next up previous
Next: Fiber Bundle Preparations Up: Notes on Timing Resolution Previous: Unfolding the Timing Resolution


Attenuation Length

To evaluate the attenuation length of the various fibers tested, it is necessary to evaluate first the ratio of the means of the left and right PMT ADC values at each position along the beam. To understand this, consider that the attenuation of light as it travels along the fiber is given by

\begin{displaymath}
I(z)=I_0(z)e^{-z/\lambda}
\end{displaymath} (8)

where $z$ is the distance from the point of impact of the beam along the fiber to the appropriate PMT, $\lambda$ is the attenuation length, and $I_0(z)$ is the amount of light produced at the interaction point.

Figure 2: ADC spectra for fiber PMT's


\includegraphics [height=94.5mm,angle=0,clip=]{adc_compare_7000series.eps}

In practice, it is found that the amount of light produced at the interaction point is a function of $z$, as can be seen in Fig. 2, where the three ADC spectra are from the TRIUMF data taken at 90 cm, 150 cm, and 210 cm with respect to the left fiber end PMT. Consequently, the ADC values for the two PMT's in question may be expressed as
\begin{displaymath}
ADC_{left}=f(z)e^{-z/\lambda}\hspace{0.2 cm} and\hspace{0.2 cm} ADC_{right}=f(z)e^{z/\lambda},
\end{displaymath} (9)

where $f(z)$ is the geometric mean calculated from
\begin{displaymath}
f(z)=\sqrt{(ADC_{left}ADC_{right})}.
\end{displaymath} (10)

Thus, a reliable method to extract the attenuation length value is to take the ratio between the two ADC values above:
\begin{displaymath}
ln(ADC_{left}/ADC_{right})=-2z/\lambda.
\end{displaymath} (11)

Plotting the ADC ratio values at different positions on a semi-log scale results in a straight line with a slope of $-2/\lambda$.


next up previous
Next: Fiber Bundle Preparations Up: Notes on Timing Resolution Previous: Unfolding the Timing Resolution
Web Master
2001-10-29