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Introduction

Optical properties have been calculated through reflectance measurements based on the Kubelka–Munk reflectance theory as applied to human enamel and dentin , maxillofacial material porcelain as well as direct restorative material . Although the study of direct restorative material used only luminous reflectance which is the CIE tristimulus value Y , and Davis et al. determined reflectance using three CIE tristimulus values ( X , Y , and Z ), most articles describe reflectance on a spectral basis, which then allows for conversion to CIE color parameters for any illuminant and observer.

The Kubelka–Munk (K–M) collection of theories, originally derived by Kubelka and Munk in 1931 and simplified by Kubelka in 1948 , model reflectance of a translucent layer with and without a backing and model transmittance through such a layer. K–M reflectance theory for layers on a backing is a mathematical model describing the reflectance resulting from two-flux radiation transfer in a homogenous and uniform medium on an opaque backing. Differential equations that led to the theory considered i and j as two light vectors traveling on opposite directions, where K and S are absorption and scattering coefficients of the translucent material respectively. An elementary layer of thickness dx is used to describe the changes in light fluxes in both directions. The decreases of the flux in the downward direction due to absorption and scattering of the downward flux are Ki dx and Si dx , respectively, and the increase in the flux in the downward direction due to scattering from the upward flux is Sj dx . Similarly, the decreases of the flux in the upward direction due to absorption and scattering of the upward flux are Kj dx and Sj dx , respectively, and the increase in flux in the downward direction due to scattering from the downward flux is Si dx . Therefore, the differential equations for the downward ( i ) and upward ( j ) fluxes are :

The solution of Eqs. (1) and (2) is based on certain assumptions : (1) the material is of a constant finite thickness, (2) illumination is diffuse and homogenous, (3) optical effects at its edges may be neglected, (4) pigment particles are uniformly distributed within the layer and are smaller than the elementary thickness, and (5) no reflection occurs at the surface of the translucent material. The solution resulted in the reflectance R of the translucent material of thickness ( X ) with scattering ( S ) and absorption ( K ) coefficients and in optical contact against a background with reflectance R _g . The simplified reflectance equation is:

where <SPAN role=presentation tabIndex=0 id=MathJax-Element-4-Frame class=MathJax style="POSITION: relative" data-mathml='a=(S+K)/K’>a=(S+K)/Ka=(S+K)/K
a = ( S + K ) / K
, <SPAN role=presentation tabIndex=0 id=MathJax-Element-5-Frame class=MathJax style="POSITION: relative" data-mathml='b=a2−1′>b=a2−1−−−−−√b=a2−1
b = a 2 − 1
, S and K are the scattering and absorption coefficients respectively, X is the thickness of the translucent material, R _g is the reflectance of the backing, and coth denotes the hyperbolic cotangent function, where coth( bSX ) is <SPAN role=presentation tabIndex=0 id=MathJax-Element-6-Frame class=MathJax style="POSITION: relative" data-mathml='coth(bSX)=(e(bSX)+e−(bSX))/(e(bSX)−e−(bSX))’>coth(bSX)=(e(bSX)+e−(bSX))/(e(bSX)−e−(bSX))coth(bSX)=(e(bSX)+e−(bSX))/(e(bSX)−e−(bSX))
coth ( b S X ) = ( e ( b S X ) + e − ( b S X ) ) / ( e ( b S X ) − e − ( b S X ) )
. If the material is infinitely thick, the K–M reflectance equation simplifies to <SPAN role=presentation tabIndex=0 id=MathJax-Element-7-Frame class=MathJax style="POSITION: relative" data-mathml='R∞=a−b’>R∞=a−bR∞=a−b
R ∞ = a − b
.

However, Eq. (3) does not take into account interfacial reflections that occur at refractive index changes as light passes from one medium to another. Frensel’s law describes light behavior as it travels between two media having different refractive indices . Interfacial reflections occur externally as the incident light passes from air to the material and also internally as the light leaves the material into air. Saunderson showed that the reflectance <SPAN role=presentation tabIndex=0 id=MathJax-Element-8-Frame class=MathJax style="POSITION: relative" data-mathml='R′’>R′R′
R ′
of material, which would be observed if reflections did occur at the front surface, is given by Eq. (4) :

where k ₁ is the fraction of incident light externally specularly reflected upon entering the material, k ₂ is the fraction of light internally diffusely reflected upon leaving the material at the front surface and R is the reflectance of the material if no interfacial reflection occurs at the front surface, what Saunderson called the internal reflectance. This correction must also be considered in determining the effective reflectance of any backing on the translucent layer and the substantial differences in these corrections for air and for a liquid matched in refractive index between the layer and the backing were shown for a dental modifier porcelain .

Duntley proposed that the correction when only half of the specular light is collected in the measurement to be:

However, no specular reflection is included in measurements with the 45° illumination and 0° observation geometry used for reflectance measurements in this study as shown in Fig. 1 . Therefore, this correction formula becomes (per Hu et al. ):

The optical arrangement the fiber optic light cable (F) transferring the illumination from a xenon arc lamp to a white reflectance standard (W) during calibration of the non-contact measuring system. The reflected light from the standard is then measured by the spectroradiometer (S).

For nonpolarized light k ₁ is a function of the incident angle about which color scientists seem to have no debate and is expressed as :

where <SPAN role=presentation tabIndex=0 id=MathJax-Element-13-Frame class=MathJax style="POSITION: relative" data-mathml='k||=[(cosi−(n2/n1)2−sin2i)/(cosi+(n2/n1)2−sin2i)]2′>k||=[(cosi−(n2/n1)2−sin2i−−−−−−−−−−−−−√)/(cosi+(n2/n1)2−−−−−−−√−sin2i)]2k||=[(cosi−(n2/n1)2−sin2i)/(cosi+(n2/n1)2−sin2i)]2
k | | = [ ( cos i − ( n 2 / n 1 ) 2 − sin 2 i ) / ( cos i + ( n 2 / n 1 ) 2 − sin 2 i ) ] 2
and <SPAN role=presentation tabIndex=0 id=MathJax-Element-14-Frame class=MathJax style="POSITION: relative" data-mathml='k⊥=[((n2/n1)2cosi−(n2/n1)2−sin2i)/((n2/n1)2cosi+(n2/n1)2−sin2i)]2′>k⊥=[((n2/n1)2cosi−(n2/n1)2−sin2i−−−−−−−−−−−−−√)/((n2/n1)2cosi+(n2/n1)2−sin2i−−−−−−−−−−−−−√)]2k⊥=[((n2/n1)2cosi−(n2/n1)2−sin2i)/((n2/n1)2cosi+(n2/n1)2−sin2i)]2
k ⊥ = [ ( ( n 2 / n 1 ) 2 cos i − ( n 2 / n 1 ) 2 − sin 2 i ) / ( ( n 2 / n 1 ) 2 cos i + ( n 2 / n 1 ) 2 − sin 2 i ) ] 2
. Here n ₂ is the index of refraction of the optically denser medium, and n ₁ is that of the less dense medium. For air n ₁ is about 1.0003 and for dental composite resin n ₂ is ∼1.5. Since n ₁ is nearly 1 so n = n ₂ / n ₁ ≈ n ₂ . The internal reflection coefficient k ₂ as theorized by Richmond for a translucent material with an index of refraction n in a vacuum is given by:

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