| Complex refractive index of fused quartz: Imaginary part |
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The imaginary part, m" of the complex refractive index of fused quartz in the near-infrared (up to 3.5 µm) depends strongly on the purity of the fused quartz (Plotnichenko et al 2000, Dombrovsky 1996b, Petrov and Stepanov 1975) and in particular on the hydroxyl content (Plotnichenko et al 2000, Petrov and Stepanov 1975). The value of the imaginary part was not always directly available from the literature and had to be recovered from spectral transmittance or emittance measurement data. Table 1 lists the references reporting experimental data for fused quartz at room temperature with the spectral range covered), and the measurements performed to recover m". The value of m" can be recovered from the spectral transmittance data, Tn(l ), at normal incidence, for a layer of thickness, l, by using the following relationship, accounting for multiple reflections (for example, Modest 2003, p. 56):
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Tn (l ) =
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(1 - ρn2) eal
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1 - ρn2 e2al
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(1)
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where ρn and a are the spectral reflectivity of the interface of the layer at normal incidence and the spectral absorption coefficient of fused quartz, respectively, and are given by
where m is the complex refractive index of fused quartz, and by
where λ is the wavelength of light in vacuum. Eq. 1 can be solved as a quadratic in the exponential factor and after some algebraic manipulation one can obtain the following expression for the imaginary part, m", of the refractive index:
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m" = - (
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) ln {
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[(1 - ρn)4 + 4ρn2Tn]½ - (1 - ρn)
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2ρn2Tn
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}
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(4)
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The imaginary part, m", of the complex refractive index, m, can also be determined from measurements of the spectral normal emittance, ε (i.e. directional emittance, for example, Modest 2003, at a direction normal to a surface), by using the following expression (Dvurechensky et al 1979),
Fig. 1 shows the imaginary part, m", of the complex refractive index of fused quartz as a function of wavelength, λ, as reported in the literature (Table 1) or derived from Eq. 4 and Eq. 5. Note, that computations of the imaginary part of the refractive index from transmittance or emittance measurements lead sometimes to negative values, particularly in a spectral region where fused quartz is very weakly absorbing (from 0.2 to 4.0 µm). Thus, in this region, the data should be used with care since the experimental uncertainty for m" is very large and m" effectively vanishes.
