## 5. Analysis: (A, μ) --> (μ |

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A simple analytic expression was developed to approximate the behavior of the data, SIGNAL(z_{f}). The expression is:

SIGNAL = MEASUREMENT/P_{o} = μ_{s} L_{focus} f exp(-a μ_{s} z_{f} 2 G)

where

- P
_{o}= total power delivered by confocal microscope [W].

The experiment also measured the reflectivity from a water/glass interface to allow calibration of the MEASUREMENT of reflected power. Hence, the term P_{o}was accounted for. The MEASUREMENT was normalized by P_{o}to yield the SIGNAL data reported here. Hence P_{o}is not included in the analysis.

- a = function of the anisotropy of scattering, g, which diminishes the effectiveness of the scattering coefficient μ
_{s}[dimensionless].

For isotropic scattering (g = 0), a(g) = 1. As scattering becomes forward-directed (g < 0), a(g) drops toward zero.

The function a(g) was determined by Monte Carlo simulations of the transport of light to the focus as a function of g and z_{f},

A exp(-μ_{s}z_{f}a G)

which omits the effect of f since this describes transport TO the focus, not backscatter FROM the focus.

- μ
_{s}= scattering coefficient [cm^{-1}]

- L
_{focus}= axial length of the focal volume in the tissue over which the transported fluence is scattered.

This is an approximation, which used L_{focus}= the distance between zero points in the axial Airy function (0.8 μm). This region is expected to dominate the scattering process.

- f = function of g, which describes the fraction of light undergoing scattering that
**backscatters**within the cone of collection of the lens system of the confocal microscope to yield observed reflectance.

The behavior of f(g) was determined by integrating the Henyey-Greenstein scattering function over the solid angle of collection of the lens aperture (half angle = 0-42 ° for NA=0.90 water-immersion lens). If g is close to 0, the value of f is maximum due to strong backscatter. As g drops and light becomes forward-directed, the value of f drops rapidly.

- 2 = the factor that accounts for the double pass IN/OUT of the tissue, as light propagtes down to the confocal volume then backscatters to the surface for collection.

- G = the geometrical factor that describes the increased average photon pathlength from the surface to the focus since the light is delivered as a focused Gaussian beam rather than as a narrow collimated beam orthogonal to the surface. The value of G varies with the numerical aperture of the lens used in the experiment. In our case, the value of G was 1.15.

The behavior of f is expressed as f/f

A water-immersion lens was used, and this analysis has neglected the slight effect of the water-tissue refractive index mismatch.