2.1. KK Relations in Continuous Spectrum Space
 In dielectric material physics, complex dielectric constant (relative permittivity) is a linear function of the susceptibility of the material, which in turn is the Fourier transformation of the response function of the dielectric material in the real time space. Therefore, must satisfy , where is the conjugate function of . Since , . Thus, the real and imaginary parts of are even and odd functions, respectively, that is, n(ω) = n(−ω) and κ(ω) = − k(−ω). This particular property leads to further simplification of the KK relations of equations (1) and (2). By multiplying the integrands of equations (1) and (2) with , and replacing the real and imaginary parts by n(ω) and k(ω), one may have the following KK relations of complex refractive index:
Equation (3) and (4) are based on angular frequency space. One may easily have the equivalent equation system on wavelength space with ω = 2πc/λ, where c is the speed of light in vacuum.
2.2. KK Analysis by Variational Function Fitting (VFF) Method
 In practical calculations, the data range is unavoidably discrete and finite. The divergence at ω′ = ωwill cause trouble in numerical integration. In order to reduce the errors caused by the finite spectrum, variants of the KK relations were introduced, e.g., FFT-based KK analysis [Bortz and French, 1989], singly subtractive KK analysis [Ahrenkiel, 1971; Bachrach and Brown, 1970] and multiply subtractive KK [Palmer et al., 1998]. The basic idea of subtractive KK analysis is to remove the n(ω) values at one or more anchor (reference) frequencies from the KK integration. The n(ω) values at anchor frequency points come from independent measurements or other previous studies. Although the subtractive form of a KK integral converges more rapidly than a conventional KK relation, the requirement to have anchor-frequency n(ω) values is not practicable for the study of leaf optics. As argued above, leaf refractive index varies with respect to leaf types and is generally hard to accurately measure. It is thus desirable to have a numerical KK analysis that allows the calculation of n(ω) just with k(ω) data, meanwhile with good convergence property. The variational function fitting method [Kuzmenko, 2005] provides efficient solutions on this aspect. The central idea of VFF method is that the analytical form of k(ω) is parameterized independently at each anchor frequency, while n(ω) is dynamically coupled to k(ω) by the KK transformation.
 The choice of functional form of k(ω) is somewhat arbitrary, but must sufficiently account for the ‘local’ spectral weight near ω.It was demonstrated that a simple triangular line-shape function which is nonzero only inside a small region adjacent toω works well if the anchor points are dense enough [Kuzmenko, 2005]. The triangular line-shape function was defined as
where ωi is the ith anchor frequency, Substituting equation (5) into equation (1) and making finite integration in the rang [ωi−1, ωi+1], one may have
Although not explicitly mentioned by Kuzmenko , this method is virtually a Rayleigh–Ritz method in direct variational analysis, where a function is approximated by a linear combination of elemental (base) functions of the same linear space. This method may yield solutions even if an analytical form for the true solution is intractable.
 Equation (6) is the contribution of the element δk which is centered on the anchor frequency ωi. To calculate the contribution of all the elements δk to the refractive index at ω, one needs sum up over all the anchor frequencies. In a discrete case, it follows that
In the case that both n(ω) and k(ω) are unknown, equations (5)–(8) may be coupled with radiative models to inverse n(ω) and k(ω) from reflectance or transmittance measurements. If k(ω) is already known, equation (8) may be used to calculate n(ω) directly from k(ω).
 The drawback of the above analysis system is that it does not take into account the spectral weight outside the frequency range [ω1, ωN]. But the same problem of extrapolations exists in all the conventional KK analysis, which actually does not have any universal solution. In Kuzmenko , two-step fitting was suggested to address the extrapolation issue. The first-step fitting uses formula-defined functionnmod(ω), e.g., Drude-Lorentz model, to match the big-picture of experimental spectra. The second-step applies the VFF methodnvar(ω) to fit the fine spectral details. The general solution is
As shown in this study, if the considered range is not large and nvar(ω) is able to describe the local spectral details properly, a simple nmod(ω) form, e.g., a linear function or constant value may suffice to account for the contribution from outside the data range.