ITAL CONVERSATION Italian 13
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Date Created: 09/12/15
PRL 103 043903 2009 PHYSICAL REVIEW LETTERS week ending 24 JULY 2009 Amplitude and Phase of Tightly Focused Laser Beams in 39Iurbid Media Carole K Hayakawa and Vasan Venugopalan Department of Chemical Engineering and Materials Science Engineering Tower University of California Irvine California 92697 USA Vishnu V Krishnamachari and Eric 0 Potmal Department of Chemistry Natural Sciences 11 University of California Irvine California 92697 USA Received 13 January 2009 published 23 July 2009 A framework is developed that combines electric eld Monte Carlo simulations of random scattering with an angular spectrum representation of diffraction theory to determine the amplitude and phase characteristics of tightly focused laser beams in turbid media For planar sample geometries the scattering induced coherence loss of wave vectors at larger angles is shown to be the primary mechanism for broadening the focal volume This approach for evaluating the formation of the focal volume in turbid media is of direct relevance to the imaging properties of nonlinear coherent microscopy which rely on both the amplitude and phase of the focused elds D01 101103PhysRevLett103043903 Image quality in laser scanning optical microscopy is related directly to the spatial distribution and strength of the focal eld Diffraction theory models the focal volume as the spatial interference of the electric eld and is suf 7 cient to describe the propagation of a focused laser beam in homogeneous media 1 However biological samples are composed of cellular and extracellular components of varying size and refractive index 23 that act as scattering centers and distort the electric eld 4 Experiments show that focused beam propagation in turbid media results in attenuation of the focal eld amplitudes and broadening of the focal spot along both lateral and axial dimensions 5 Current formulations of diffraction theory being limited to homogeneous media cannot describe these important changes to the focal volume Monte Carlo methods can be used to solve the radiative transport equation by simulating light propagation as the transport of photons that undergo scattering at discrete locations within the turbid medium Monte Carlo studies have shown the depth dependence and resolution of two and threeiphoton excited fluorescence images to scale with the photon density in the focal region 56 However because particleibased Monte Carlo methods do not model the waveicharacteristics of light they are unable to model explicitly the amplitude and phase of tightly focused laser beams Complete knowledge of the focal eld characteristics in turbid media is crucial for analysis of the resolution an signal strength in all forms of focused laser microscopy including uorescence microscopy and optical coherence tomography Moreover the imaging properties of nonlini ear coherent imaging methods including multiharmonic generation and coherent antiAStokes Raman scattering mi croscopy CARS in turbid samples cannot be understood without an evaluation of the amplitude and phase of focal elds 7 Finiteidifference time domain FDTD methods 00319007o91o34o439o34 04390371 PACS numbers 4225 p 4262Be can model explicitly the electric eld propagation in turbid media 89 However PDTD calculations are computai tionally expensive and require the exact location and ref fractive index of the scattering centers within tissue properties that are generally unknown Consequently mod eling the general mechanisms that underlie the formation of the focal volume in turbid media remains a major challenge Here we introduce a general framework that combines fully vectorial diffraction theory for focused elds with electric eld Monte Carlo EMC simulations to determine the amplitude and phase of tightly focused elds in turbid media The propagation of optical wave fronts is charac7 terized by tracking the direction of the wave vector k its axial location depth within the medium and the path length traveled between successive Mie scattering events using an EMC approach 1011 The connection with diffraction theory is made through the coherent angular dispersion function CADF which describes the ampli7 tude loss and phase retardation associated with an optical wave front that enters a slab of speci ed thickness with direction k and exits the slab with direction k The CADF is employed within an angularispectrum representation of diffraction theory to calculate the full diffractionilimited focal eld We decompose the incoming light into a set of plane waves each characterized by x and y components of the waveivector k Using an angularispectrum representation the electric eld in the Vicinity of the focal volume is ifeiikff Ed kk 277 acecase fax y 1 gtlt eiltkxxkyykzzgt k dkXdk 1 ENC y z where f is the focal length of the lens and Ear is the 2009 The American Physical Society PRL 103 043903 2009 PHYSICAL REVIEW LETTERS week ending 24 JULY 2009 refracted eld at the lens surface The refracted eld Egr is Written in spherical coordinates as where G6 I6 is the CADF per unit solid angle For nonscattering media Egr EfaI and Eq 1 is identical to the wellknown diffraction integral 112 G6 NBC is E faIW 2 W2 G6 I6 Efar6 used to modify the refracted eld Efar to include the effects 75 0 9 0 of scattering see Fig 1a For an aplanatic lens the X sin6d6d 2 vector components of Efar6 can be expressed in terms of the incident eld Emc6 as n 12 cos cos6 cos y sin sin y Emw gt A vcosa sin cosacos y cosqs sinw 7 311149745 3 n2 sin6 cos y where y is the angle of polarization of the incident eld with respect to the x axis and ml and n2 are the refractive indices of the media before and after the refraction at the lens respectively The CADF is determined by an EMC simulation Similar to the situation encountered in the optical micro scope plane waves k are launched uniformly over a hemi sphere 0 S 15 lt 27739 0 S cos6 lt 1 and propagate through a planar slab of thickness T see Fig 1b Each plane wave k is launched at given polar and azimuthal angles 6 and subject to scattering in the medium The associated electric eld is characterized by the coordinate system Ifl f s where s is the unit propagation direction of the photon and Ifl and are unit vectors in the direction of the parallel and perpendicular components of the polar ized electric eld E EHIfl Ei Upon collision with a scatterer whose probability per unit path length is characterized by the scattering coef cient MS of the medium the wave s coordinate system is updated to Ifl s The polar scattering angle 6 between the incoming and the outgoing wave is determined by sampling the angular distribution function 176 11 S1 2 lSz l2 Qscax2 27739 p gt A p lt1gtgtdlt1gt 4 Egar 67 FIG 1 a Schematic of the diffraction geometry The wave front of the initial eld EfaI is modi ed to EEK which captures the effects of a given medium The diffraction pattern of the scattered eld is determined in the vicinity of the focal spot 0 b Schematic of the geometry used to calculate Gk Ik Waves launched from a Lambertian source symbolized by the semi circle at angle 0 are allowed to scatter in a medium of thickness T and the amplitude and phase at each exit angle 0 is deter mined where 176 CD is the scattering phase function S1 and Sz are the angledependent elements of the amplitude scatter ing matrix Q5ca is the scattering ef ciency and x is the particle size parameter 13 The azimuthal scattering angle CI is found using the rejection sampling method of the conditional probability 17CIgtI6 176 CIgt 176 The components of the normalized scattered eld E are calcu lated as El l 1 526 cosCD 526 sinCI E WltSi smq S1 COSCDltEJ 5 where N 6 CD is the angledependent normalization fac tor 11 Multiple scattering events introduce changes in both the propagation direction and polarization of the wave Hence after traversal through the slab the propaga tion angles 6 W of the outgoing wave front may differ from those of the incident wave front The phase of the scattered elds is determined by the path length of the Wave front as it passes through the slab and the phase shifts associated with the Mie scattering events For a given polarization direction 45 the CADF is determined by the total coherent eld E6I6 W6 obtained from the averaged sum of plane waves launched at 6 and exiting at 6 where W6 is the total intensity exiting in the 6 direction We also de ne an incoherent angular dispersion function IADF determined by IE6I6 W6 This approach fully characterizes electric eld propaga tion in any medium in which the scattering events can be treated independently Here we wish to apply our frame work to simulate the speci c case of random scattering as it best captures the evolution of the focal eld with depth in turbid media In this case the resulting electric eld is determined for numerous realizations of particle arrange ments resulting in a positionindependent incoherent back ground in addition to the coherent elds that form the focal volume While this methodology can be applied to a sam ple with xed scatterers the resulting predictions would reveal spatial interference patterns speckle and obscure the overall variation of the focal eld characteristics We consider planar slabs of 10 intralipid solution a highly turbid medium with scattering coef cient 113 2733 cm 1 and absorption coef cient 11 002 cm 1 0439032 PRL 103 043903 2009 PHYSICAL REVIEW LETTERS week ending 24 JULY 2009 at 800 nm Spherical lipid particles of radius 01913 um are chosen to match the experimentally deter mined rst moment of the angular distribution function g 0636 These properties result in a scattering mean free path ls 366 um and a transport mean free path 1 100 um We consider a waterimmersion objective lens with numerical aperture 11 and linearly xpolarized light de ned by E 1 and E i 0 Upon exiting the slab we determine the component of the light polarized parallel to the incident wave front For the EMC simula tions we launched 109 plane waves resulting in a relative error of lt3 in the IADF In Fig 2 twodimensional representations of scattering induced angular dispersion are shown for turbid planar slabs of thickness T 1 um and 150 um The graphs provide a measure of the probability by which optical Wave fronts that enter the slab at a speci c angle 6 will exit the slab at another angle 6 The IADF is shown in Figs 2a and 2b For the 1 um slab due to the limited number of scattering events the IADF has diagonal ele ments of essentially constant magnitude a result expected for nonscattering media By contrast propagation through the 150 um thick slab involves multiple scattering events and results in a signi cant dispersion of the angular distri bution of the optical wave fronts off the diagonal Twodimensional representations of the CADF for the 1 um and 150 um slabs are shown in Figs 2c and 2d respectively For the 1 um thick slab the coherent fraction of the light is largely represented by the waves that have not undergone scattering as evidenced by the diagonal in the graph For propagation through the 150 um slab an almost similar trend is observed indicating that the coher ent amplitude predominantly stems from unscattered light FIG 2 color Incoherent and coherent angular dispersion in scattering media IADF for slab thicknesses of a T 1 um and b T 150 um CADF for slab thicknesses of c T 1 um and d T 150 um Graphs are plotted on a logarithmic scale Note that for both slabs the offdiagonal elements of the CADF Figs 2c and 2d are much smaller than their IADF counterparts Figs 2a and 2b This indicates that waves exiting at an angle 6 different than their entrance angle 6 undergo different phase delays and thus do not constructively interfere Moreover the coherent amplitude of the larger entrance angles 6 is strongly attenuated This is due to the longer path lengths that large 6angle waves experience through the medium which increases the like lihood of scattering degrades coherence and reduces the effective numerical aperture of the focused coherent radiation Figure 3 shows the calculated focal elds After travers ing the 1 um thick slab the focal eld resembles the well known amplitude of the Airy disk as shown in Fig 3a The phase pro le of the focal eld shown in the inset is nearly identical to the spatial phase associated with the Airy disk Figure 3b shows the axial distribution of the focal eld which resembles the focal volume known from standard diffraction theory Thus for thin turbid samples the diffractionlimited focal volume is qualitatively unaffected The situation is very different for propagation across the 150 um thick slab First as shown in Fig 3c the central peak in the focal plane is broadened and the diffraction rings prominent at lower depths have virtually disappeared Similar losses in spatial con nement along the axial direc tion are shown in Fig 3d where the focus is much more extended and dispersed as compared to the focal eld produced after traversal across a thin slab Fig 3b Surprisingly while the rst diffraction maximum has broadened the phase pro le still retains its sharp features as seen in the inset of Fig 3c The wellbehaved phase pro le is achieved because the actual diffractionlimited Volume is the result of spatial interference of the inphase components of the focal eld The dispersion of the focal volume at greater depths is due to the greater attenuation z Hm FIG 3 color Focal eld amplitude for a 1 um thick slab a b and a 150 um thick slab cd The amplitudes in the focal plane xy are given in a and 0 whereas b and d depict the xz cross sections The insets show the phase pattern with values 0 blue and 7739 orange and have the same dimensions as the amplitude plots 0439033
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