
3.3.6 Fiber Gratings Metrology
C. M. Wang
A. H. Rose
A narrowband passive component (e. g. fiber grating) is characterized
by a collection of performance attributes.
Many of these attributes, such as bandwidth and center wavelength are
measured at x dB below the ``plateau'' level of the
reflectance/transmittance response.
When there are outlying measurements, the lower and upper
xdB wavelengths, based on the xdB transmission/reflectance response
y_{x} calculated as
may be misleading. We need a robust estimate of y_{x} representing the plateau level of the response curve. We use a statistic, called shorth (Andrews et al. Robust Estimates of Location: Survey and Advances, 1972), which is similar to the median (in robustness) but has a convenient geometrical interpretation. The shorth of , is the midpoint of the shortest interval that includes half of y_{i}. If we fit a horizontal line to , the mean of y_{i}is the line that minimizes the sum of the squared residuals. The shorth of y_{i} is the line that minimizes the median of the squared residuals. Another important problem in fiber grating measurements is the determination of wavelength spacings. If there are no outlying measurements, is the estimate of the plateau level of the curve. We can determine the proper sample size, hence the proper wavelength increment, based on the desired precision of this plateau estimate. Under some regularity conditions, it can be shown that the standard deviation of y_{x} is approximately equal to , where is the maximum possible measurement error of y_{i}. We can then equate this standard deviation to a threshold value to obtain the sample size required.
Once we have a ``good'' estimate of the xdB transmission/reflectance
response, the lower and upper xdB wavelengths can be
calculated by interpolation.
It can be shown that the maximum error of the lower x dB wavelength
(similar result for the upper wavelength) is given by
where h is the wavelength increment, and y^{} and y^{+} are the first two consecutive responses such that . An appropriate wavelength increment h can be obtained by requiring the maximum error of be less than a threshold value, say, , or The result indicates that when the response curve is slow varying in regions where y_{x} is located ( y^{+}  y^{} is small), or is large, we need a smaller increment.
Figure 15: This graph displays the estimated plateau of the reflectance/transmittance curve based on the mean (dotted horizontal line) and the shorth (solid horizontal line) of y_{i}. It also shows the 0.5dB wavelengths based on the shorth (dotted vertical lines) and the mean (solid vertical lines).
Date created: 7/20/2001 