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wave propagation through a multiply scattering medium is an important problem for different disciplines , from optics to acoustics and microwaves  @xcite , since it finds important applications from fundamental physics to biomedical imaging  @xcite .    for a monochromatic coherent input beam , the medium typically generates a speckle pattern  @xcite - the result of complex interference between the scattered waves - that is usually considered a major drawback for imaging and focusing . in optics , wavefront shaping by spatial light modulators ( slms ) has emerged as a powerful tool to control the speckle and thus light propagation through such complex media  @xcite . different approaches have been proposed , in particular via optimization methods  @xcite , to focus light through  @xcite or into  @xcite a complex medium . however , a more general way to understand light propagation is to describe it via its transmission matrix  ( tm )  @xcite , which in essence describes the fact that propagation through a disordered medium , however complex , remains a linear process , and therefore can be described by a linear operator @xmath0 linking input to output modes , i.e. it can be described as @xmath1 . recently , we introduced a method to measure the monochromatic transmission matrix of a complex medium and to subsequently demonstrate focusing and imaging through it  @xcite . matricial methods are now widely used to characterize disordered media  @xcite or multimode fibers  @xcite , and even focusing inside complex media  @xcite .    generally speaking , the medium has a spectrally dependent response , with a spectral correlation bandwidth , @xmath2 , that is inversely related to the confinement time of light in the medium @xmath3  ( @xmath4 )  @xcite.the monochromatic treatment is , therefore , only valid as long as the bandwidth of the source is contained within this spectral correlation bandwidth . consequently , when light from a broadband source passes through a multiply - scattering medium , its different spectral components generates different speckle patterns . this is the case where a short pulse - shorter than the average confinement time through the medium - is scattered in space and spread in time . thanks to the inherent coupling between spatial and spectral degrees of freedom in a scattering medium  @xcite , it is still possible to achieve spatiotemporal focusing through it by optimising the spatial input wavefront  @xcite .    in this work , we show that we are able to measure the multispectral transmission matrix  ( mstm ) of a complex medium -  i.e. the tridimensional matrix that is essentially a set of monochromatic tms  - for several chosen wavelengths . first , we show that we are able to retrieve the spectral correlation bandwidth of the medium from the spectral correlations in the mstm . then , exploiting the mstm , we can achieve efficient and deterministic spatiospectral control of a broadband and spatially coherent light source , here a femtosecond pulse , by spatial - only wavefront shaping . the mstm gives mores flexibility than both the optimization  @xcite and the pulse shaping approach  @xcite to control separately and individually spatial and spectral components of the output fields . we demonstrate selective spatial focusing of a given wavelength as well as multispectral focusing . finally , we show that the medium can be turned into a controllable dispersive optical element to focus different spectral components of the pulse at different spatial positions , similarly to a grating  @xcite .     system ( @xmath5  mm ) conjugates the slm with the back focal plane of a low numerical aperture objective  ( o@xmath6 , na = 0.3 ) that focuses the laser beam on a strongly scattering medium  ( m ) . the transmitted light is then imaged on a ccd camera after polarization selection by a linear polariser  ( p ) . a spectrometer  ( s ) is used to monitor the output wavelength of the laser . ( b - d )  examples of the speckle patterns at the output of a zno sample with spectral correlation bandwidth @xmath7 nm for three totally uncorrelated wavelengths : ( b )  @xmath8  nm , ( c )  @xmath9  nm , and ( d )  @xmath10  nm . the white circles highlight the decorrelation of one bright speckle grain at the central wavelength @xmath9  nm . ( e )  normalised spectral correlation functions @xmath11 of different strongly scattering media obtained by correlating different speckle images acquired while scanning the laser wavelength from @xmath12  nm to @xmath13  nm with a step of @xmath14  nm . the full width half maximum  ( fwhm ) of these curves gives the spectral correlation bandwidth @xmath2 of the different samples : tio@xmath15 with @xmath16  nm  ( circles ) , zno with @xmath17  nm ( squares ) , zno with @xmath18  nm  ( diamonds ) , and zno @xmath19  nm  ( triangles ) . for the zno samples the correlation bandwidth was lowered by increasing the thickness of the scattering layer . the shaded areas represent one standard deviation around the average values.,width=306 ] in essence , in the monochromatic tm measurement approach , a set of input wavefronts , that describes a complete basis of the input modes , is sent through the medium . part of the beam remains unmodulated and generates a stationary speckle that serves as a reference  @xcite . using phase - shifting interferometry , the amplitude and phase of the output signal can then be retrieved from a set of speckle images . in order to measure the full mstm , the laser is operated in cw mode at a given wavelength followed by the procedure for measuring the monochromatic tm of the medium  @xcite ; this step is then reiterated at every desired wavelength by changing the laser wavelength . in this 3d matrix , the output field at the @xmath20-th pixel on the ccd camera is connected to the input field by    @xmath21    where @xmath22 is the element of the mstm linking the @xmath20-th output ccd pixel to the @xmath23-th input slm pixel at frequency @xmath24 . just as in  @xcite , the reference is an unknown speckle whose amplitude and phase information is encoded in the @xmath25 coefficients . since this reference is frequency dependent , the information on the relative local phase between monochromatic tms can not be accessed in this type of measurement . this measurement , therefore , precludes exact temporal control but still allows spatiospectral control of a broadband light source . [ fig:1](a ) shows a simplified scheme of the experimental setup used to measure the mstm . a titanium - sapphire laser  ( ti : sa , spectra - physics mai - tai ) illuminates a two - dimensional phase - only spatial light modulator ( slm , hamamatsu x10468 - 04 ) . the laser is meant to work in mode - locked pulsed regime  ( central wavelength tunable between @xmath26  nm and @xmath27  nm , pulse duration @xmath28  fs and corresponding bandwidth @xmath29  nm ) but it can also be unlocked and operated as a continuous wave  ( cw ) , tunable laser between @xmath26  nm and @xmath27  nm with a coherence length of @xmath30  cm . with such a coherence length , the laser in cw can be considered monochromatic for the scattering media studied in this work . a spectrometer  ( hr400 ocean optics ) is used to monitor the spectrum and the mode of operation of the laser , with a spectral resolution of 0.2  nm . the illuminated slm pixels are grouped into macro - pixels , each of which induces a controllable phase shift @xmath31 between @xmath32 and @xmath33 . the light is reflected by the slm and passes through a strongly scattering layer of zno or tio@xmath15 powder of different thicknesses  @xcite . figs . [ fig:1](b - d ) show examples of measured speckle patterns transmitted through the same zone of one sample at different wavelengths but for a same input wavefront : these speckles are strongly wavelength - dependent . by correlating speckle images taken while tuning the wavelength of the laser in cw mode by steps of 0.2  nm , the spectral correlation bandwidth of the medium can be quantified  @xcite . [ fig:1](e ) shows these correlation curves for samples of different thicknesses and materials . their full - width - half - maximum defines the spectral correlation bandwidth @xmath2 of the corresponding medium . [ fig:2](a ) shows that the correlation between different spectral components of the mstm is consistent with the spectral correlation bandwidth of the medium as measured in fig . [ fig:1](e ) . in order to maximise the information and minimise the amount of measurements required to control a polychromatic pulse of bandwidth @xmath34 , the spectral sampling of the mstm can be chosen to match the spectral correlation bandwidth @xmath2 of the medium . the elements of the mstm are sufficiently uncorrelated and yet contain enough information to describe accurately the scattering process . here , therefore , the medium response is sampled by a mstm composed of @xmath35 monochromatic tms with @xmath36 ( fig . [ fig:2](b ) ) .    in the following , all measurements are performed on a sample with spectral correlation bandwidth of 1.6  nm , corresponding to an average confinement time of 625  fs . the input light is a broadband pulse with a bandwidth of  @xmath37  nm , corresponding to a fourier - limited duration of  @xmath28  fs . therefore , an experimental @xmath38 is chosen . the number of input macro - pixels is set to 256 , corresponding to @xmath39 macro - pixels , and , on the ccd camera , an observation region , approximately corresponding to @xmath40 speckle grains , is selected .     among different monochromatic transmission matrices  ( tms ) for a zno medium  ( @xmath7  nm ) measured in the wavelength range between @xmath12  nm and @xmath13  nm with a step of @xmath14  nm . this correlation curve is consistent with the spectral correlation curve @xmath11 calculated by directly correlating speckle intensity images  ( fig.[fig:1](e ) ) and here represented by the red dashed line . the shaded areas represent one standard deviation around the average values . ( b )  in order to maximise the information about the scattering process and minimise the measurement time , the mstm is a 3d matrix composed of @xmath41 monochromatic matrices recorded at regular intervals within the bandwidth @xmath42 of the polychromatic pulse propagating though the scattering medium.,width=306 ] the first most straightforward use of the mstm is as a set of monochromatic tms to focus light at a specific wavelength through a strongly scattering medium by phase conjugation , as shown in fig . [ fig:3 ] . this corresponds to selecting a vertical section of the mstm  ( in the ccd - slm pixels plane ) in fig . [ fig:2](b ) . in this case , only the fraction of the input beam at the chosen wavelength  ( within the spectral correlation bandwidth ) will be effectively controlled  @xcite . [ fig:3](a - i ) show focusing images of monochromatic waves at various wavelengths  ( columns ) using the phase mask given by a vertical section of the mstm recorded at a different wavelength  ( rows ) . as expected , a focus appears only when the incident wavelength matches the wavelength at which the monochromatic tm was recorded . nm , ( b , e , h ) @xmath9  nm , and ( c , f , i )  @xmath10  nm , using the phase mask given by a vertical section of the mstm , in the ccd - slm pixels plane ,  ( fig.[fig:2](b ) ) recorded at ( a - c ) @xmath8  nm ,  ( d - f ) @xmath9  nm , and  ( g - i ) @xmath10  nm . a focus appears only when the tm recorded at the corresponding wavelength of the beam is selected in ( a ) for @xmath8  nm , in ( e ) for @xmath9  nm , and in ( i ) for @xmath10  nm , respectively.,width=306 ]      just like it is possible to focus in the monochromatic case in several points simultaneously by adding coherently the corresponding phase masks on the slm  @xcite , it is also possible to focus several wavelengths at the same position by displaying a coherent sum of different phase - conjugate wavefronts for different wavelengths extracted from the mstm . when one considers a single output position , an elegant way to formalize the above solution is to take a horizontal section of the mstm as shown in fig . [ fig:4](a ) . this section is a two - dimensional matrix @xmath43 that links the spectral components of the input slm pixels to the spectral components at a given output spatial position @xmath44 . in a quasi - monochromatic case , when the bandwidth of the polychromatic source is much smaller than the central wavelength , the dependency in @xmath24 at the input can be neglected . in this configuration , eq . [ mstm ] reduces to    @xmath45    this problem is equivalent to the one treated in  @xcite but now the output dimension is spectral rather than spatial . for a given desired output spectrum vector @xmath46 , the input vector can be obtained by phase conjugation as @xmath47 .    in this case , with homogeneous illumination and phase - only control , the phase of the calculated @xmath48 needs to be displayed on the slm . in order to focus spatially one spectral component @xmath24 , @xmath49 is set to 1 while all the other components are set to zero : the resulting wavefront is identical to what is obtained by phase conjugation of the monochromatic tm at @xmath24 . in order to focus all spectral components with the same weight , the imposed condition is @xmath50 . using a phase masks that focuses several wavelengths simultaneously , and scanning the wavelength in cw mode , figs . [ fig:4](b - d ) show that all wavelengths are indeed focused at the same spatial position .    . ( b - d )  focusing images for a monochromatic beam at various frequency for this optimal phase mask : ( b ) @xmath8  nm , ( c ) for @xmath9  nm , and ( d ) @xmath10  nm . the focusing efficiency is reduced by a factor roughly @xmath51 compared to fig  [ fig:3](a)(e ) and ( i ) since the phase mask tries to focus many wavelengths simultaneously.,width=306 ]    , number of controlled wavelength : ( a ) sbr= 7.2 for @xmath52 , ( b ) sbr = 26.1 for @xmath53 , and ( c ) sbr = 30.7 for @xmath54 . , width=306 ]    this approach becomes even more relevant for broadband illumination . [ fig:5 ] shows , for an ultrashort pulse , how the intensity at the focal point increases over the background when the number @xmath35 of different spectral components set to one in @xmath55 , all other components being set to zero . in fig . [ fig:5](c ) , all 6 independent spectral components of the broadband pulse measured with the mstm are controlled , showing clear spatial focusing . the signal to backroung ratio  ( sbr ) , defined as the ratio between the average value of all the pixels inside the fwhm and the average value of the speckle image before focusing , is  30.7 . since these are linear images , they are not sensitive to possible fluctuations of the temporal duration of the pulse at this position . when we use the speckle as reference , the relative phase between the different spectral components is unknown , and no temporal compression of the pulse occurs .          as a last example , fig . [ fig:6 ] demonstrates another possible use of the mstm by focusing different wavelengths to different positions . in this way the scattering medium is used as a deterministic dispersive optical element  @xcite . in fig . [ fig:6](a ) , three different wavelengths of a broadband pulse are focused at three arbitrary positions along a line . scanning the wavelength of a monochromatic laser reveals that each focus corresponds indeed to a different wavelength within the pulse ( 797  nm in fig . [ fig:6](b ) , 800  nm in fig . [ fig:6](c ) and 803  nm in fig . [ fig:6](d ) ) . these results can be interpreted as a generalized grating . in terms of resolution and efficiency , the main difference between our approach and a conventional dispersive element is that , in our case , scattering by the medium itself determines both spectral resolution and spatial localization . spectral resolution is given by the spectral correlation bandwidth of the medium , linked to the confinement time of the light in the medium , whereas the latter is diffraction - limited , i.e. limited by the effective numerical aperture of the medium . on the other hand , the efficiency of the spatio - spectral focusing , given by the sbr , is ultimately limited by how many degrees of freedom are controlled , i.e. how many slm pixels and how many speckle grains are controlled simultaneously . in practice , the limitation in number of degrees of control is due to the measurement time , which is limited by the stability of the experimental setup and the medium . here the measurement time is of the order of 15 minutes for 256 input modes and 6 wavelengths . this can easily be scaled up by increasing the stability and the modulation speed . the spectral control is , in this case , deterministic  @xcite . interestingly , unlike a grating , the spatial dispersion of the different spectral foci can be arbitrary as demonstrated in fig . [ fig:6 ] , since each independent wavelength can be focused at any spatial position . in conclusion , we introduced a method to measure the spatially and spectrally resolved transmission matrix , or mstm , of a multiply scattering medium . compared to other approaches based on optimization , the main advantage of our technique is that it is deterministic . exploiting the information contained in the mstm , we are able to selectively focus one or many spectral components of a broadband light source to one or many arbitrary spatial positions . as a proof of principle , we demonstrate multispectral focusing of a short pulse , and show the medium can be turned into a deterministic dispersive optical element , thus generalizing the concept of grating . here we demonstrate spatiospectral control , and our results are valid not only for short laser pulse , but also for temporally incoherent sources such as sleds or supercontinuum sources . further control of the spectral phase with a plane wave as reference should enable deterministic spatiotemporal control of an ultrashort pulse . 99 p. sebbah , _ waves and imaging through complex media _ ( springer , 2001 )    a. ishimaru , _ wave propagation and scattering in random media _ ( wiley - ieee press , 1999 )    m. stoytchev and a. z. genack , phys . b , * 55 * , 14 , ( 1997 ) . p. sheng , _ introduction to wave scattering , localization , and mesoscopic phenoma _ ( academic press , new york , 1995 ) .    lihong v. wang and hsin - i wu , _ biomedical optics : principles and imaging _ ( wiley , 2007 ) . j. goodman , _ speckle phenomena in optics _ ( roberts and company publishers , 2010 ) . a. p. mosk , a. lagendijk , g. lerosey and mathias fink , nat . phot . * 6 * , 283292 ( 2012 ) . i. vellekoop and a. mosk , optics letters , * 32 * , 12 ( 2007 ) . i. vellekoop , a. lagendijk and a. mosk , nature photonics , * 4 * , 320 - 322 ( 2010 ) . i. vellekoop , e. van putten , a. lagendijk and a. mosk , optics express , * 16 * , 67 - 80 ( 2008 ) .    c. w. j. beenakker , rev . phys . * 69 * , 731 ( 1997 ) . s. m. popoff , g. lerosey , r. carminati , m. fink , a. c. boccara , and s. gigan , phys . lett . * 104 * , 100601 ( 2010 ) . s. m. popoff , g. lerosey , m. fink , a. c. boccara , and s. gigan , nat . * 1 * , 81 ( 2010 ) . y choi , td yang , c fang - yen , p kang , kj lee , rr dasari , m. s. feld and w. choi , phys . . lett . * 107 * , 2 , 023902 ( 2011 ) . m kim , y choi , c yoon , w choi , j kim , qh park and w choi , nature photonics * 6 * , 581 - 585 ( 2011 ) . t. izmr and k. dholakia , optics express , * 19 * , 18871 - 18884 ( 2011 ) .    c yoon , y choi , m kim , j moon , d kim and w choi , opt . lett . * 37 * , 21 ( 2012 ) .    i. n. papadopoulos , s. farahi , c. moser and d. psaltis , biomed opt express . * 4 * , 260270 ( 2013 ) .    s. bianchi , v. p. rajamanickam , l. ferrara , e. di fabrizio , c. liberale , and r. di leonardo , optics letters , * 38 * , 4935 - 4938 ( 2013 ) . t. chaigne , o. katz , a. c. boccara , m. fink , e. bossy and s. gigan , nature photonics * 8 * , 5864 ( 2014 ) . y. choi , t. r. hillman , w. choi , n. lue , r. r. dasari , p. t. c. so , w. choi and z. yaqoob , physical chemistry chemical physics * 16 * , 27074 - 27077 ( 2014 )    e. small , o. katz , y. guan , and y. silberberg , opt . * 37 * , 16 ( 2012 )    f. van beijnum , e. g. van putten , a. lagendijk , and a. p. mosk , opt . * 36 * , 3 ( 2011 )    s. faez , p. m. johnson , and ad lagendijk , phys . lett . , * 103 * , 053903 ( 2009 )    n. curry , p bondareff , m. leclercq , n. f. van hulst , r. sapienza , s. gigan , and s. grsillon , opt . lett . * 36 * , 17 ( 2011 ) .    f. lemoult , g. lerosey , j. de rosny , and m. fink , phys . , * 103 * , 173902 ( 2009 ) . j. aulbach , b. gjonaj , p.johnson , a. mosk , and a. lagendijk , phys . , * 106 * , 103901 ( 2011 ) . o. katz , e. small , y. bromberg , and y. silberberg , nature photon . , * 5 * , 372 ( 2011 ) . h. p. paudel , c. stockbridge , j. mertz and t. bifano , optics express , * 21 * , 14 ( 2013 ) . d. j. mccabe , a. tajalli , d. r. austin , p. bondareff , i. a. walmsley , s. gigan , and b. chatel , nat . * 2 * , 447 ( 2011 ) . i. freund , phys . a , * 168 * , 4965 ( 1990 ) .    j. h. park , c. park , h. yu , y. h. cho and y. k. park , opt . lett . , * 37 * , 15 ( 2012 ) . p. garca , r. sapienza ,  . blanco , and c. lpez , adv . * 19 * , 2597 ( 2007 ) . b. redding , s. f. liew , r. sarma and h. cao , nature photon . , * 7 * , 746751 ( 2013 ) . this work was funded by the european research council ( grant no . 278025 ) , anr rococo and programme emergence(s ) from the city of paris . o.k . was supported by the marie curie intra - european fellowship for career development ( ief ) . we thank david martina and mickael mounaix for helpful discussions . s.g . ( gigan ) conceived the project . d.a . , s.g . and s.g . planned the experiment . d.a . performed the experiment and analyzed the data . s.p . proposed the analytic formalism for spatiospectral control . , g.v . and s.g . ( gigan ) wrote the manuscript . all authors discussed the results and revised the manuscript .
we present a method to measure the spectrally - resolved transmission matrix of a multiply scattering medium , thus allowing for the deterministic spatiospectral control of a broadband light source by means of wavefront shaping . as a demonstration 
 , we show how the medium can be used to selectively focus one or many spectral components of a femtosecond pulse , and how it can be turned into a controllable dispersive optical element to spatially separate different spectral components to arbitrary positions .