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3.1.9 Wavelength Calibration

Jack C.M. Wang

Statistical Engineering Division, ITL

Sarah L. Gilbert

William Swann

Optoelectronics Division, EEEL

Many new high capacity systems use several laser transmitters, operating at slightly different wavelengths, to increase the transmission capacity of a single fiber, a technique known as wavelength division multiplexing (WDM). This requires that the wavelengths of the individual lasers be well known and controlled. A NIST-developed SRM is a fiber-connected gas absorption cell that permits quick wavelength calibration of instruments, such as optical spectrum analyzers, used in the development of WDM systems. The measurements used in the calibration consist of wavelength and corresponding absorption power. The calibration is carried out by fitting a model, called Voigt, to the data.

A Voigt density is obtained by convolving a Gaussian density with a Lorentzian (Cauchy) density. Specifically, if U is a Gaussian random variable with parameters $\mu$ and $\sigma$, and V is a Lorentzian random variable with parameters $\mu$ and $\lambda$ and is independent of U, then W = (U+V)/2 is distributed as a Voigt with density function given by

\begin{displaymath}f(w) = \frac{2\lambda}{\pi^{1.5}} \int_{-\infty}^{\infty}
...u^2)}{\lambda^2 + \left[2(w-\mu)-\sqrt{2}\sigma u\right]^2}du.

For data-fitting purpose, a Voigt model must be general enough to allow for an arbitrary translation of the data. The Voigt model, relating absorption power (y) and wavelength (x), is given by

\begin{displaymath}y = \beta_0 + \beta_1\int_{-\infty}^{\infty} \frac{\exp(-u^2)}
{\beta_2^2 + \left[2(x-\beta_3) - \beta_4^2 u\right]^2}\;du

where $\beta_0,\, \beta_1, \cdots, \beta_4$ are parameters of the model. The parameters of interest are the wavelength that attains the maximum absorption power ($\beta_3$), and the relative height and width of the absorption power spectrum (functions of $\beta_2$ and $\beta_4$). Since both power and wavelength are subject to measurement errors, a Fortran program, utilizing a nonlinear errors-in-variables regression procedure, has been developed to estimate the parameters and their standard errors.


Figure 9: The top figure displays the density functions of Gaussian, Lorentzian and Voigt ($\mu=2$, $\lambda=1.5$, and $\sigma=1.2$). It shows that a Voigt possesses the peak feature of a Gaussian and the tail profile of a Lorentzian. The bottom plots the scatterplot of wavelength vs. power and the Voigt (solid curve) model fitted by the errors-in-variables regression. The ``$\circ$'' points were used to fit the model and ``$\times$'' points were not used.

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Date created: 7/20/2001
Last updated: 7/20/2001
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