Abstract:
Imaging systems may include a light source that directs light towards a scattering medium. A digital imaging system receives transmitted light that is transmitted through the scattering medium substantially without being scattered, and also receives multiple components of scattered light that are passed through the scattering medium after being scattered. The digital imaging system then outputs image intensity information related to the transmitted light and the scattered light. An imaging controller determines a property of the scattering medium, based on spatial correlations related to the image intensity information.

Description:
STATEMENT OF GOVERNMENT INTEREST 
       [0001]    The following description was made in the performance of official duties by employees of the Department of the Navy, and, thus the claimed invention may be manufactured, used, licensed by or for the United States Government for governmental purposes without the payment of any royalties thereon. 
     
     TECHNICAL FIELD 
       [0002]    This description relates to characterization of media scattering properties. 
       BACKGROUND 
       [0003]    Conventional methods exist to determine various media scattering properties, and, in particular, to determine optical properties of scattering media. Scattering media include, for example, fluid media in which particles are distributed or suspended. For example, particles suspended in a fluid may scatter incident light according to a certain angular distribution, and/or may attenuate transmission of the incident light to an identifiable extent. As a result, an intensity distribution or other characteristics of light passing through such a scattering media may be affected, and these effects may be measured and analyzed. 
       SUMMARY 
       [0004]    According to one general aspect, a system includes a light source that is operable to direct light towards a scattering medium. The system also includes a digital imaging system that is operable to receive transmitted light that is transmitted through the scattering medium substantially without being scattered, and to receive interference effects among multiple components of scattered light that are passed through the scattering medium after being scattered thereby, and furher operable to output image intensity information resulting from interference among such components. The system also includes an imaging controller that is operable to determine a property of the scattering medium, based on spatial correlations related to the image intensity information. 
         [0005]    According to another general aspect, a sequence of images that are formed as a result of source light that has traveled through a scattering medium having particles suspended therein is received, the images containing image intensity information for each of the images. Spatial correlations between intensities of at least two points in each image are determined, and effects of transmitted light that is passed through the scattering medium substantially without scattering are accounted for, based on the spatial correlations. Interference effects of interference among components of scattered light that is scattered by the particles are analyzed, based on the spatial correlations, and a property of the scattering medium is determined, based on the accounting for the effects of transmitted light, the analyzing of the interference effects, and on the spatial correlations. 
         [0006]    According to another general aspect, an imaging controller includes an imaging system controller that is operable to determine imaging characteristics of a digital imaging system at which a detecting system detects both scattered light from a light source that has passed through a scattering medium and transmitted light from the light source that has passed through the scattering medium. The imaging controller further includes an image analyzer that is operable to receive image intensity data from the digital imaging system, determine spatial correlations that are associated with the image intensity data, based on interference between components of the scattered light, and further based on the transmitted light, and determine a property of the scattering medium, based on the spatial correlations. 
         [0007]    The details of one or more implementations are set forth in the accompanying drawings and the description below. Other features will be apparent from the description and drawings, and from the claims. 
     
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]      FIG. 1  is a block diagram of an imaging system for determining media scattering properties. 
           [0009]      FIG. 2  is a block diagram of an implementation of the imaging system of  FIG. 1 . 
           [0010]      FIG. 3  is a flowchart illustrating processes associated with an operation of the systems of  FIGS. 1  or  2 . 
           [0011]      FIG. 4  is a flowchart illustrating examples of operations of the systems of  FIGS. 1  or  2 . 
           [0012]      FIG. 5  is a flowchart illustrating techniques for determining properties of a scattering medium. 
       
    
    
     DETAILED DESCRIPTION 
       [0013]      FIG. 1  is a block diagram of an imaging system  100  for determining media scattering properties. In  FIG. 1 , a scattering medium  102  is placed between a light source  104  and digital imaging components  106 . The digital imaging components  106  detect sequences of digital intensity image data over a period of time for output to an imaging controller  108 . By analyzing the sequences of digital intensity image data, the imaging controller  108  is able to determine various properties, including scattering and other optical properties, of the scattering medium  102 . 
         [0014]    As part of such determinations, the imaging controller  108  is operable to remove unwanted components of light received at the digital imaging components  106 . For example, some of the light output by the light source  104  may be transmitted through the scattering medium  102  and other elements of the system  100  without being scattered. Such transmitted light may be unnecessary or detrimental to the analysis and characterization of the scattering and other optical properties of the scattering medium  102 . 
         [0015]    Additionally, some of the light that is output by the light source  104  may be scattered by elements of the system  100  other than the scattering medium  102 . Such light may be referred to as stray light, and, like the transmitted light just referenced, may be unnecessary or detrimental to the operation of the system  100 . 
         [0016]    The imaging controller  108  is operable to account for such unwanted light components as, for example, the transmitted light and the stray light. As a result, the system  100  is operable to analyze a wide range of types of scattering media, and to determine scattering and other optical properties thereof, with a minimal number of components being required by the system  100 . That is, for example, the system  100  may not require, but may use in some implementations physical components for removing the effects of the transmitted light and/or the stray light. 
         [0017]    In some implementations, the scattering medium  102  may include a fluid in which certain types of particles are suspended. In such cases, the imaging system  100  may be operable to determine scattering and other optical properties of the scattering medium  102  over a wide range of fluid types and characteristics, and for a variety of types of the suspended particles. 
         [0018]    In particular, and as discussed in more detail below, some examples of the scattering medium  102  in which particles are suspended in fluid may be assumed to scatter light from the light source  104  weakly. That is, light from the light source  104  may be assumed to be scattered at most once during its transmission through the scattering medium  102  before passing through to the digital imaging components  106 . 
         [0019]    Such an assumption may be made, for example, when the scattering medium  102  is “optically thin.” For example, the scattering medium  102  may be considered to be optically thin when including a fluid with a relatively low density of scattering particles, or when a path length L of the scattering medium  102  is limited to a certain distance. 
         [0020]    In other implementations, such an assumption may not be safely made, e.g., when the scattering medium  102  is considered “optically thick.” In these implementations, due to, for example, a relatively large density of suspended particles, or a relatively long path length L, multiple scattering may occur within the scattering medium  102 . 
         [0021]    That is, light that is incident on, and scattered by, a first particle within the scattering medium  102  may subsequently be incident on, and scattered by a second particle (and, subsequently, a third, fourth, or more particles) within the scattering medium  102 . The imaging controller  108  is operable to account for such multiple scattering within the scattering medium  102 . As a result, the system  100  is operable to analyze a wide range of such types of scattering media, and to determine scattering and other optical properties thereof. 
         [0022]    In the system  100 , a window  110  is placed between the scattering medium  102  and the light source  104 . An aperture  112  having a diameter D 1  is defined by system components  114 , such that the light from the light source  104  passes through the aperture  112  prior to being transmitted through the window  110 , and, subsequently, through the scattering medium  102 . 
         [0023]    An assembly  116  is placed on the opposite side of the scattering medium  102  from the window  110 , as shown. As described in more detail below, the assembly  116  and the window  110  may be movable relative to one another. As a result, an effective thickness, volume, and/or density of the scattering medium  102  may be determined in a desired manner. Thus, the path length L through the scattering medium  102  may be determined. 
         [0024]    Within the assembly  116 , a mask  118  may be used to define a near-field region  120 . Light from the light source  104  thus passes through the window  110  and the scattering medium  102 , and may converge at least in part at the near field region  120  for detection by the digital imaging components  106 . 
         [0025]    In some implementations, a distance of the near-field region  120  from the scattering medium  102  and/or the light source  104  is such that all desired light that is scattered by the scattering medium  102  is captured by a detector or detector array associated with the digital imaging components. Typically, such capture may be performed without movement of the detector or detector array relative to the scattering medium  102 . In these implementations, calibration of multiple detectors or detector arrays relative to one another may not be required, and, similarly, complications resulting from movement of the detector or detectors arrays may be avoided. 
         [0026]    The digital imaging components  106  are operable to detect sequences of images, and, more particularly, are operable to detect intensity data with respect to such images. In so doing, operations of the digital imaging components  106  may be controlled by a digital imaging components (DIC) controller  122  within the imaging controller  108 . For example, the DIC controller  122  may be operable to determine a number of images within a particular sequence, an exposure time of each image within the sequence, and/or a time interval in between each consecutive pair of images within the sequence. The image sequences, and information about the image sequences, may be stored within an image sequences database or memory  124  within the imaging controller  108   
         [0027]    Then, also within the imaging controller  108 , an image analyzer  126  is operable to receive the image data from the memory  124 , or directly from the DIC controller  122 , and determine various properties of the scattering medium  102 , including, for example, properties of particles suspended within the scattering medium  102 . For example, the image analyzer  126  may determine an extinction coefficient  128  of the scattering medium  102 , an angular distribution  130  of light scattered by the scattering medium  102 , or diffusion rates  132  of the suspended particles. These properties are discussed in more detail below, and other properties of the scattering medium  102  also may be determined using the system  100 . 
         [0028]    In performing the analyses necessary to determine the properties  128 ,  130 , and  132 , the image analyzer  126  may use information related to physical or properties of the system  100 . For example, a diameter adjuster  134  may be used to change and/or determine a diameter of the aperture  112  (e.g., by adjusting the components  114 ). A light source adjuster  136  may be used to change and/or determine properties of the light source  104 . Also, a path length adjuster  138  may be used to change and/or determine the path length L of the scattering medium. Of course, other adjusters besides the adjusters  134 ,  136 , and  138  may be used, and, moreover, the adjusters  134 ,  136 , and  138  may be partially or wholly located outside of the imaging controller  108 . 
         [0029]      FIG. 2  is a block diagram of an implementation  200  of the imaging system  100  of  FIG. 1 . In  FIG. 2 , the sample scattering medium  102  is placed between the window  110 , for receiving the illumination beam from the light source  104  as described above, and the assembly  116 . 
         [0030]    In the implementation of  FIG. 2 , the assembly  116  includes a light shield  202  with an attached extension window  204 . The assembly  116  includes a microscope objective  206 , which, in the representation of  FIG. 1 , is part of the digital imaging components  106 . 
         [0031]    The sample region, that is, the region containing the scattering medium  102 , may be open for sampling ambient scattering media, or may represent a container for holding samples for in vitro measurements. The path length L may be varied as referenced above by a micrometer screw  208 , which moves the end of the assembly  116  toward or away from the window  110 . The end of the assembly  116  may be attached to a suitable mounting, such as, in the example of  FIG. 2 , a translation stage  210 , which allows the motion toward or away from the window  110 . The shield  202  is designed to extend sufficiently, for example, by incorporating a bellows  212 , to allow the motion of the end of assembly  116 . In some implementations, the shield  202  need not be attached to the extension window  204 . The shield  202  may instead be fixed in place as the end of the assembly  116  extends to reduce the path length L in the scattering medium  102 , between the window  110  and the extension window  204 . 
         [0032]    A laser  214  provides illumination as part of the light source  104 . An intensity of the laser  214  may be stabilized and/or monitored using an intensity monitor  216 . The intensity monitor  216  in  FIG. 2  employs a beamsplitter  218 , and a detector within the intensity monitor  216 . 
         [0033]    The illumination intensity of the laser  214  may be adjusted by, for example, a circular linear-wedge neutral-density filter  220 . Light from the laser  214  may be spatially filtered, expanded, and collimated with a spatial filter assembly  222  to produce a uniform illumination over the diameter of the aperture  112  in front of the window  110 . 
         [0034]    In  FIG. 2 , laser illumination from the laser  214  enters the sample region containing the scattering medium  102 , where a portion is transmitted through, and a portion is scattered by, the scattering medium  102 . Transmitted laser illumination and scattered light that reaches the extension window  204  in the assembly  116  is transmitted by the extension window  204  toward the microscope objective  206 , and at least a portion passes through the planar near-field region  120 . 
         [0035]    In some implementations, the diameter of the region  120  may be limited to a size of a field of view of the microscope objective  206 . The size and shape of the region  120  may further be limited by other components of the imaging system  200 , as described in more detail below. 
         [0036]    In the implementation of  FIG. 2 , the microscope objective  206  may be an oil immersion type with an infinity conjugate, so that its front focal plane also is its object plane. A high numerical aperture and high angular aperture may be used to obtain scattered light angular distribution over a large angular range, and to improve the system transfer finction. The sine of the angular aperture is equal to the numerical aperture divided by the refractive index of the immersion oil for which the objective is designed. 
         [0037]    Immersion oil may be used to fill the volume within assembly  116  between the extension window  204  to the microscope objective  206 . In some implementations, a material for the extension window  204  may be a glass that matches the refractive index of the immersion oil for which the objective  206  is designed. In these cases, the angular aperture within the extension window  204  may therefore be the same as the angular aperture of the microscope objective  206 . 
         [0038]    In some implementations, a diameter of an aperture  224  in the shield  202  where the extension window  204  may attach may be one sufficient to allow full illumination of the numerical aperture of the microscope objective  206  over the region  120 . In some implementations, a diameter of the aperture  112  may be at least as large as needed to provide full illumination of the numerical aperture of the microscope objective  206  when the refractive index of the scattering medium matches the refractive index of the immersion oil. 
         [0039]    If the diameter of the aperture  112  is at least this large, then the range of measurement may only be limited by the angular aperture of the microscope objective  206 . However, for scattering media  102  with a smaller refractive index, a larger angular range of measurement may be possible, if, for example, the diameter of the aperture  112  is larger. For example, if the diameter of the aperture  112  is large enough, the numerical aperture of the microscope objective  206  divided by the refractive index of the scattering medium  102  is the sine of the maximum scattering angle that can be measured. 
         [0040]    In other implementations, where the diameter of the aperture  112  is not as large, e.g., if the refractive index of the scattering medium  102  is equal to or smaller than the numerical aperture of the microscope objective  206 , then the maximum angular range of measurement of the scattered light distribution may be smaller. The maximum angular range of measurement of the scattered light distribution may be, for example, from 0 degrees (forward scattering) to at least the arctangent of the difference between the diameters of the apertures  112  and of the near-field region  120 , divided by twice the sum of the path length L and the illustrated distance Z, as shown in Eq. (1): 
         [0000]      θ max =arctan[( D 1 −Dp )/2( L+Z )]  Eq. (1) 
         [0041]    In Eq. (1), D 1  is the diameter of the aperture  112 , Dp is the diameter of the region  120 , and Z is the distance between this region and the far surface of the extension window  204 . As L decreases and Z increases, the actual angular range may become larger. 
         [0042]    The diameter D 1  of the aperture  112  may be changed in a repeatable manner to facilitate measurement of the extinction coefficient  128 . For example, the collimation lens of the spatial filter assembly  222  may be a zoom lens capable of adjusting a size of the illumination beam of the laser  214  in proportion to the chosen diameter D 1  of the aperture  112 , thereby maintaining efficient illumination of the aperture. When the diameter D 1  of the aperture  112  is set at a minimum slightly larger than the diameter Dp of the region  120 , the intensity of scattered light reaching the region  120  may be reduced without changing an intensity of the transmitted light that reaches the region  120  without scattering, as referenced above. 
         [0043]    The ability to modify the diameter D 1  of the aperture  112  (e.g., using the diameter adjuster  134 ) allows improved discrimination between the effects of the scattered light and the transmitted light, and also allows the laser intensity to be increased. Both effects may be useful, for example, in measuring the transmitted intensity as the optical thickness of the scattering medium  102  increases (so that the transmission consequently decreases). Thus, the aperture  112  may be set to the diameter D 1  when measuring the angular distribution of scattered light and particle difflusion rates, and reduced to the minimum diameter, as needed, to aid in extinction coefficient measurement. 
         [0044]    When measuring scattering angular distribution with the aperture  112  set to the diameter D 1 , some additional scattered light may enter through the extension window  204  in a direction that does not take the scattered light through the region  120 . Such light may be considered unwanted noise, and the mask  118  prevents the unwanted light from reaching the microscope objective  206  either directly or after reflecting one or more times from the surfaces of the extension window  204  or other parts of the shield  202 . In some implementations, the mask  118  may be placed around the region  120  in the front focal plane of the microscope objective  206 . Light entering the extension window  204  that misses the region  120  is intercepted directly by the mask  118 , and light that passes through the region  120  at an angle too large to be accepted by the microscope objective  206  either is absorbed by the objective  206  or reflected back into the mask  118 . 
         [0045]    Further in  FIG. 2 , a positive lens  226  and a digital imaging system  228  connected to a computer  230 , where the computer  230  may implement some or all of the imaging controller  108 . In some implementations, the lens  226  may be a part of the digital imaging system  228 , or, in the context of  FIG. 1 , may be a part of the digital imaging components  106 . 
         [0046]    A distance of the lens  226  behind the microscopic objective  206  may be flexible. For example, when an infinite conjugate objective is used, this distance need only be small enough that the light arriving at the lens  226  from the objective  206  can pass through an aperture that is small enough to make aberrations insignificant for the purposes of measurement. The digital imaging system  228  may include a detector or detector array  232  positioned at a focal plane of the lens  226 . The lens  226  and the detector array  232  may be centered so that their respective optical axes correspond to that of the objective  206 . 
         [0047]    In  FIG. 2 , the infinity conjugate microscope objective  206  accepts and collimates light from the region  120 , and the lens  226  focuses that light on the detector array  232 , forming an image of the region  120  on the detector array  232 . In some implementations, the image may fill the detector array  232 . The choice of the focal length of the lens  226  may allow such implementations to be realized, based on a size of the detector array  232  and an angular field of view of the microscope objective  206 . 
         [0048]    The largest dimension of the detector array  232  (for example, the diagonal) divided by the focal length may be equal to twice the tangent of the angle from the center to the edge of the field. For example, if a specification of the microscope objective  206  provides a field number and designed tube lens focal length of the objective  206 , then twice the tangent of the angle from the center to the edge of the field is equal to the field number in millimeters divided by the focal length in millimeters of the tube lens for which the objective  206  is designed. 
         [0049]    In some implementations, though, an exact match may not be used. In these implementations, then the focal length of the lens  226  may be larger, or the size of the detector array  232  may be smaller, and a somewhat smaller portion of the region  120  may be used. In these implementations, the region  120  may be regarded as the image of the detector array  232  made by the lens  226  and the microscope objective  206 . The magnification may be given by the ratio of the microscope objective  206  focal length divided by the lens  226  focal length. Therefore, the diameter of the aperture  224  in the shield  202 , and the size of opening in the mask  118 , may be adjusted to accommodate actual dimensions of the region  120 . 
         [0050]    Continuing the example in which the objective  206  is an infinity conjugate objective, the resolution of the detector array  232  may be given by a minimum pixel spacing equal to a wavelength in air of the laser  214 , divided by four times the numerical aperture of the microscope objective  206 , and multiplied by the ratio of the focal length of lens  226  to the focal length of the microscope objective  206 . For example, a green He-Ne laser has a wavelength in air of 0.5435 micrometers. If the numerical aperture of the microscopic objective is 1.20, and a focal length of the microscopic objective is 3.6 millimeters, then a lens  226  focal length of 210 millimeters would produce a minimum pixel spacing of 6.61 micrometers. In other implementations, a larger pixel spacing may be used. 
         [0051]    The computer  230  may be connected to the intensity monitor  216  to receive a readout of the setting of the circular linear-wedge neutral-density filter  220 , the path length L, and the diameter D 1  of the aperture  112 . The computer  230  also may be used to control the acquisition of images by the digital imaging system  228 , as described above, including, for example, an exposure time of individual images and/or time between images, and also may be operable to store and analyze the sequences of images taken, as illustrated above with respect to  FIG. 1  and described in more detail, below. 
         [0052]      FIG. 3  is a flowchart illustrating processes associated with an operation of the system  100  or the system  200 , or one or more of the other variations thereof just described, or other implementations described in more detail, below. In  FIG. 3  and in the following equations, the resulting data to be analyzed includes sequences of digital intensity image data of the form I(m x , m y ; j), representing the intensity value recorded at position (x(m x ), y(m y ))=(m x Δx, m y Δy) and time t j =jΔt, with m x  and m y  being integer column and row numbers of the pixel position, and j being the number of the image in the time sequence. 
         [0053]    The parameters Δx, Δy are the effective pixel spacing in the x and y directions (e.g., the pixel spacing in the x and y directions of the detector array  232  divided by the ratio of the focal length of lens  226  to the focal length of the microscope objective  206 ), and the parameter At represents the time interval between successive sample images. In the following equations, spatial averaging is indicated by enclosing the quantity averaged in angle brackets (                     
         [0000]    ), and time averaging is indicated by overbars. 
         [0054]    Thus, an average intensity image obtained by averaging of a sequence of images at each pixel may be shown as in Eq. (2): 
         [0000]          I ( m   x   ,m   y ) =    I ( m   x   ,m   y   ;j )   Eq. (2) 
         [0000]    and the average over all pixels of the time-average image may be shown as in Eq. (3): 
         [0000]                    I             =           I( m   x   ,m   y   ;j )             Eq. (3) 
         [0055]    An autocorrelation image sequence may be expressed as a sequence of spatial averages of the products of intensity at pixel (m, n) multiplied by the intensity at the shifted pixel (m+Δm, n+Δn), if the autocorrelation of an image is dependent only on the shift (Δm, Δn). An example of such an autocorrelation image sequence is shown in Eq. (4): 
         [0000]      R 1 (Δm x ,Δm y ;j)≡         I(m x ,m y ; j)I(m x +Δm x ,m y +Δ m   y ;j)           Eq. (4) 
         [0056]    If needed, corrections for dark current can be obtained ( 302 ) from a sequence of intensity images, D(m, n; j), measured with no laser illumination, as shown in Eqs. (5) and (6): 
         [0000]    
       
         
           
             
               
                 
                   
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         [0057]    A sequence of intensity images may be used to calculate a time-averaged intensity image, the spatial average of the time-averaged intensity image, and a time-averaged correlation image ( 304 ), as shown in Eqs. (7), (8), and (9), respectively: 
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                     9 
                     ) 
                   
                 
               
             
           
         
       
     
         [0058]    In Eqs. (6) and (9), the intensity autocorrelation image sequence is calculated first, and then the sequence is averaged to produce the time average of the individual intensity autocorrelation images. Conversely, an autocorrelation image of the average intensity image of Eq. (7) may be computed ( 306 ) as shown in Eq. (10): 
         [0000]        R     I   raw   (Δ m   x ,Δm y )=             I   raw   ( m   x   ,m   y )    I   raw   ( m   x +Δm x   ,m   y +Δ m   y )           Eq. (10) 
         [0059]    The equations for using the dark current corrections are shown in Eqs. (11), (12), and (13): 
         [0000]                    I             =             I   raw             −             D               Eq. (11) 
         [0000]          R   I (Δ m   x ,Δ m   y ) =    R   I   raw ( 66   m   x ,Δ m   y )  −    R   D (Δ m   x ,Δ m   y ) −2         Ī                       D               Eq. (12) 
         [0000]        R Ī(Δ m   x ,Δ m   y )= R   I raw   (Δ m   x Δ m   y )−           D             2 − 2           Ī                     D             Eq. (13) 
         [0060]    In the above, Eqs. 11 and 12 provide information about the intensity fluctuations resulting from the combination of transmitted laser illumination and total scattered light falling on the region  120  ( 308 ). As described, the total scattered light includes light scattered by the scattering medium  102  in the volume between the window  110  and the assembly  116 , and the stray light scattered by imperfections in the rest of the instrument. When the fluctuations resulting from the light scattered by the medium are uncorrelated in amplitude and phase from one image to the next, this decorrelation can allow separation of the effects of the two scattered light contributions ( 310 ). In this case, Eq. (13) may provide an estimate of the unchanging contribution to the fluctuations that results from the stray light. 
         [0061]    The contributions of the scattered light of interest and the stray light may be separated using different techniques. For example, an estimate of the average intensity            B z,≦ resulting from the total scattered light that passes through the region  120  to the microscope objective  206  and recorded may be written as in Eq. (14): 
         [0000]                  B           =         Ī( m   x   ,m   y ) 2 −[  2{overscore (I )}( m   x   ,m   y ) 2 −R I (0,0) ] 1/2             Eq. 14) 
         [0062]    An estimate of the intensity A of the transmitted laser illumination that passes without scattering may be written as in Eq. (15): 
         [0000]        A =             I             −             B               Eq. (15) 
         [0063]    An estimate of the average intensity          N          resulting from the stray light component of the total scattered light may be written as in Eq. (16): 
         [0000]                  N             =             Ī ( m   x   ,m   y ) 2  −[2 Ī ( m   x   , m   y ) 2   −R   Ī (0,0)] 1/2             Eq. (16) 
         [0064]    As a result, an estimate of the average intensity            S            resulting from light scattered by the scattering medium  102  and recorded may be written as in Eq. (17) ( 312 ): 
         [0000]                    S             =             B             −           N             Eq. (17) 
         [0065]    An estimate of the intensity autocorrelation image resulting from the stray light component may be written as in Eq. (18) ( 314 ): 
         [0000]        R   N (Δ m   x ,Δ m   y )= R     I-S   (Δ m   x ,Δ m   y )= R Ī(Δ m   x ,Δ m   y )+           S             2 −2           S                     Ī           Eq. (18) 
         [0066]    With the above results for intensity fluctuation statistics, various scattering properties may be calculated. For example, the transmitted laser illumination intensity A is related to the incident intensity A 0 , the extinction coefficient ξ, and the path length L through the sample by the Bouguer-Lambert-Beer law, as shown in Eq. (19): 
         [0000]      A=A 0 e −ξL    Eq. (19) 
         [0067]    As described above, the systems  100  and/or  200  of  FIGS. 1  and/or  2  provide for determination of the path length L. A calibration to determine A 0 , or a set of measurements of A(L) as a function of varying path lengths L with the illumination kept constant (or adjusted for drift of the laser intensity as monitored by the intensity monitor IM) allows for the solution of Equation 19 for the extinction coefficient ξ. 
         [0068]    Angular scattering properties of the scattering medium  102  may be obtained from the Fourier transformation of the amplitude autocorrelation function, J s (Δm x , Δm y ), which is given by Eq. (20): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             J 
                             s 
                           
                            
                           
                             ( 
                             
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   m 
                                   x 
                                 
                               
                               , 
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   m 
                                   y 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               
                                 
                                   
                                     R 
                                     I 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         Δ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           m 
                                           x 
                                         
                                       
                                       , 
                                       
                                         Δ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           m 
                                           y 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 _ 
                               
                               - 
                               
                                 
                                   〈 
                                   
                                     B 
                                     _ 
                                   
                                   〉 
                                 
                                 2 
                               
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               
                                 R 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   
                                     Δ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       m 
                                       x 
                                     
                                   
                                   , 
                                   
                                     Δ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       m 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 〈 
                                 N 
                                 〉 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     20 
                     ) 
                   
                 
               
             
           
         
       
     
         [0069]    The right-hand side of Eq. (20) is the difference between J b (Δm x , Δm y ), the total scattered light amplitude autocorrelation function, and J n (Δm x , Δm y ), the stray light amplitude autocorrelation function, given respectively by Eqs. (21) and (22): 
         [0000]        j   b ( Δm   x   ,Δm   y )=√{square root over ( R   I ( Δm   x   ,Δm   y )−           B             2 )}− A    Eq. (21) 
         [0000]        J   n (Δ m   x   ,Δm   y )=√{square root over ( R   N ( Δm   x   ,Δm   y )−           N             2 )}− A    Eq.(22) 
         [0070]    Computation of the angular distribution of scattered light from an amplitude correlation function estimate is illustrated below for the case of the sample scattered light amplitude correlation finction obtained from Eq. (20). The stray light angular distribution computation is analogous. 
         [0071]    To compute the angular distribution of scattered light from the amplitude correlation function(s), the spatial frequency distribution σ(q x , q y ), of the scattered light component from the volume of the scattering medium  102  may be calculated using a two-dimensional discrete Fourier transformation, illustrated in Eq. (23): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           σ 
                            
                           
                             ( 
                             
                               
                                 q 
                                 x 
                               
                               , 
                               
                                 q 
                                 y 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         Re 
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           [ 
                           
                             
                               ∑ 
                               
                                 
                                   Δ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     m 
                                     x 
                                   
                                 
                                 = 
                                 0 
                               
                               
                                 
                                   M 
                                   x 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   
                                     Δ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       m 
                                       y 
                                     
                                   
                                   = 
                                   0 
                                 
                                 
                                   
                                     M 
                                     y 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   
                                     J 
                                     s 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         Δ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           m 
                                           x 
                                         
                                       
                                       , 
                                       
                                         Δ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           m 
                                           y 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       
                                         i 
                                          
                                         
                                           ( 
                                           
                                             
                                               Δ 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 m 
                                                 x 
                                               
                                                
                                               
                                                 υ 
                                                 x 
                                               
                                             
                                             + 
                                             
                                               Δ 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 m 
                                                 y 
                                               
                                                
                                               
                                                 υ 
                                                 y 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       
                                         M 
                                         x 
                                       
                                        
                                       
                                         M 
                                         y 
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     23 
                     ) 
                   
                 
               
             
           
         
       
     
         [0072]    In Eq. (23), the function Re[ ] represents the real part of the argument, while M x  and M y  are the respective maximum values of the integer shifts Δm x  and Δm y , v x  is an integer between −M x /2 and M x /2, v y  is an integer between −M y /2 and M y /2, i is the square root of −1, and e is the base of the natural logarithms. On the left-hand side of Eq. (23), the discrete spatial frequency components q x  and q y  are given by Eqs. (24) and (25), respectively: 
         [0000]    
       
         
           
             
               
                 
                   
                     q 
                     x 
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           υ 
                           x 
                         
                       
                       
                         
                           M 
                           x 
                         
                          
                         Δ 
                          
                         
                             
                         
                          
                         x 
                       
                     
                     ≡ 
                     
                       
                         υ 
                         x 
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         q 
                         x 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     24 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     q 
                     y 
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           υ 
                           y 
                         
                       
                       
                         
                           M 
                           y 
                         
                          
                         Δ 
                          
                         
                             
                         
                          
                         y 
                       
                     
                     ≡ 
                     
                       
                         υ 
                         y 
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         q 
                         y 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     25 
                     ) 
                   
                 
               
             
           
         
       
     
         [0073]    Eqs. (24) and (25) also define the spacing in spatial frequency, Δq x  and Δq y , of the points (q x , q y ). The function σ(q x , q y ) can be corrected for the modulation transfer function of the microscope objective  206 , or for the overall system transfer function of the imaging system  100  and/or  200 , or variations thereof, obtained by a suitable calibration procedure using Eq. (26): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       σ 
                       c 
                     
                      
                     
                       ( 
                       
                         
                           q 
                           x 
                         
                         , 
                         
                           q 
                           y 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       σ 
                        
                       
                         ( 
                         
                           
                             q 
                             x 
                           
                           , 
                           
                             q 
                             y 
                           
                         
                         ) 
                       
                     
                     
                       MTF 
                        
                       
                         ( 
                         
                           
                             q 
                             x 
                           
                           , 
                           
                             q 
                             y 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     26 
                     ) 
                   
                 
               
             
           
         
       
     
         [0074]    In Eq. (26), σ c (q x , q y ) is the corrected function, and MTF(q x , q y ) is a modulation transfer function of the microscope objective  206 . As just referenced, an overall instrument function can be substituted for the modulation transfer function in Eq. (26), if such a finction has been determined by suitable calibration. 
         [0075]    The relationship between points (q x , q y ) in spatial frequency and angular scattering variables (θ, φ) is given by Eqs. (27) and (28), respectively: 
         [0000]    
       
         
           
             
               
                 
                   θ 
                   = 
                   
                     arc 
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   q 
                                   x 
                                   2 
                                 
                                 + 
                                 
                                   q 
                                   y 
                                   2 
                                 
                               
                               ) 
                             
                             
                               1 
                               / 
                               2 
                             
                           
                           
                             
                               n 
                               m 
                             
                              
                             
                               k 
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     27 
                     ) 
                   
                 
               
             
             
               
                 
                   ϕ 
                   = 
                   
                     
                       arc 
                        
                       
                           
                       
                        
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               q 
                               x 
                             
                             
                               q 
                               y 
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       ϕ 
                       0 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     28 
                     ) 
                   
                 
               
             
           
         
       
     
         [0076]    In Eq. (28), φ 0  represents the angle between the desired reference azimuth and the detector array axis chosen as the x direction. In some implementations, it may be assumed that scattering by the scattering medium  102  is symmetric with respect to a change of φ by π radians. 
         [0077]    For diffusion rate calculations, it may be convenient to express the scattering distribution as a function of the momentum transfer wavevector Q instead of the polar angle θ . The momentum transfer wavevector is defined as the difference between the wavevectors of the incident and scattered light. The magnitude Q of Q in terms of the polar angle may be given by Eq. (29): 
         [0000]    
       
         
           
             
               
                 
                   Q 
                   = 
                   
                     2 
                      
                     
                         
                     
                      
                     
                       n 
                       m 
                     
                      
                     
                       k 
                       0 
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           θ 
                           2 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     29 
                     ) 
                   
                 
               
             
           
         
       
     
         [0078]    The characteristic time for intensity fluctuation patterns to become uncorrelated in amplitude and phase between images, as required for validity of Eq. (13), is given by Eq. (30): 
         [0000]    
       
         
           
             
               
                 
                   
                     t 
                     d 
                   
                   = 
                   
                     1 
                     
                       DQ 
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     30 
                     ) 
                   
                 
               
             
           
         
       
     
       In Eq. (30), D represents the difflusion coefficient. 
       [0079]    For Brownian motion of 1-micron radius spherical particles in water (viscosity=0.01 poise) at 300 Kelvin, D=2.2×10 −9  cm 2 /sec. Eq. (30) may be used to provide an indication of the maximum exposure time for each intensity fluctuation image when Q corresponds to the largest scattering angles to be measured, which, in some implementations, may be limited to no more than π/2 radians. 
         [0080]    Eq. (30) also may provide a minimum time between exposures when Q corresponds to the minimum scattering angle to be resolved. This minimum time can be estimated by setting Q equal to Δq x  or Δq y  in Eq (30). Diffusion rates can be obtained with Eq. (30) by experimentally determining one or both of these limiting times by making a comparison of the results of analyzing different sequences of intensity fluctuation images where the exposure time and/or the time between exposures is varied. 
         [0081]    In some implementations, the above calculations may be somewhat differently expressed as a calculation of a relationship between the intensity correlation and the amplitude correlation with respect to the near field scattering measurements. That is, the transmitted incident light field and the scattered light field may be treated as two components, and then the field amplitude correlation of the scattered field may be calculated from the measurements of the intensity fluctuations. The transmitted incident light field may include the effects of fixed sources of stray light. 
         [0082]    In this type of analysis, and using capital letters for intensity values and lower case letters for corresponding amplitudes, an intensity that is resultant from a combination of two fields may be expressed as in Eq. (31): 
         [0000]        I=|a+b|   2    Eq. (31) 
       Eq. (31) maybe re-written as Eq. (32): 
       [0083]        I=A+B+ 2Re(ab*)   Eq. (32) 
         [0084]    Then, the expected ensemble averaged and time averaged intensity may be shown as in Eqs. (33), (34), and (35), respectively: 
         [0000]                  I             =             |a+b|   2             Eq. (33) 
         [0000]      Ī=  |a+b| 2     Eq. (34) 
         [0000]                    I             =           |a+b| 2               Eq. (35) 
         [0085]    Using known relationships between moments of intensity of the sum of a speckle field (as the scattered field may be considered) and a constant field, the respective mean intensity contributions, A and &lt;B&gt;, to the speckle pattern resulting from the combination of a constant transmitted field with a speckle field from scattered light may be determined using Eqs. (36) and (37): 
         [0000]        A =[2           I             2   −             I   2           ] 1/2    Eq. (36) 
         [0000]                  B             =             I           −[2           I             2   −             I   2           ] 1/2    Eq. (37) 
         [0086]    The result of Eq. (37) may be obtained from Eq. (32) using the fact that A=|a| 2  is constant, while b is a circular Gaussian random variable with zero mean. The averages indicated may be obtained, by example, from pixel-by-pixel averages of measured intensity and measured intensity squared over a sufficiently long sequence of images that have statistically independent realizations of the speckle field. The constant field may not be independent of position, but may be constant in time. 
         [0087]    To obtain the best estimate of            B            and A ij  for pixel i, j, the time average of the intensity Ī ij  and squared intensity  I ij   2    for all pixels, may be determined, and then the estimates of          B ij            may be calculated from Eq. (39) using the time averages as estimates of the ensemble averages. Finally, if the scattered light is assumed to be statistically stationary, then the estimates          B ij            of          B          may be averaged over all pixels. The resulting estimate of            B            may then be used with Eq. (32) to recalculate the estimates of A ij  from the per-pixel intensity time averages. 
         [0088]    In order to obtain the mutual intensity of the speckle field, the spatial intensity correlation may be used, as shown in Eq. (38): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             R 
                             
                               ( 
                               
                                 I 
                                 - 
                                 A 
                               
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 1 
                               
                               , 
                               
                                 
                                   y 
                                   1 
                                 
                                 ; 
                                 
                                   x 
                                   2 
                                 
                               
                               , 
                               
                                 y 
                                 2 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           〈 
                           
                             
                               ( 
                               
                                 
                                   I 
                                   1 
                                 
                                 - 
                                 
                                   A 
                                   1 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   I 
                                   2 
                                 
                                 - 
                                 
                                   A 
                                   2 
                                 
                               
                               ) 
                             
                           
                           〉 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 〈 
                                 
                                   [ 
                                   
                                     
                                       B 
                                       1 
                                     
                                     + 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         Re 
                                          
                                         
                                           ( 
                                           
                                             
                                               a 
                                               1 
                                             
                                              
                                             
                                               b 
                                               1 
                                               * 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       B 
                                       2 
                                     
                                     + 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         Re 
                                          
                                         
                                           ( 
                                           
                                             
                                               a 
                                               2 
                                             
                                              
                                             
                                               b 
                                               2 
                                               * 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 〉 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 
                                   〈 
                                   
                                     
                                       B 
                                       1 
                                     
                                      
                                     
                                       B 
                                       2 
                                     
                                   
                                   〉 
                                 
                                 + 
                                 
                                   2 
                                    
                                   
                                     〈 
                                     
                                       
                                         
                                           B 
                                           1 
                                         
                                          
                                         
                                           Re 
                                            
                                           
                                             ( 
                                             
                                               
                                                 a 
                                                 2 
                                               
                                                
                                               
                                                 b 
                                                 2 
                                                 * 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       B 
                                       2 
                                     
                                      
                                     
                                       Re 
                                        
                                       
                                         ( 
                                         
                                           
                                             a 
                                             1 
                                           
                                            
                                           
                                             b 
                                             1 
                                             * 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   〉 
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 4 
                                  
                                 
                                   〈 
                                   
                                     
                                       Re 
                                        
                                       
                                         ( 
                                         
                                           
                                             a 
                                             1 
                                           
                                            
                                           
                                             b 
                                             1 
                                             * 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       Re 
                                        
                                       
                                         ( 
                                         
                                           
                                             a 
                                             2 
                                           
                                            
                                           
                                             b 
                                             2 
                                             * 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   〉 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     38 
                     ) 
                   
                 
               
             
           
         
       
     
         [0089]    The expectation value in the middle term of the last equality of Eq. (38) may be shown to be zero using, for example, an expression for a joint probability density function for intensity and phase. 
         [0090]    If the fields are stationary, the intensity autocorrelation may be rewritten as a function of the difference vector (Δx, Δy)=(x 2 −x 1 , y 2 −y 1 ), and the first term may be written as the intensity autocorrelation of the speckle field, yielding Eq. (39): 
         [0000]        R   (I−A) (χ 1 ,y 1   ;χ   2   ,y   2 )= R   B (Δχ ,Δy )+4           Re ( a   1   b   1   * ) Re ( a   2   b   2   * )   Eq. (39) 
         [0091]    A relation between the intensity autocorrelation and the amplitude autocorrelation, or mutual intensity J b (x 1 , y 1 ; x 2 , y 2 )=(b 1 b 2   * ) of a complex circular Gaussian speckle field that allows information about the field autocorrelation of the scattered light to be derived from the measurement of the intensity and its autocorrelation is shown in Eq. (40): 
         [0000]        R   B (χ 1   ,y   1 ;χ 2   ,y   2 )=           B   1                       B   2             +|J   b (χ 1   ,y   1 ;χ 2   ,y   2 )| 2    Eq. (40) 
         [0092]    Evaluating the expectation taken in the last term of Eq. (39) yields Eq. (41): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
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                     ( 
                     41 
                     ) 
                   
                 
               
             
           
         
       
     
         [0093]    The averages in terms dropped within the development of Eq. (41) may be shown to have zero expectation. The two remaining terms can be written in terms of the mutual intensity J b . 
         [0094]    Then, substituting Eqs. (40) and (41) into Eq. (39) leads to Eqs. (42) and (43): 
         [0000]        R   ΔI (χ 1   ,y   1 ;χ 2   ,y   2 )=           B             2   +|J   b (χ 1   ,y   1 ;χ 2   ,y   2 )| 2 +2 Re[a   1   *   a   2   J   b (χ 1   ,y   1 ;χ 2   ,y   2 )]  Eq. (42) 
         [0000]        R   ΔI (χ 1   ,y   1 ;χ 2   ,y   2 )+ A   1   A   2   −             B             2   =|a   1   a   2   *   +J   b (χ 1   ,y   1 ;χ 2   ,y   2 )| 2    
         [0000]      [ R   ΔI (χ 1   ,y   1 ;χ 2   ,y   2 )+ A   1   A   2   −             B             2 ] 1/2   =|a   1   a   2   *   +J   b (χ 1   ,y   1 ;χ 2   ,y   2 )|  Eq. (43) 
         [0095]    In Eq. (42), the terms R ΔI  (x 1 , y 1 ;x 2 ,y 2 ) and          B           2  represent terms that may be calculated from the measurements taken by the computer  230 . The remaining two terms of the first line of Eq. (42), both written in terms of the mutual intensity J b , represent terms referred to as the homodyne and the heterodyne components, respectively. 
         [0096]    That is, the homodyne component |J b  (x 1 , y 1 ; x 2 , y 2 )| 2  represents components of the scattered light resulting from scattering of light from the laser  214  or other light source  104 , and, in particular, interference between such components. In contrast, the heterodyne component 2Re[a 1   * a 2 J b (x 1 , x 2 ; y 1 , y 2 )] is related to interference between the transmitted light that is not scattered by the scattering medium  102  interfering with components of the scattered light. 
         [0097]    The homodyne component may become small or negligible relative to the heterodyne component when, for example, a density of the scattering medium  102  is low, and/or in other situations in which an amount of scattered light is small relative to the transmitted light. Conversely, the heterodyne component may be removed, for example, either by a high density scattering medium  102  or by a beam stop or other physical mechanism for preventing the transmitted light from reaching the detector array, or by a high density of scattering material minimizing the transmission of light without scattering, i.e., the factors a 1   *  a 2  each approach zero. 
         [0098]    Using the above Eqs. (31)-(42), perhaps in conjunction with some or all of the Eqs. (1)-(30) above, the spatial correlation function of the electric field at the detector array may be determined from the intensity variations measured there. For example, the expressions            
       (I 1 −A 1 )(I 2 −A 2   
       [0099]    in Eq. (38) represent the intensity variation at two pixels in a given image. Thus, a correlation of the intensity variation as a function of positions of a pair of pixels [x 1 , y 1 ; x 2 , y 2 ] may be determined without requiring an assumption that the stray light component of the total scattered light is statistically spatially stationary. 
         [0100]    As a result, with respect to Eq. (42), the final line of Eq. (42) may be solved for the spatial correlation fimction of the electric field by, for example, suitable averaging or estimation techniques to eliminate measurement uncertainties and/or unknown phase factors in the stray light terms. For example a maximum likelihood estimate may be made of the combination of electric field autocorrelation J b (x 1 , y 1 ; x 2 , y 2 ) and constant factors a 1 *a 2  that produced the sequence of images, taking advantage of the fact that Jb may be dependent on its two position arguments (x 1 , y 1 ) and (x 2 , y 2 ) only through the difference (Δx, Δy). 
         [0101]      FIG. 4  is a flowchart illustrating examples of operations of the systems of  FIGS. 1  or  2 , using selected ones of the Eqs. (1)-(42). Using the example of  FIG. 2 , calibration of the system  200  may be implemented and automated through the use of the computer  230  ( 402 ). For example, making a measurement with the system  200  with no scattering medium being present (or with a blank medium) and allowing storage of the resulting intensity distribution will allow the user to have a comparative system baseline. 
         [0102]    Periodically, the user will be able to acquire an intensity distribution with no scattering medium, for comparison with the system baseline that has previously been stored. This comparison may generally require that these two measurements be performed with the same system settings. Finding the difference between these two distributions and analyzing the resultant signal may allow the user to monitor the status of the system and facilitate detection of any changes that may possibly alter the ability of the system to perform an accurate measurement. 
         [0103]    Before each series of measurements, it would be desirable for the user to construct a series run baseline ( 404 ) to use for comparison with the previous baseline(s) and/or subsequent experimental measurements. The new baseline may be used to aid in the analysis of the data and to allow a check for any discrepancies within the system  200 . At the discretion of the user during routine use, this series baseline comparison may be performed when the user feels a check is required or suspects the equipment is operating in a faulty manner. 
         [0104]    If certain changes have been made to the optical setup, such as, for example, changing certain optical components, adjusting any optical settings, or changing any beam diameters, a new system baseline measurement should be performed to allow the user to determine the status of the system. If any discrepancies are noted, the necessary adjustments may be made. 
         [0105]    During normal use, a sample is introduced into the volume between the window  110  and the assembly  116  ( 406 ). The path length L is adjusted as desired ( 408 ), and the intensity of the laser  214  is then adjusted ( 410 ) to provide adequate intensity for the digital imaging system  228 , without causing saturation. 
         [0106]    The computer  230  then commands acquisition of a sequence of images at an exposure time t 1  and a time between exposures t 2 , and stores the sequence for analysis ( 412 ). Details of the settings, including exposure time, and time between exposure, depend on the particular set of measurements desired. The following discussion includes examples for determining settings and measurements that may be needed for a given application. 
         [0107]    As should be understood from  FIG. 1 , the path length L, the source light parameters, and the image acquisition settings all may be controlled by the corresponding modules of the imaging controller  108 , as described therein. In  FIG. 2 , such modules may be included within, and/or implemented by, the computer  230 , and/or by other, separate hardware or software components. 
         [0108]      FIG. 5  is a flowchart illustrating techniques for determining properties of the scattering medium  102 . For example, such techniques may be employed as part of the path length adjustments ( 408 ), source light adjustments ( 410 ), or the acquiring, storing, and analyzing of the intensity image data itself ( 412 ), depending on, for example, which property of the scattering medium is to be determined ( 502 ). 
         [0109]    For example, to measure an extinction coefficient  4 , the diameter of the aperture  112  may be reduced to limit the scattered light recorded in the image sequences ( 504 ). Also, the size of the illumination beam from the spatial filter assembly  222  may be proportionally reduced by adjusting the zoom setting to increase the intensity of illumination ( 506 ). 
         [0110]    The exposure time ti may be set so as to be short compared with the minimum diffusion time expected ( 508 ), and the time between exposures t 2  may be set to be long compared with the longest diffusion time expected ( 510 ). In some implementations, image sequences of approximately 100 images are used. 
         [0111]    A first sequence may be determined using a desired concentration of the scattering medium  102 , with the path length L set to a suitably large value, e.g., its maximum travel ( 512 ). A suitably large path length L may be determined to be one that achieves an optical thickness (the product ξL in the exponent of Eq. (19)) satisfactorily large compared with the uncertainties of the measurements. In some implementations, the optical thickness may not be so large that the measurement breaks down because of excessive multiply-scattered light. 
         [0112]    A second series may be acquired with a significantly smaller optical thickness ( 514 ), either by, for example, reducing the path length L, or by using a blank sample in place of the scattering material. Additional sequences at intermediate values of the path length L ( 516 ) may allow determination of the extinction coefficient from a straight-line fit of the logarithm of the transmitted light A(L) plotted against the path length L ( 518 ). Deviations from a straight line at large values of L may indicate breakdown of the measurement because of excessive multiply-scattered light. Results at those large values of L may be discarded. 
         [0113]    For a measurement of the angular distribution of scattered light, the aperture a1 may be set to a relatively large diameter ( 520 ), such that any particles away from the optical axis capable of scattering light in the direction of the region  120  may be illuminated by the incident beam. Also, the spatial filter assembly  222  may be set so that the illumination is uniform over the aperture  112  ( 522 ). The exposure time, time between exposures, and number of images needed are determined in the same way as for the measurement of an extinction coefficient ( 524 ). Sequences of more than 100 exposures may reduce the variance of the measurements. Alternatively, if lower angular resolution is acceptable, the variance also may be reduced by smoothing the results. 
         [0114]    Depending on the optical thickness chosen for the measurement ( 526 ), a measurement of the angular distribution of scattered light for a sample may allow determination of the point-spread function ( 528 ) or phase function ( 530 ) of the sample. To obtain the phase function, the optical thickness must be small compared with 1. A phase function measurement may thus utilize a small setting for the path length L, a dilute sample, or both. A proper normalization of the angular distribution obtained from the measurement may yield the phase function. 
         [0115]    More generally, the angular distribution measurement is the point-spread function less the distribution of the transmitted light, which is known from, for example, the illumination system characteristics and the results of Eq. (15). The point-spread function is dependent on the scattering material and the optical thickness. Therefore, a length of the path length L may be chosen that results in the optical thickness representative of conditions for which the point-spread finction is desired. 
         [0116]    Diffusion rates may be obtained by determining the effect of reducing the time between exposures when measuring the angular distribution of scattered light ( 532 ), or of increasing the exposure time of the individual images ( 534 ). In the first case, one approach is to perform the angular distribution measurement with additional exposures made by decreasing the time between exposures and increasing the number of exposures to maintain the same overall acquision time. The results obtained from subsets of exposures with larger and smaller times between exposures may then be compared. 
         [0117]    For example, if the time between exposures is reduced by a factor of 5, 500 exposures may be acquired instead of 100. A subset comprising every fifth exposure is used for 1 calculation of the angular distribution and compared with the results from a subset comprising the first 100 exposures. The spatial frequency at which the measurement based on the second subset deviates significantly from the measurement based on the first indicates the diffusion rate. Other settings will be similar to the settings used for a phase function determination. Suitable calibration may allow difflusion rate measurements for samples of higher optical thickness. 
         [0118]    The effects of increasing the exposure time of the individual images ( 540 ) may be evaluated with two or more recorded sequences. In one sequence, the settings should be those appropriate for sequences used in measuring angular distribution of scattered light. In the rest of the sequences, the settings should be the same except that the exposure time should be increased from one sequence to the next, so that there are two or more sequences, each with a fixed exposure time different from that of the others. The analysis technique for determining the diffusion rate is similar to that for analyzing the effect of shorter time between exposures described above. In this case, the deviation between measurements can be expected to start at higher spatial frequencies for small increases in exposure time, and progress to lower spatial frequencies as the exposure time increases further. In the previous case, the deviations start at lower spatial frequencies and appear at higher and higher spatial frequencies, as the time between exposures is reduced. 
         [0119]    Data useful for retention as baseline or calibration records include measurements as described in the above examples for a suitable standard sample, such as a suspension of known materials, particle size, shape, and concentration. Such reference standards are available from, for example, suppliers that use a repeatable synthesis technique. Polystyrene microsphere suspensions in water of a specified, uniform size are an example of an available reference standard material. In addition, the time-average images may be retained along with the supporting data, including settings, needed for the stray light calculations in Eqs. (16), (18), and (22). 
         [0120]    This information may be useful for monitoring changes in the stray light and/or in establishing the extent to which the stray light signal is repeatable. For example, the stray light effects may be repeatable over a series of measurements where certain settings are unchanged. Combining these retained records may then improve the accuracy of the stray light corrections or reduce the number of images required for a sequence. 
         [0121]    These features may be applicable for at least two of the examples above: extinction measurements made at fixed settings by varying the particle concentration of the sample, and diffusion rate measurements using sequences where the setting change is only to the exposure time of individual images. Additional gains are possible if these records establish that stray light effects are repeatable if the same settings are used as for previous measurements. For example, the stray light results may be the same whenever the setting of the path length L is returned to a previously used value. 
         [0122]    The techniques of  FIGS. 3-5  may be implemented in other variations of the systems of  FIG. 1  and/or  2 . For example, in some other implementations, the microscope objective  206  may be fitted with a mask in the form of a straight-edged mask located in the back focal plane of the microscope objective  206 . Such a mask may be used, for example, by adjusting the position of the edge with a micrometer to bisect the back focal plane at the center of the focused transmitted beam on the optic axis of the microscope objective  206 . The described analysis of the angular distribution of scattered light from the intensity fluctuation measurements may be modified to account for the elimination of negative spatial frequencies in one direction. Such an alternative allows the distance Z from the scattering sample to the region  120  to be reduced without complicating the calibration of the measurement, and allows use of scattering media having relatively large optical thicknesses than may otherwise be available. 
         [0123]    In an alternative arrangement, illumination from the opposite side of the scattering medium  102  may be used to measure angular distribution of backscattered light about the direction of retroreflection (180 degrees). For example, the laser  214 , spatial filter assembly  222 , and intensity monitor  216  with beamsplitter  218  may be arranged such that the beamsplitter  218  is placed between the microscope objective  206  and the lens  226  to direct one beam axially into the back of the microscope objective  206 . 
         [0124]    In such implementations, the aperture  112  may not be needed. Instead, for example, a focusing system may be placed between the spatial filter assembly  222  and the beamsplitter  218  so that the illumination forms a cone of light with its vertex in the back focal plane of the microscope objective  206 . After passing the back focal plane, the cone of light expands again and is then collimated by the microscope objective  206 . 
         [0125]    The focusing system may be designed to compensate for aberrations caused by the beamsplitter  218  and the microscope objective  206  when used for collimating a source in the back focal plane instead of in the front focal plane discussed with respect to  FIGS. 1 and 2 . The focusing system may have an entrance pupil large enough to make the collimated light beam diameter as large as possible upon exiting the end of the microscope objective  206  in the direction of the front focal plane. The collimated light beam diameter may match the size of the aperture  112  at the end of the microscope objective  206 , and a long working distance microscope objective should be used. This and similar implementations also may be used, for example, to analyze light from randomly rough surfaces of solids. 
         [0126]    A maximum thickness of fluid samples may be smaller than the working distance of the microscope objective  206  when the maximum angular range of backscatter from the retroreflection direction is to be measured. The spatial filter assembly  222  may have a fixed focal length collimating lens instead of a zoom lens. The size of the expanded beam may be sufficient to provide uniform illumination within the entrance pupil of the focusing system. This arrangement can be combined with the use of a mask, as described in implementations, above. 
         [0127]    Dependence of scattering on polarization may be obtained with the addition of polarizing optics and a second imaging system. For example, a polarizing beamsplitter may be placed between the microscope objective  206  and the positive lens  226 . The second output of the beamsplitter  218  illuminates another positive lens and digital imaging system. A polarizer in front of the aperture  11   2  polarizes the light and illuminates the scattering medium  102 . 
         [0128]    In another implementation, the window  10  may be made movable to adjust the path length L. In this case, the window  110  may be the bottom of a vessel for containing the scattering medium  102 . The vessel may be set on top of the stage of a digital microscope, and the collimated laser illumination may be introduced from beneath the stage, directed upwards by a mirror, for example. 
         [0129]    The assembly  116  may be a simple canister forming the shield  202 , with the extension window  204  at the bottom and the mask  118  inside. The assembly  116  may be made to fit over a microscope objective and fixed in place, so that the bellows  212  or translation stage  210  is not required. The assembly  116  and the vessel with the window  110  may be attachments for a digital microscope that has a means for directing the illumination upwards from under the stage through the window  110 . 
         [0130]    The digital microscope in these implementations may substitute for the combination of microscope objective  206 , lens  226 , and digital imaging system  228 . Other components may be arranged as described above. In these implementations, the components of a digital imaging system may be implemented as a conversion for a digital microscope that can be attached and removed again, allowing the digital microscope to be used for multiple purposes. 
         [0131]    In another implementation, reflecting surfaces parallel to the collimated laser illumination may be used to confine the scattered light, from the window  110  to the near-field region  120 . These surfaces would form a prism with a cross section matching the area of the region  120  that images on the two-dimensional detector array  232 . Within the scattering medium  102 , between the window  110  and the extension window  204 , a reflecting coating may be applied. From the extension window  204 , the reflecting surfaces may be used as an interface between a glass prism and air, and the mechanism of reflection may be total internal reflection. In this implementation, the outer shield  202  of the assembly  112  may not be required. These implementations may allow a smaller illuminating beam to be used, resulting in high light efficiency, and reducing a size of the spatial filter assembly  222 . 
         [0132]    Imaging systems as described herein allow characterization of various scattering properties of suspensions of small particles within a scattering medium. These properties may include, as in the examples above, the extinction coefficient, the effect on scattering of particle diffusion rates in the medium (and from the scattered intensity distribution as a function of angle), the volume scattering function, and/or the point-spread function versus optical thickness of the sample. 
         [0133]    The imaging systems described herein, and related implementations, thus provide advantages including the ability to obtain stray light information (i.e., stray light due to imperfections or surface contamination of components of the optical train) directly from the sequence of image data made with the sample of scattering material to be measured, so that separate measurements of these factors may not be required. Other advantages may include insensitivity to nonuniformity in the response of the detector array, good angular resolution, and flexibility in the positioning of the region  120  where the measurements are made. 
         [0134]    In addition, such imaging systems allow the combined use of intensity fluctuations resulting from interference between different components of the scattered light and fluctuations resulting from the interference of components of the scattered light with transmitted light. As a result, minimal system elements may be used, e.g., a beam stop in the back focal plane of the microscopic objective may not be required. Also, the optical thickness of the scattering medium need not be limited to values smaller than one scattering mean-free path length. Rather, measurements can be made with samples at higher concentrations of scattering particles, longer path lengths, or both, resulting in significantly greater optical thickness. Specifically, for example, scattering extinction coefficients and point-spread functions can be obtained for samples of optical thickness greater than 1. 
         [0135]    Further advantages include the ability to adjust the path length L within the scattering medium in order to make measurements at a desired optical thickness, or at more than one value of optical thickness without changing the sample concentration. Also, optical thickness may be measured, along with particle diffusion times. Further, when the positive lens  226  is a part of the sensor assembly  116 , the resulting imaging system may allow flexibility in choosing the separation of the lens  226  and the assembly  116  from the microscope objective  206 . 
         [0136]    As a result, the described imaging systems and variations thereof may be used to characterize optical properties of scattering media. Optical characteristics may be discovered that could be exploited for detection of specific types of particles, such as, for example, biological agents. Also, types of particles may be distinguished from other particles, e.g., background particulates. Additional examples of applications include monitoring of optical properties of scattering media. In ocean optics, for example, the local point-spread finction could be monitored in aid of improving imaging through turbid water. 
         [0137]    A number of implementations have been described. Nevertheless, it will be understood that various modifications may be made. Accordingly, other implementations are within the scope of the following claims.