   ( [ avip1ip1 ] ) , see appendix a ] : @xmath55 making use of eqs .  , and for two sources with the same amplitude @xmath56 
   , we obtain @xmath57 \nonumber \\ & = \frac{\omega^4}{8 ^ 3 \ , \epsilon_0 ^ 2}|p|^4   \left\langle   \rho_{12}^2\right\rangle   \ . 
   \label{eq : fluct_iu}\end{aligned}\ ] ] this equation shows that the intensities corresponding to two different speckle spots ( @xmath58 ) in the angular speckle pattern formed by two incoherent point sources are strongly correlated , with a correlation given by the fluctuations @xmath59 of the cdos . 
   we end up with the surprising result that for two incoherent sources a cross - term survives averaging , and induces infinite range correlations in the speckle pattern . 
   this correlation is formally very similar to that given by the @xmath52 contribution in the case of a single source , the fluctuations of the cdos replacing the fluctuations of the ldos . 
   since the cdos is a two - point quantity , connecting in this specific situation the two source points , the speckle correlations encode the distance between the two sources . 
   another implication of eq . 
   is that in the presence of multiple scattering , cdos fluctuations can be accessed from measurements of the directional intensity , and not only from measurements of the total power @xmath14 as suggested initially by eq .  . 
   more precisely , we will now show that the speckle contrast measured in a single speckle spot ( hereafter denoted by @xmath60 ) contains the same information as the contrast @xmath61 in eq . 
   , that assumed a measurement of the total emitted power . to proceed , we define the speckle contrast for a detection in a given polarization channel @xmath47 as @xmath62 in a similar way as the contrast of the total emitted power in eq .  . from eq .  , and the relation = one immediatly obtains _ 
   s= = _ p [ sc2 ] or equivalently _ 
   [ scfinal ] this equation is the second important result in this paper . 
   it shows that for two incoherent sources in a disorder medium , the speckle contrast measured in a single speckle spot , and computed from both temporal averaging over the fluctuations of the sources _ and _ configurational averaging over disorder , is proportional to @xmath63 that characterizes the fluctuations of the cdos connecting the two sources.this means that the information on the interdistance between the sources is encoded in the speckle contrast @xmath60 measured at a single point . 
   changes in the speckle contrast could therefore be used to detect changes in the interdistance between the two sources . 
   compared to a measurement of the total emitted power @xmath14 , a measurement of the speckle contrast @xmath39 would require a simpler instrumentation , but at the cost of a substantial reduction of the signal level . 
   note that a parallel detection of the fluctuations in several speckle spots could be performed using a ccd camera . in practice , a compromise between simplicity in instrumentation and signal - to - noise ratio should be found . 
   the contrast in eq . 
   ( [ scfinal ] ) would decrease to zero when increasing the interdistance . 
   this change occurs on a range given by the width of the cdos ( green function ) considered as a function of the interdistance @xmath64 . 
   this width depends on the microscopic structure of the disordered medium , providing in principle resolution capabilities beyond the free - space diffraction limit . 
   the determination of the absolute value of the interdistance from the speckle contrast would require an expression of @xmath63 . 
   the crucial issue in this case is to find the expression of @xmath65 in a disordered medium . 
   this can be done at least in the weak scattering limit , as discussed in the next section . 
   another consequence of eq . 
   ( [ scfinal ] ) , or eq .  , is that it provides a way to measure @xmath66 in a complex medium from the speckle noise recorded at a single point , which in the context of green s function retrieval might provide an alternative to the two - point field - field correlation technique , as discussed previously in section ii . 
   when the sources are embedded in a weakly disordered , homogeneous and isotropic medium , it is possible to give an explicit approximate expression of the fluctuations of the cdos @xmath67 \langle ( \mathrm{im } g_{12})^2 \rangle$ ] . in the limit @xmath68 , where @xmath69 is the effective refractive index , @xmath70 ( with @xmath71 the wavelength in vacuum ) and @xmath72 the boltzmann scattering mean free path , the averaged green function is given by the well known result for homogeneous and isotropic media and we can write @xmath73 as @xmath74 where @xmath75 is the effective wavenumber . 
   the description of the scattering medium by a complex effective wavenumber breaks down when the distance @xmath76 between the emitters approaches the size of the homogeneities ( more precisely the correlation length of disorder , see the discussion in ref . @xcite and references therein ) . 
   although this might look like a severe limitation , this approximation is in practice very robust and has been shown to model accurately light diffusion through biological tissues . 
   to lowest order , we can further approximate @xmath77 which , from eqs .  , or , and provides an explicit expression of the intensity correlation function , or the speckle contrast , in terms of the relative distance between the sources . 
   such an explicit expression should be of practical interest , e.g. , for the sensing of the interdistance between emitters embedded in biological tissues . 
   let us point out that in terms of the detection of _ changes _ in the interdistance ( without measuring its absolute value ) , the method suggested in this paper does not requite any explicit expression of @xmath63 and relies only on eqs . 
   ( [ eq : varp_time_2 ] ) , ( [ eq : expr_sigmap ] ) or ( [ scfinal ] ) . 
   the results derived in this paper suggest the intriguing possibility that intensity measurements at only one point in a speckle pattern produced by two incoherent sources can provide information about the relative distance between the sources . 
   moreover , this information is in principle not limited by diffraction . 
   it can be extracted from the speckle contrast @xmath60 obtained after a proper time and configurational averaging process . 
   the results also suggests an alternative approach to the green function retrieval technique . in the later the statistical isotropic ambient noise 
   is detected by two receivers , while in the method suggested here the green function is encoded in the noise due to two fluctuating incoherent sources measured at a single point . 
   finally , let us note that for fluorescent emitters , the quantum optical equivalent of eqs .  , or might be established in terms of decay rate or photocount statistics of a coupled system , and fluctuations could be associated with super - radiant and sub - radiant states . 
   this quantum treatment is left for future work . 
   we acknowledge helpful discussions with m. fink and f. scheffold . 
   this work was supported by the spanish mec through grant no . 
   fis2012 - 36113 , by the comunidad de madrid through grant no . 
   s2009/tic-1476 ( microseres project ) , by labex wifi ( laboratory of excellence within the french program `` investments for the future '' ) under references anr-10-labx-24 and anr-10-idex-0001 - 02 psl*. jjs acknowledges an ikerbasque visiting fellowship and rc and gc the hospitality of the dipc at donostia - san sebastian ( spain ) where part of this work was done . 
   the far - field intensity radiated by two dipoles emerging in a given direction @xmath41 , per unit solid angle , and for a polarization state @xmath42 , can be written as i_(*u * ) = _ r r^2 | _ 
   j=1 ^ 2 * e*_*g*(*r*,*r*_j ) * p*_j |^2 with @xmath45 . assuming that the @xmath40 solid angle is divided in a finite number @xmath78 , of ( solid angle ) pixels , the total power emitted by the two sources is given by p & = & _ 4 i(*u * ) * u*= _ _ 
   u^n i_(*u * ) which can be written in compact matrix form , p & = & ^^ where @xmath79 and @xmath80 is a @xmath81 matrix ( @xmath82 comes from the two orthogonal polarizations of the field for each observation angle in the far - field ) . by using the singular value decomposition ( svd ) , the @xmath80 matrix can be factorized as & = & ^ + ^&= & ^2 ^ where @xmath83 is an unitary @xmath84 matrix , @xmath85 is a @xmath81 rectangular diagonal matrix with nonnegative real numbers on the diagonal and @xmath86 is a @xmath87 unitary matrix . 
   equation in the main text can now be written as @xmath88= \frac{4\pi}{n } [ \hat{{{\bf p}}}^\dagger\mathbf{v } \mathbf{\sigma}^2 \mathbf{v}^\dagger \hat{{{\bf p } } } ]   \nonumber \\ & =   \frac{\mu_0 \ , \omega^3}{2 } \sum_{j , j ' = 1}^2   p_{j'}^*p_j \mathrm{im } g_{jj ' }   \nonumber \\ & =   \frac{\mu_0 \ , \omega^3}{2 }   \begin{pmatrix } { { \bf p}}_1^\dagger & { { \bf p}}_2^\dagger \end{pmatrix } \begin{bmatrix } \mathrm{im}{{\bf g}}(11 ) & \mathrm{im}{{\bf g}}(12 ) \\ \mathrm{im}{{\bf g}}(21 ) & \mathrm{im}{{\bf g}}(22 ) \end{bmatrix } \begin{pmatrix } { { \bf p}}_1 \\ { { \bf p}}_2 \end{pmatrix } \label{eq : mat1}\end{aligned}\ ] ] where @xmath89 . in terms of these new matrices , 
   the intensity at a given angle @xmath41 and polarization @xmath47 can be computed as @xmath90 where @xmath91 corresponds to the @xmath92 component of the dipole @xmath93 ( with @xmath94 $ ] and @xmath95 ) . 
   if we assume that the radiated field is statistically isotropic , we can consider @xmath96 as a random unitary matrix statistically independent of the @xmath85 and @xmath97 matrices ( isotropy hypothesis ) . 
   this is one of the key assumptions in the macroscopic scaling approach of transport theory @xcite . by using the averages over the unitary group ( evaluated by mello in ref . 
   @xcite ) , u_b u_^ u_u_ b = _ b b _ * u**u* _ and the ensemble average of the intensity is given by @xmath98 which , from eq . 
   , gives @xmath99 with @xmath100 and @xmath101 i.e. , as expected , under the isotropy hypothesis , the intensity at a given angle is simply proportional to the average of the total radiated power , @xmath102 . after averaging eq . 
   ( [ eq : isum ] ) over the temporal fluctuations of the sources we have @xmath103 and @xmath104 is simply the sum of the individual intensities [ eq . ] i.e. the non - diagonal boxes of @xmath105 do not contribute ( no crosstalk term ) . 
   the same applies for @xmath106    let us now consider eq . 
   , & = & _ _*u*,*u* , [ exact3 ] with @xmath107 given by eq . . 
   computation of eq . involves again @xmath103 together with averages of 4 elements of an unitary random matrix @xcite @xmath108 it is easy now to find : @xmath109 which , in the large @xmath110 limit , gives eq .   in the main text . 
   following a similar procedure , one also obtains @xmath111 which , in the large @xmath110 limit , gives eq .  . 
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summary: we study the fluctuations of the light emitted by two identical incoherent point sources in a disordered environment . 
   the intensity - intensity correlation function and the speckle contrast , obtained after proper temporal and configurational averaging , encode the relative distance between the two sources . 
   this suggests the intriguing possibility that intensity measurements at only one point in a speckle pattern produced by two incoherent sources can provide information about the relative distance between the sources , with a precision that is not limited by diffraction . 
   the theory also suggests an alternative approach to green s function retrieval technique , where the correlations of the isotropic ambient noise detected by two receivers are replaced by a measurement at a single point of the noise due to two fluctuating incoherent sources .