Refractive Index Approximation

 

 

In his book “Optical Aberration Coefficients”, H.A. Buchdahl suggests a version of dispersion equation based upon &omega:

     &omega = &delta&lambda / ( 1 + 2.5 &delta&lambda )

where &delta&lambda is &lambda – &lambda_D.  This definition of &omega has a zero at the D-line and a singularity at about 187 nm.  An alternative dispersion equation base is:

     H = &deltaf² / ( 1 - 0.04 &deltaf² )

where &deltaf² = f² – f_e² and f = 1 / &lambda.  This base has the singularity at about the same wavelength as the Buchdahl base, but, of course, has its zero at the e-line.  Relative to f², the value of H is invertible, in that:

     f² = f_e² + H / ( 1 + 0.04 H )

the first few derivatives of which are:

        df²/&deltaH  =         1  / ( 1 + 0.04 H )²
        d²f²/&deltaH² = -2 * 0.04  / ( 1 + 0.04 H )³
        d³f²/&deltaH³ =  6 * 0.04² / ( 1 + 0.04 H )⁴

If S is a function of f², then the derivatives of S (with respect to H) are then:

    S’   =   &deltaf²/&deltaH * &deltaS/&deltaf²
    S’’  =  (&deltaf²/&deltaH)² * &delta²S/&delta(f²)² +
                &delta²f²/&deltaH²  *  dS/&deltaf²
    S’’’ =  (&deltaf²/&deltaH)³ * &delta³S/&delta(f²)³ +
              3 &deltaf²/&deltaH * &delta²f²/&deltaH² * &delta²S/&delta(f²)² +
                &delta³f²/&deltaH³ * &deltaS/&deltaf²

the attempt here being Taylor series estimation of S:

   S* = S_e + S_e’ H + S_e’’ H² / 2 + S_e’’’ H³ / 6 …

within a small region about the e-line.  According to Buchdahl, using his original base allowed good approximations with very few terms.  This note will ignore beyond the third derivative.

We are interested in two related forms of S; the six-term Schott dispersion equation (also known as the Cauchy dispersion equation):

    n² = A₂ (f²)^-1 + A₀ + A_-2 f² + A_-4 (f²)² + A_-6 (f²)³ + A_-8 (f²)⁴

The coefficients, A₂…A_-8, are specific to the glass type.(Note: these names were originally based upon &lambda instead of f; hence the apparent sign inversion.)

and the three-term Sellmeier dispersion equation:

    n² – 1 =  E₁ + E₂ + E₃

where E_i = B_i / ( 1 – C_i f² ).  The coefficients, B_i and C_i, are specific to the glass type.In both of these, “S” is the refractive index, n.

The derivatives of the Cauchy equation are found from:

  2*n*&deltan/&deltaf²    =    -A₂ (f²)^-2 + A_-2 + 2A_-4 f² + 3A_-6 (f²)² + 4A_-8 (f²)³
  2*n*&delta²n/&delta(f²)² = 2*(A₂ (f²)^-3 + A_-4 + 3A_-6 f² + 6A_-8 (f²)² - (&deltan/&deltaf²)²)
  2*n*&delta³n/&delta(f²)³ = 6*(-A₂ (f²)^-4 + A_-6 + 4A_-8 f² – &deltan/&deltaf²*&delta²n/&delta(f²)²)

The derivatives of the Sellmeier equation can be from from:

  2*n*&deltan/&deltaf²    =    E₁ D₁  + E₂ D₂  + E₃ D₃
  2*n*&delta²n/&delta(f²)² = 2*(E₁ D₁² + E₂ D₂² + E₃ D₃² - (&deltan/&deltaf²)²)
  2*n*&delta³n/&delta(f²)³ = 6*(E₁ D₁³ + E₂ D₂³ + E₃ D₃³ - &deltan/&deltaf²*&delta²n/&delta(f²)²)

where D_i = C_i / ( 1 – C_i f² ).

Typically, glass catalogs can list the refractive index n, the Abbe number v, the dispersion &Deltan, the partial dispersion P_F,e and the partial dispersion P_g,F.  The value of n desired here is not necessarily n_D or n_d, but the one that satisfies the equation n = 1 + v&Deltan.  The values P_F,e and P_g,F may be provided as &DeltaP_F,e and &DeltaP_g,F, using the equations supplied in Schott catalogs:

   P_F,e = 0.4884 – 0.000526 v + &DeltaP_F,e
   P_g,F = 0.6438 – 0.001682 v + &DeltaP_g,F

If none of these is available, it may be assumed the glass is a “normal” glass, with &DeltaP_F,e = &DeltaP_g,F = 0.  If only one of the partial dispersions is missing, the other may be derived from the empirical formula:

     &DeltaP_g,F = 4.0 * &DeltaP_F,e

These values may be related to the derivatives by:

  n               =  S_e

  &Deltan*( P_F,e – 1 )    =       H_C S_e’ + H_C² S_e’’/2 + H_C³ S_e’’’/6
  &Deltan*P_F,e            =       H_F S_e’ + H_F² S_e’’/2 + H_F³ S_e’’’/6
  &Deltan*( P_F,e + P_g,F )  =       H_g S_e’ + H_g² S_e’’/2 + H_g³ S_e’’’/6

This matrix form is invertible. Its inverse may be used to derive values of S_e, S_e’, S_e’’, and S_e’’’ from the glass catalog values.

The values of H_C, H_F and H_g used in the matrix above need not change when a minimum focus wavelength other than the e-line is used.  They will simply represent analogous translations from that new wavelength, and not necessarily back to the C, F and g lines themselves.