6.9 Problems.
6 :1 Consider the interface between air (nair= 1.00) and samples of the following
three materials: hemoglobin (nHb= 1.37), dentin (nd= 1.50), and tooth
enamel (nt= 1.73). Show that the Brewster angles are 53.9°, 56.3°, and
60.0°, respectively.
6 :2 Verify the plots of the parallel and perpendicular reflection coefficients
shown in Fig.6.3for light incident from air onto a material that has a
refractive index of 1.33.
6 :3 Consider a dermis skin tissue sample in which the absorption coefficient is
12 cm−^1 at 308 nm and 2.7 cm−^1 at 633 nm. Show that the absorption
lengths for each of these wavelengths at which I(x)/I 0 = 1/e = 0.368 are
0.833 mm and 3.702 mm, respectively.
6 :4 Consider an aorta tissue in which the absorption coefficient is 2.2 cm−^1 at
1320 nm. Show that the penetration depth is 4.55 mm.
6 :5 Because of its varying structure the absorption coefficient of skin changes
with depth. For depths of 1 mm or less the absorption coefficient at 633 nm
is 0.67 cm−^1 , for depths between 1 and 2 mm the absorption coefficient at
633 nm is 0.026 cm−^1 , and for depths > 2 mm the absorption coefficient at
633 nm is 0.96 cm−^1. Let a collimate light beam with a power level of 100
mW be incident on a thick skin tissue sample. If scattering effects are
ignored, show that the optical powers at depths of 1, 2, and 3 mm are 93.5,
93.2, and 84.6 mW, respectively.
6 :6. Consider a 1-mm thick opticalfilter that has a 10 cm−^1 absorption coefficient
at a HeNe laser wavelength of 633 nm. Suppose a collimated 5 mW light
beam is incident perpendicular to the surface of thefilter. Show that the
power level of the attenuated beam that emerges from thefilter is 1.84 mW if
the attenuation is due only to absorption.
6 :7. In the visible spectrum, the Rayleigh scattering effect is significantly smaller
in the red region (around 680 nm) compared to the blue region (around
450 nm). Show that the ratio Is/I 0 of Rayleigh scattering is 5.2 times smaller
at 680 nm than at 450 nm.
6 :8. Verify the values of the Rayleigh scattering cross section and the geometric
cross section given in Example 6.7.
6 :9. Consider a spherical particle that has a radiusa= 10 nm, a refractive index
ns= 1.57, and that is in a background material of refractive index nb= 1.33.
Show that the scattering efficiency at a wavelength λ= 400 nm is
Qs= 6.83× 10 −^5.
6 :10. Plot the albedo versus the scattering coefficient for attenuation coefficients of
αa= 0.1, 1.0, and 10 cm−^1. Let the albedo on the vertical axis range from 0
to 1 and let the scattering coefficients on the horizontal axis be on a loga-
rithmic scale ranging from 0.1 to 1000 cm−^1.
6 :11. If the optical albedo of a biological tissue isa= 0.9, show thatαs=9αa.
192 6 Light-Tissue Interactions