Biomedical Optics

Chapter 7.2.3 - Sensing of Optical Properties and Spectroscopy: Error Expression

7.2.3.   Error Expression

While the signal is from the unscattered light, the unwanted bias is from the
scattered light. The unscattered light, assumed to be collimated, is completely
collected by the detector. The scattered light contains both single- and multiple-
scattered light. When μsd < 1, multiple-scattered light is negligible; hence, only
the single-scattered light is considered here. If g → 1, the number of single-
scattered photons can be estimated by N1 from Eq. (7.5). Thus, the relative error
due to the detected bias is approximately

 

where the numerator and denominator represent the numbers of received single-
scattered and unscattered photons, respectively. Although specular reflections
on both slab surfaces are neglected explicitly, the unscattered and the collected
single-scattered photons experience similar specular reflections. Therefore, the
contributions of specular reflections to the numerator and the denominator in
Eq. (7.11) partially offset each other.

Substituting the expressions for N0 and N1 from Eq. (7.5) into Eq. (7.11), we
obtain

 

Although derived using the preceding approximate analytical expressions,
Eq. (7.12) can be validated by the accurate Monte Carlo method (see Problem
7.3).

Substituting Eq. (7.9) into Eq. (7.12), we obtain

 

which is valid if both θ'd << 1 − g and 1 − g << 1 hold. Therefore, the sample
thickness must be limited on the basis of θ'd as well as the optical properties so that
the relative error is controlled within a given level. Although unknown initially,
μs can be first measured and then checked for error using Eq. (7.12). If the error
is unacceptable, one can modify the sample thickness—or the concentration of
scatterers if the sample is liquid—and repeat the measurement.

Example 7.2. Apply Eq. (7.13) to a realistic case. Known parameters include
nrel = 1.37, g = 0.99, and θd = 1 mrad (milliradian). Assume μs ≈ 100 cm-1.

If εr ≤ 1% is desired, Eq. (7.13) leads to d ≤ 0.037 cm.

 

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