Appendix

Statistical Design and Analysis for Calibrating the Transmittance of Filters for Spectrophotometer Linearity Testing

For four sets of filters, each set containing seven filter types of various transmittances, an assigned transmittance value needed to be determined when a triangular passband of given width is superimposed. Additionally, the polarization and location of the spot size on the filter had an unknown influence on the transmittance value. Although a commercial instrument can scan the entire spectrum at little cost, the measurements are an order of magnitude less accurate than those obtained using the high accuracy instrument. For this reason, a design was developed to determine:

(1) The values to be measured for purposes of calibration, and

(2) The existence of a possible effect due to polarity or the effect of location on the filter.

The motivation behind the choice of design is discussed in section A below. In the event of (2), the calibration must be reconsidered. On the basis of these extremely precise measurements, an effect due to location was statistically significant. This suggests that either:

(1) The calibrated values can be used only for the central location on each filter; or

(2) Additional measurements will be required to determine the magnitude of changes in transmittance due to spot location.

Since the observed differences are small (<0.2%; see table 4), the location effect may not interfere with practical usage of these filters. However, measurement as near to the central location on each filter as possible is recommended. The conclusions regarding location and polarity effects are presented in section B, and the estimation of the transmittance curve leading to the calibration values is shown in sections C and D. The method of extension for general passbands is given in section E.

A. The Design

Most organizations using these filters will have triangular passbands on their measuring equipment. Therefore, adequate determination of the transmittance using this passband is essential. However, other passband shapes may be used, and for these, a more general

In all that follows, we will assume that the passband h has been normalized in this way.

In order for this average value to be relatively insensitive to wavelength specification, f(λ) needs to be a smooth, slowly-varying function. In mathematical terms, we search for a neighborhood where the first two derivatives of the spectrum are fairly small. This suggests a region where f(A) is nearly constant, or, at worst, a quadratic having very small degree of curvature. A locally linear spectrum with a noticeably nonlinear slope would yield a seriously biased estimate, whereas a locally quadratic spectrum, centered at a peak or trough, would have a negligible linear term and therefore a bias which depends primarily on the second-order term (hopefully

A design which minimizes the variance-covariance matrix Var(e) of the parameter estimates is given in [9]. The design would place the two observations each at λ + 10.5 nm in addition to the one at λ. If our function is truly quadratic, such a design would be optimal.

For many reasons, however, we modify the optimal design which permits more flexibility in our choice of model. Such a design is suggested in [10]. The design recommends two different values rather than repeating them at the endpoints.

Closely related to this design is the one which minimizes the maximum variance of the best linear unbiased estimate of the function f(x) given by eq (5). Reference [11] shows that the five points should be placed at the zeros of the polynomial

10.5 (1 - x2) P1'(x),

B. Results: Polarity and Location

The reported results are based on measurements taken on four sets of filters. A complete series of 15 measurements across the 21-nm range at 1.5 nm spacing was taken on the master set, as a check for the adequacy of the five-point design used on the other three sets. In addition, polarity and location were tested on each filter in all sets. The results of these tests are reported in this section; transmittance calculations are reported in section C.

=

547.0,

For set 1, six measurements were taken at λ at both polarity 1 and 2 at the center of the filter (0,0) and at placements of two units to the right (2,0) and above (0,2) the center. This permits a 3 × 2 factorial analysis for detecting differences due to location and polarity.

For sets 2, 3, and 4, t-tests on location and polarity were calculated. The test on location was deemed significant if the comparison of either (0,0) and (0,2) or (0,2) and (2,0) indicated a chance of less than 1% under the hypothesis of no effect.

None of the tests of polarity was significant at the 0.01 level. (One filter from one set gave significance of 0.025. Out of 28 such tests, the chance of obtaining one or more

P1'(x) = 17.5x3- 7.5x.

surprising.) For location, however, the following sets showed significant difference (level given in parentheses):