The spatial backscattering impulse-response at length-scales shorter than a transport mean free path
is highly sensitive to the shape of the scattering phase function due to the fact that rays exiting within this regime have undergone relatively few scattering events. Typically this measurement is accomplished using a lens to focus the illumination beam to a near impulse on the sample surface. A CCD camera then detects the spatial distribution of backscattered light at the sample surface. With this design, the full azimuthal backscattering distribution can easily be measured for each polarization channel [1
]. Still, a true impulse can never be achieved and information at short length-scales is lost to an optical mask (~mm diameter) needed to reject specular reflections. Thus in media such as biological tissue with long
on the order of 1 mm, information about the shape of the phase function is extremely difficult to measure experimentally. As an alternative to overcome these difficulties, here we report a simple bench-top instrument using polarized enhanced backscattering (EBS) to first measure the backscattering distribution in the angular domain and subsequently convert to the spatial domain through inverse Fourier transformation.
EBS is an angular intensity peak centered in the exact backscattering direction that is formed by the summed diffraction pattern from all sets of time-reversed path-pairs within a scattering medium (a direct consequence of the reciprocity theorem) [2
]. Conceptually, it is useful to think of each time-reversed path-pair as a single Young’s double pinhole experiment. For a single time-reversed path-pair with rays exiting at a particular spatial separation, the diffraction pattern is the Fourier transform of two delta functions (cosine pattern). For a semi-infinite medium under plane wave illumination the distribution of spatial separations for all time-reversed path-pairs is given by the spatial backscattering impulse-response for multiply scattered light
in the exact backward direction p(x,y)
. As a result, in an idealized case the EBS peak IEBS
) is simply the Fourier transform of p(x,y)
. However, as we will demonstrate the dependencies are more complex:
denotes the Fourier transform operation, pc
specifies the degree of phase correlation between the forward and reverse path, s
represents the effective distribution of rays which remain within the illumination spot, c
is the spatial coherence function, and mtf
is the modulation transfer function of the optical instrument. Within the
represent instrinsic sample properties while s, c
, and mtf
represent extrinsic instrument properties. Each of these functions will described in further detail in the following paragraphs.
The experimentally measurable peff
is found by computing the inverse Fourier transform of the EBS peak:
For sample characterization, the functions p
are of prime interest. In principle these intrinsic parameters can be found by dividing peff
by functions s, c
, and mtf
. However, in practice this will amplify noise and a reliable result can be difficult to obtain. Alternatively, a more stable approach which we follow in this Letter is to convert theory into the experimentally observable peff
before comparison with experiment.
Our experimental instrument is shown in . An illumination source (xenon lamp with finite spatial coherence length Lsc or broadband laser with Lsc = ∞) is collimated by lens L1 and polarized by linear polarizer P before being directed onto the scattering sample via a non-polarizing 50/50 beam splitter B. Backscattered light is then collected by B and passes through a linear analyzer A before being focused onto a CCD camera with a lens L2. A liquid crystal tunable filter attached to the CCD separates the backscattered light into its component wavelengths. All polarization combinations are achieved by rotating P and/or inserting a quarter wave plate QWP between B and the sample.
EBS instrument. Collimating lens L1 (f = 200 mm), linear polarizer P, mirror M, beam splitter B, quarter wave plate QWP, linear analyzer A, Fourier lens L2 (f = 100 mm), liquid crystal tunable filter LCTF.
Information regarding the processing of experimental and Monte Carlo simulated EBS data can be found in detail in other publications [3
]. It should be noted that all experimental data is scaled by a constant multiplicative factor of 0.65 to obtain a full match with Monte Carlo simulation [4
]. This type of one parameter fit is common in EBS literature [5
]. In this Letter, all experiments and Monte Carlo simulations are for aqueous latex microsphere suspensions (10% solids by weight) which are diluted in DI water to achieve different values of
After retrieving the experimental EBS peak, the spatial distribution of backscattered light is computed using Eq. 2
. According to the properties of the discrete Fourier transform, the spatial resolution Δs
is related to the angular resolution Δθ
where λ is the illumination wavelength and n
is the number of pixels in the x
direction. Our instrument collects angles up to 7.5776° with 0.0074° resolution. Within the visible range of illumination this corresponds to a spatial extent of >1500 μm with <10 μm resolution.
Strictly speaking the reciprocity theorem is only fully valid for polarization preserving channels (i.e. linear co-polarized xx
or helicity preserving ++
]. For these channels, each ray has a time-reversed partner which exits with the same accumulated phase and pc
= 1 for all x,y positions. However, in the orthogonal polarization channels (i.e. linear cross-polarized xy
or opposite helicity +-
) not all scattering rotations are reversible. As a result, the forward and reverse paths may exit with different phases and pc
does not necessarily equal 1. Following the calculations of Ref. [6
in the orthogonal channels can be approximated from the degree of linear (
) and circular (
shows function pc
for the four measured polarization channels. For very short length-scales, rays undergo relatively few scattering events and pc
is nearly 1. As the exit radius increases, the average number of scattering events also increases. As a result, a greater proportion of rays travel through irreversible paths and pc
→ 0 as r
→ ∞ [2
]. shows the excellent agreement between experiment and simulation for peff
measured from each of the 4 polarization channels.
Fig. 2 Polarization sensitivities. (a) Function pc for the 4 measured polarization channels as a function of exit radius. (b) Experimental (symbols) and simulation (lines) peff. Sample : 0.65μm diameter sphere with
= 205 μm at 633 nm. Spot (more ...)
Since realistic experiments have finite illumination beam extent, an effective distribution of rays which remain within the illumination spot (and therefore possess a time-reversed partner) must be found. This can be obtained by computing the normalized autocorrelation function ACF
of the illumination pattern A
incident on the scattering sample [4
In our case, a circular beam is used and A is a top hat function (i.e. A = 1 within the illumination spot and 0 otherwise). For each spot size, s
begins at 1 and decreases to 0 when the path-pair separation is larger than the beam diameter. shows function s
for 2 illumination spot sizes. shows excellent agreement between the experiment and simulation peff
for 2000 and 7000 μm spot sizes.
Fig. 3 Illumination beam spot size sensitivities. (a) s as a function of exit radius for 2 beam spot sizes. (b) Comparison between experiment (symbols) and simulation (lines) for the different beam spot sizes using ++ polarization. Sample : 0.65 μm diameter (more ...)
Another consideration for the measurement of peff
is function c
which describes the ability for path-pairs to interfere when using a partial spatial coherence source. For a typical laser source, Lsc
is sufficiently large that over the extent of the beam spot size it can be considered spatially coherent (i.e. c = 1). However, even for illumination with very short Lsc
(e.g. sunlight or xenon source) an EBS peak is formed. In this case, function c
can be found according to the van-Cittert Zernike theorem. For our instrument, function c
takes the form of a first-order Bessel function of the first kind J1
shows function c
for illumination with 4 different Lsc
possible with our instrument. shows the agreement between the experiment and simulation peff
for each of these Lsc
Fig. 4 Sensitivities to the spatial coherence of the illumination. (a) Function c for different Lsc. (b) Comparison between experiment (symbols) and simulation (lines) for different Lsc using ++ polarization. Sample : 0.65 μm diameter sphere with
= (more ...)
For any imaging system the mtf describes the ability to capture information from different spatial frequencies. As such, the measurement of peff will also be modulated by the mtf. To determine our system’s mtf (shown in ) we computed the Fourier transform of the point spread function measured by placing a mirror in the sample plane and imaging the laser onto the CCD.
As a final demonstration of the sensitivity of our technique to fine details in the scattering phase function, we measure two suspensions of microspheres with the same
(1000 μm) and anisotropy factor g
(0.87) but different size parameter ka
is the wavenumber in water and a
is the microsphere radius). displays the two samples’ phase functions. Although they are generally similar in shape, the sample with higher ka
has many higher frequency oscillations in accordance with Mie theory. In we demonstrate that these higher frequency oscillations have a profound effect on the shape of peff
at short length-scales. These slight differences are readily distinguishable using the technique presented in this Letter.
Fig. 5 Sensitivity of EBS to the phase function. (a) Phase function for two microsphere samples with the same g but different ka. (b) Corresponding peff for ++ polarization shows a large difference in shape at short length-scales. ka ~ 4 : 0.65 μm diameter (more ...)
In summary we have presented EBS as a useful method to measure the backscattering impulse-response at small length-scales where information about the phase function is preserved. We first discussed the various sample dependent and experimental variables which affect the measurement and then provided a demonstration of its extraordinary sensitivity to the shape of the phase function.