3.3.6 Fiber Gratings Metrology

C. M. Wang
Statistical Engineering Division, ITL

A. H. Rose
Optoelectronics Division, EEEL

A narrow-band 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

$$y_{-x} = \max(y_i,\; i = 1, 2, \cdots) - x$$

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 $y_i,\; i=1,2,\cdots,n$, is the midpoint of the shortest interval that includes half of y_i. If we fit a horizontal line to $y_i,\; i=1,2,\cdots,n$, the mean of y_iis 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, $\max(y_1,\, y_2, \,\cdots \, y_n)$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 $\Delta y/(n+2)$, where $\Delta y$ 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

$$\Delta \lambda_L \approx \frac{\Delta y}{y^{+} - y^{-}} h$$

where h is the wavelength increment, and y^- and y⁺ are the first two consecutive responses such that $y^{-} \le y_{-x} \le y^{+}$. An appropriate wavelength increment h can be obtained by requiring the maximum error of $\lambda_L$ be less than a threshold value, say, $\epsilon$, or

   $$h \le \frac{\epsilon (y^{+} - y^{-})}{\Delta y}.$$

The result indicates that when the response curve is slow varying in regions where y_-x is located ( y⁺ - y^- is small), or $\Delta y$ 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
Last updated: 7/20/2001
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