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Refractive Index, Abbe Value and Dispersion Formula

The optical properties are defined by two primary parameters, the refractive index and Abbe number. The refractive index is a measure of the bending power of a light beam when passing from one medium into another. The Abbe or v-number named after German physicist Ernst Abbe, is a measure of the dispersive power and is defined as:



Where nd, nF and nC are the refractive indices of the material at wavelengths of the Fraunhofer d-, F- and C- spectral lines (587.6 nm, 486.1 nm and 656.3 nm respectively). Besides , the dispersion based on e-line is often used as: 



   Glass Type  nd  Vd Density (g/cm3)
Glass Code  
  N-BK7 1.51680 64.17 2.51 517642  
  N-SF11 1.78472 25.68 3.22 785257  
  Table 1: Examples of glass code and primary parameters  

Table 1 shows some basic information of glass code, two primary optical parameters and density. Table 2 shows spectral lines and designated letters for them. The refractive indices of optical glasses are given at these wavelengths with the N-BK7 as an example.

  Wavelength (nm)
Designated Letter
 Element N-BK7 Refractive Indices
Spectral Line
  1013.98 t Hg 1.50731 Infrared mercury line  
  852.11 S Cs 1.50980 Infrared cesium line  
  706.52 r He 1.51289 Red helium line  
  656.27 C H 1.51432 Red hydrogen line
  643.85 C' Cd 1.51472 Red cadmium line  
  632.8 632.8 He-Ne Laser
1.51509 Helium-neon gas laser
  589.29 D Na 1.51673 Yellow sodium line      (center of the  double line)
  587.56 d He 1.51680 Yellow helium line  
  546.07 e Hg 1.51872 Green mercury line
  486.13 F H 1.52238 Blue hydrogen line  
  479.99 F' Cd 1.52283 Blue cadmium line  
  435.83 g Hg 1.52668 Blue mercury line  
  404.66 h Hg 1.53024 Violet mercury line  
  365.01 i Hg 1.53627 Ul traviolet mercury line  
  Table 2: Wavelengths used for spectral lines  

The refractive index at a wavelength other than the spectral lines can be calculated from a dispersion formula. SCHOTT uses Sellmeier dispersion formula:



where λ is the wavelength in um ,and B1, B2, B3, C1, C2 and C3 are coefficients to be determined for each optical glass type. Table 3 gives the parameters of N-BK7 and N-SF11. HOYA’s dispersion formula is derived from a series expansion of the theoretical formula:



where λ is the wavelength in um ,and A0, A1, A2, A3, A4 and A5 are coefficients. The accuracy of both formulas for calculating refractive index at a wavelength in the visible and near infrared range has an order of 10-6.

   Constants of dispersion
 B1  B2  B3  C1  C2  C3  
  N-BK7 1.03961212 0.231792344 1.01046945 0.00600089867 0.0200179144 103.560653  
  N-SF11 1.73759695 0.313747346 1.89878101 0.013188707 0.0623068142 155.23629  
  Table 3: Constants of dispersion parameters of N-BK7 and N-SF11  


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