Egan Polynomial Approximation for R1
Another approximation for R1 is Egan and Hilgeman's [11] polynomial
fit of the data of Orchard [46]. Orchard's data was generated using
Walsh's formula (B.41). The polynomial is
|
R1Egan = -0.4399 + 0.7099 n-1 - 0.3319 n-2 + 0.0636 n-3.
|
(B42) |
where n=ni/nt<1. If n>1 then Equation (B.14) should
be used.
In Table B.4 values for R1 calculated using the various approximations
are presented. Walsh's or Egan's method are superior to the approximations
of Star and Keizer. Since the approximate methods are not sufficiently
faster, their loss in accuracy dictates that they should not be used.
Table B.4:
The first moment of the Fresnel reflection
calculated using various approximations. The values for
R1 when ni/nt<1 are obtained using Equation (B.14),
except for the Keizer approximation which uses Equation (B.39).
The analytic Walsh values are identical to the numerical (exact)
values. The Egan polynomial approximation is much better than
either the Keijzer or Star approximations.
| ni/nt |
R1exact |
R1Walsh |
R1Egan |
R1Keijzer |
R1Star |
| 0.50 |
0.161 |
0.161 |
0.161 |
0.132 |
0.000 |
| 0.60 |
0.116 |
0.116 |
0.116 |
0.100 |
0.000 |
| 0.70 |
0.081 |
0.081 |
0.083 |
0.073 |
0.000 |
| 0.80 |
0.053 |
0.053 |
0.053 |
0.050 |
0.000 |
| 0.90 |
0.027 |
0.027 |
0.026 |
0.029 |
0.000 |
| 1.00 |
0.000 |
0.000 |
0.002 |
0.000 |
0.000 |
| 1.10 |
0.195 |
0.194 |
0.193 |
0.175 |
0.174 |
| 1.20 |
0.337 |
0.336 |
0.336 |
0.311 |
0.306 |
| 1.30 |
0.445 |
0.444 |
0.445 |
0.418 |
0.408 |
| 1.40 |
0.530 |
0.529 |
0.530 |
0.504 |
0.490 |
| 1.50 |
0.597 |
0.596 |
0.597 |
0.573 |
0.556 |
| 1.60 |
0.651 |
0.651 |
0.651 |
0.630 |
0.609 |
| 1.70 |
0.696 |
0.696 |
0.696 |
0.677 |
0.654 |
| 1.80 |
0.733 |
0.733 |
0.732 |
0.717 |
0.691 |
| 1.90 |
0.764 |
0.764 |
0.764 |
0.750 |
0.723 |
|
| 
