LIGHT REFLECTION FROM HAIR 305 Tip , Root //Y'b • Cortex ............. ,7 ......... r- ............... Di Laser ND Figure 4. Principles of light reflection and backward scattering on and in a hair fiber, graphically defining the parameters used in equations 1-6. The relevant components of keratin morphology are indicated. Angles were chosen to illustrate the principles of light reflection and refraction, not for physical correctness. ND is the ,ormal direction with respect to the fiber axis and in the horizontal plane. A first fraction of light S is specularly reflected at the receptor angle %: % = ½,r- 2•b (1) where •b is the tilt angle of the cuticle cell with respect to the fiber axis. Values for the tilt angle given in the literature are between 2.5 ø (11) and 3 ø (1,4,11). For our experi- mental setup, using an incident angle of ½• = 40 ø and assuming •b = 2.5 ø leads to an expectation value for the receptor angle of the specularly reflected light of % = 35 ø. Analogous geometric considerations apply for light traveling in the tip-to-root (TR) direction (11). A second fraction of light D s is diffusely scattered and reflected at and near the fiber surface, namely at surface roughnesses (12), at the various interfaces between the cuticle cell layers of human hair (11), the interface of cuticle and cortex, and at optical imper- fections of the cortex, such as voids and inclusions. When the hair is colored, the intensity of light reflected from within the hair is diminished, leading to a decrease in intensity of diffusely reflected light with the darkness of the hair. If the hair surface would be an ideal, diffuse reflector, scattering would occur omnidi- rectionally, so that the intensity of the diffusely reflected light would be uniform. Since, in view of Figure 2 and the other GP curves presented below, this is obviously not the case, it can be assumed (11) that the geometrical dimensions of the scattering centers are comparable to or greater than the wavelength of the incident light. Surface structures of suitable dimensions are the cuticle scale edges. Due to the random nature of the scattering and reflection process leading to diffuse reflection and due to the non-uniform nature of the effect, the mean receptor angle 3/d

306 JOURNAL OF COSMETIC SCIENCE for diffusely reflected light may be expected to coincide with the incident angle of 40 ø , so that: 'Yd = e• (2) A further part of the incident beam is refracted into the fiber according to Snell's or Descartes' law. Taking the cuticle inclination angle into account, this yields the rela- tionship: n•c = sin(e• - 0)/sin(e• - 0) (3) so that e R = sin-l[1/nK sin(e• - 4))] + 4) (4) where eR is the refraction angle, that is, the angle at which the light enters the fiber with respect to the normal direction and nK is the refractive index of keratin, taken as nK = 1.55 (11). Together with the values e• = 40 ø and 4) = 2.5 ø, as given above, equation 4 yields e• = 26 ø. Inside the hair the light is scattered and partly absorbed by hair pigment and color and is thus wavelength-filtered, depending on the hair color and its intensity. Diffuse reflection takes place at structural inhomogeneities within the cortex. For lightly colored hair, light may be diffusely reflected (11) at or in the medulla, a more or less continuous and hollow, tube-like structure in the fiber interior. An increase in this type of light reflection is considered to play an important role in loss of shine in lightened Japanese hair (13). In very blonde or white hair a significant amount of light may be reflected at the backside of the fiber, which is the hair/air interface opposite to the point of incidence. Assuming that the light passes through the fiber axis (symmetrical passage) and taking the opposite tilt direction of this reflecting surface into account, the back-side reflection will occur at: % = e• + 20 (5) yielding the expectation value •b = 31 ø This beam reaches the surface and, with the principles underlying equation 4, is re- fracted out of the fiber according to: y, = sin-linK sin(% + 4))] - 4) (6) where % is the receptor angle for this internally reflected light, with an expectation value of 55 ø. When this third component of light reemerges from the fiber, it has the color of the hair and is experimentally observed as a separate peak in the GP curve (1,2,4). Stammet a/. (11) consistently observed that the location of this peak was shifted to higher angles by 10%. Due to its width, the peak in the GP curve associated with this type of light is considered as being a specific fraction of diffusely reflected light, termed D , since it originates from internal reflection. It is important to note that this model does not take into account the complex, layered morphological structure of the cuticle cell (10). Equal refractive indices are assumed for the morphological components. The two main components, namely exo- and endocuticle are indicated in Figure 4.