A. R. Forouhi and I. Bloomer deduced dispersion equations for the refractive index, n, and extinction coefficient, k, which were published in 1986 and 1988. The 1986 publication relates to amorphous materials, while the 1988 publication relates to crystalline. Subsequently, in 1991, their work was included as a chapter in The Handbook of Optical Constants. The Forouhi–Bloomer dispersion equations describe how photons of varying energies interact with thin films. When used with a spectroscopic reflectometry tool, the Forouhi–Bloomer dispersion equations specify n and k for amorphous and crystalline materials as a function of photon energy E. Values of n and k as a function of photon energy, E, are referred to as the spectra of n and k, which can also be expressed as functions of the wavelength of light, λ, since E = hc/λ. The symbol h is the Planck constant and c, the speed of light in vacuum. Together, n and k are often referred to as the "optical constants" of a material (though they are not constants since their values depend on photon energy). The derivation of the Forouhi–Bloomer dispersion equations is based on obtaining an expression for k as a function of photon energy, symbolically written as k(E), starting from first principles quantum mechanics and solid state physics. An expression for n as a function of photon energy, symbolically written as n(E), is then determined from the expression for k(E) in accordance to the Kramers–Kronig relations which states that n(E) is the Hilbert transform of k(E). The Forouhi–Bloomer dispersion equations for n(E) and k(E) of amorphous materials are given as:
k ( E ) = A ( E − E g ) 2 E 2 − B E + C {\displaystyle k(E)={\frac {A(E-E_{\text{g}})^{2}}{E^{2}-BE+C}}\ }
n ( E ) = n ( ∞ ) + ( B 0 E + C 0 ) E 2 − B E + C {\displaystyle n(E)=n(\infty )+{\frac {(B_{0}E+C_{0})}{E^{2}-BE+C}}\ }
The five parameters A, B, C, Eg, and n(∞) each have physical significance. Eg is the optical energy band gap of the material. A, B, and C depend on the band structure of the material. They are positive constants such that 4C − B2 > 0. Finally, n(∞), a constant greater than unity, represents the value of n at E = ∞. The parameters B0 and C0 in the equation for n(E) are not independent parameters, but depend on A, B, C, and Eg. They are given by:
B 0 = A Q ( − B 2 2 + E g B − E g 2 + C ) {\displaystyle B_{0}={\frac {A}{Q}}\ \left({\frac {-B^{2}}{2}}\ +E_{g}B-{E_{g}}^{2}+C\right)}
C 0 = A Q [ ( E g 2 + C ) B 2 − 2 E g C ] {\displaystyle C_{0}={\frac {A}{Q}}\ \left[({E_{g}}^{2}+C){\frac {B}{2}}\ -2E_{g}C\right]}
where
Q = 1 2 ( 4 C − B 2 ) 1 2 {\displaystyle Q={\frac {1}{2}}\ (4C-B^{2})^{\frac {1}{2}}}
Thus, for amorphous materials, a total of five parameters are sufficient to fully describe the dependence of both n and k on photon energy, E. For crystalline materials which have multiple peaks in their n and k spectra, the Forouhi–Bloomer dispersion equations can be extended as follows:
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