Effect of index of refraction mismatch on the recovery of optical properties of cylindrical inhomogeneities in an infinite turbid medium

Optical inhomogeneities embedded in a turbid medium are characterized not only by their absorption and reduced scattering coefficients, but also by their index of refraction relative to the background medium. Although in diffusion theory it is impossible to separate the index of refraction from the absorption and reduced scattering coefficients in an infinite homogeneous medium, application of boundary conditions for an inhomogeneity adds enough information to separately determine these optical properties. A mismatched index of refraction affects diffuse photon propagation in two ways: photons travel at a different speed inside the inhomogeneity, and photons entering and leaving the inhomogeneity are influenced by Fresnel reflections at the surface of the object. We have integrated these two effects into the analytical solution to the diffusion equation for a cylinder in an infinite medium. Theoretical results are compared with experimental data, and the effect of index of refraction mismatch is evaluated for different combinations of optical properties.


INTRODUCTION
Index of refraction (n) of tissues is experimentally important in the determination of optical properties such as specular reflectivity, angular change in beam direction at tissue interfaces, and acceptance angles of optical fibers in tissue. The index of refraction has been determined for many different human tissues with a majority of values ranging from n= 1 .35to n=l .45 at a wavelength of 632.8 nm.1'2'3'4 Values for tissue indices of refraction have both been measured in vitro using a quartz optical fiber', and estimated based on the water content of tissue. 4 These different values of n not only affect light propagation at the surface of diffuse media but also at any inhomogeneity inside the medium.
While the effect of an index of refraction mismatch at the surface of a turbid medium has been studied, little experimental work has been done to verify the effect of an index mismatch at a boundary of an inhomogeneity inside a turbid medium. Haskell et al.5 predicted that the index of refraction will modify boundary conditions inside a turbid medium such that only the normal component of the flux will be continuous across boundaries, but not the fluence rate. Some of the light incident on the boundary will undergo Fresnel reflection depending on the index of refraction mismatch between the object and background media. We have included this modified boundary condition in the analytical solution for a cylinder in an infinite turbid medium and compare our theoretical solution with experimental data.

THEORY
We can describe the propagation of photons in a scattering medium with the Boltzmann transport equation. This equation can be simplified by a diffusion approximation to yield, in the frequency domain, for homogeneous media:6 (v2 + k2 )?(r,w) = - (1) where, * Further author information S.A.W. (correspondence): Email: sawalker@uiuc.edu; WWW: http:llwww.physics.uiuc.edu/groups/fluorescence/ phone: (217)244-5620; Fax (217)244-7187 k2 a (2) vD Here c1(r, 0)) is the photon fluence rate, o is the angular frequency of source modulation, q(r, co) is the source power per unit volume, and V C I n is the speed of light in the scattering medium (n is the index of refraction of the medium), J.ta is the absorption coefficient and D=lI(3j.t)7 is the diffusion coefficient (jig is the reduced scattering coefficient of the medium). The general form of the solution for a point source at position r5 is: (r,r,€o) = exp(ikoutIr_rsI) (3) 4icDrwhere S is the power of the source in photons/s. If we wish to solve the Helmholtz equation (Eci 1) in the presence of an infinite cylinder we can expand the general solution (Eq. 3) in terms of modified Bessel functions.ö The fluence rate inside the cylinder ( ()), and the fluence rate resulting from the scattering of the incoming wave by the cylinder ( scatt) will also be expanded similarly. This procedure follows the approach presented by other researchers for spherical We can determine the coefficients in the expansion by imposing the following boundary conditions (p=a represents the cylinder surface):5 1. scatt as p-4O 2. 'T? is finite everywhere (5) 3.
( (6) 4. D0 -4'out = -at p=a (7) p Where the subscripts "out' and "in" indicate whether the corresponding quantity is evaluated outside or inside the cylinder respectively. Here, p is the radial distance from the center of the cylinder, and a is the radius of the cylinder. R12 and R2 1 effective reflection coefficients describing the fraction of emitted light that is reflected either out of or into the cylinder at a boundary due to Fresnel reflections (For the index of refraction mismatch in our experiment cy1inder1 .8 vs. rbackgroundl33 R120.45 and R21=O.057). Boundary condition 3 shows a discontinuity in the fluence rate at the object surface which depends on the effective reflection coefficients R12 and R21.5 The discontinuity in 1 is due to the fact that not all of the light incident on the cylindrical boundary is transmitted into (or out of) the cylinder.
After matching the expanded solutions using the boundary conditions above one can arrive at an analytical solution for the case of an infinite cylinder in a turbid medium. This solution accounts for the effects of the index of refraction mismatch between the cylinder and the background medium in two ways: first the speed of light is changed in the inhomogeneity, and second Fresnel reflections cause a discontinuity in the fluence rate across the boundary between the cylinder and the background medium.

MATERIALS AND METHODS
Experimental measurements were conducted in a quasi-infinite geometry using a frequency domain spectrometer and an XYZ positioning scanner. A 120 MHz radio frequency signal was amplified and sent to a 50 mW laser diode (2=793 nm) coupled to a 1 mm fiber optic conduit to channel the near infrared light into the turbid medium. The coupling efficiency with the laser diode was 20% giving an output power of about 10 mW. Detected light was collected by a 0.3 cm diameter fiber optic bundle and processed with frequency domain methods to measure the DC intensity, AC amplitude, and phase of the photon density wave. Measurements took place inside a large glass container of Intralipid fat emulsion mixed with black India ink. The volume of the container was 16 L. The concentrations of Intralipid and India ink were adjusted to give background optical coefficients .taO=O.O79 cm4, t5o =8.0 cm4 as measured by the multidistance protocol.11 The index of refraction of the medium is 1.33 (water).
To mimic infinitely long cylinders, we cast 5 cylinders each 10 cm long with 3 different radii (0.25 cm, 0. 5 Table 1 . Measured values of absorption and reduced scattering coefficients and index of refraction for experiment materials. Cylindrical inhornogeneities were made from hot melt glue, and the background medium was Intralipid fat emulsion mixed with India ink. The value for index of refraction of the background medium is the value for water.
Each cylinder was made from a large block of Hot Melt glue which was characterized using a semi-infinite medium multidistance 12 Due to the optical coefficients of the background medium, the probability of a photon traveling from source to detector about the end of a cylinder is negligible. Thus we treat each cylinder as being infinitely long.
We performed an experiment to measure the index of refraction of the glue. After forming the clear glue into a right angle prism we directed light from a He-Ne laser (2=632.8 nm) into the prism and measured the critical angle for total internal reflection. From this measurement we employed Snell's law to extract the index of refraction of the glue.
A set of experiments was performed to compare the theoretical model with experimental data. A single cylinder was located midway between source and detector, (as shown in Fig. 2). 4.1 Fit of 1-ia ' s ' and n for a given cylinder radius Table 2 shows the values of absorption and reduced scattering coefficient that we recover assuming no index of refraction mismatch between the cylinders and the background medium.  Using boundary conditions that do not take into account Fresnel reflections, it is virtually impossible to separate the effect due a change in the index of refraction from changes in the absorption and scattering parameters.4 In this case the independently measured quantities are essentially np. and Ja " since these variables appear together in the diffusion approximation to the Boltzmann, transport equation. Hence, the same reduced 2 values will be obtained for different combinations of values of jta ' Fts , and n.  Table 3. Best fit values of absorption and reduced scattering coefficients and index of refraction of cylindrical inhomogeneities (see table 1  In general, independently measured values of the absorption and reduced scattering coefficients as well as the index of refraction of each cylinder are recovered within experimental error. The reduced 2 value for each fit serves as a measure of goodness of fit (experimental errors are 0.2% error in the AC counts and 0. 1 degrees in the phase). Each cylinder is assumed to be at a known position with a vertical orientation. Note that the improvement in the reduced 2, with the introduction of Fresnel reflections, is largest for the less absorbing cylinders of material type A-. The actual plot of experimental and theoretical values is shown for the 1.5 cm diameter cylinder of material type A-in Fig. 2 (a) and (b).

CONCLUSION
Previously, highly scattering inhomogeneities have been characterized by two parameters, their absorption and reduced scattering coefficients. The index of refraction of an inhomogeneity could not be independently determined from optical measurements in a highly scattering medium. Any error in the assumed index of refraction affected the recovered values of absorption and reduced scattering coefficients because detected photons traveled at a different than expected speed inside the inhomogeneity. However, there is an added effect due to an index of refraction mismatch: photons entering and leaving the inhomogeneity are also influenced by Fresnel reflections at the surface of the object. This surface effect effectively sets photon migration theory apart from the diffusion theory of other non-interacting particles, such as neutrons, and allows us a possibility for independently determining the index of refraction of an inhomogeneity. In this paper, we have corrected the boundary conditions for the diffusion approximation to the Boltzmann transport equation to include Fresnel surface reflections in the case of a mismatch in the refractive index between an infinite cylinder and its surrounding medium. These new boundary conditions lead to better fit of experimental data than with the old boundaiy conditions without introducing any significant complication. The model including Fresnel reflections allows us to separate the effects of a material's index of refraction from its absorption and scattering coefficients, effectively creating another parameter which should be separately determined to fully characterize the optical properties of a highly scattering inhomogeneity.