Topics in Particle and Dispersion Science

Measuring attenuation of light: Optimum pathlength

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Provided that the medium does not scatter light or, if it does, that the detector acceptance angle has been suitably restricted, there is an optimum optical pathlength (optical thickness), which minimizes the relative error, dc/c, of the measurement of the attenuation coefficient, c. Note that we are not interested here in the evaluation of the actual mangitude of dc/c. Rather, we aim to show that dc/c assumes a minimum at a certain optical thickness. Indeed, by solving the transmission equation (see Lambert's law):

Φ(z) = Φ0 exp(-cz)

(1)

where Φ is the power of light, for c:

c = - (1/z) [ ln Φ(z) - ln Φ0 ]

(2)

and differentiating c with respect to Φ, we can express the relative differential dc/c (relative error) as follows:

dc / c	= -1 / (cz) dΦ / Φ(z)
	= -1 / [cz exp(-cz) ] dΦ / Φ0	(3)

The relative error of the attenuation coefficient, c, given the relative power measurement error, dΦ/Φ₀, assumes the minimum, where the function x exp(-x) assumes the maximum, i.e. at x = 1 (Fig.1). Given that x = cz, this implies that the relative error of c assumes a minimum for the medium thickness z = 1 / c. It follows that a similar optimum exists for the transmission of light by the sample in the determination of concentration of the scattering/absorbing centers in a medium through the Beer-Lambert law (for example, Olsen E 1975; see also problem: Optimum transmission of a medium for the determination of concentration of scattering/absorbing centers).

Note that the derivation of the condition for the optimum pathlength does not involve the shape and other properties of the volume scattering function. The only assumption used there is the applicability of the Lambert law. However, the scattering function does affect the applicability of the Lambert law through both the scattering correction and the onset of multiple scattering (see Single and multiple scattering).

Eq. 3 can be expressed in terms of the transmission, T, as follows (see also, Waltz SC et al 2004, Malm WC et al 1986)

dc / c = (1 / lnT ) (dT / T )

(4)

The above discussion applies to a situation when the fluctuation of the number of photons (photon noise) is negligible and the error dΦ in measuring the power of light is dominated by instrumental factors. If the number of photons in the transmitted power is so small that the photon noise contributes substantially to dΦ, this contribution must be accounted for. Farsiu S et al (2007) derived the lower bound of the variance of the attenuation coefficient in this case. Transforming their formula to our notation, we obtain:

<dc>/<c> = [ e + exp(-cz)]1/2 / [(qΦp0)1/2 cz exp(-cz)]

(5)

where e is the average of an instrumental error (noise) relative (0 ≤ e ≤ 1) to the number rate of the incident photon flux, Φ_p0, and q is the number of measurements of the attenuation of light. The dependence of <dc>/<c> on cz (Fig. 2) is similar in form to that for dc/c expressed by the previously discussed Eq. 3.

CITATION: Swanson N. L., Jonasz M. 2007. Measuring attenuation of light: Optimum pathlength (www.tpdsci.com/Tpc/AtnCfMsOptmPlen.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 21-Nov-2007 Modified: 30-Dec-2007 Peer-reviewed: 22-Dec-2007

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