Integrating cavity ring-down spectroscopy (ICRDS) and the direct measurement of absorption coefficients

Integrating cavity spectroscopy is a highly sensitive technique for weak absorption measurements [1–5]. Most recently, it has been used to demonstrate enhanced detection of extremely low concentrations of biological waste products in water supplies [6]. Basically, an integrating cavity is a closed container whose interior wall has a very high diffuse reflectance (Lambertian). If light energy is supplied to an integrating cavity containing an absorbing sample, the equilibrium optical energy in the cavity provides a measure of the sample absorption. Since the diffuse reflecting wall of the cavity produces an isotropic illumination of the sample, scattering in the sample cannot change this isotropic illumination. Scattering also cannot change the amount of light energy in the cavity. As a result, absorption measurements are independent of scattering. The many reflections of the light from the cavity walls mean the light also makes many transits through the sample, i.e. the effective path length through the sample is much greater than the dimensions of the sample, resulting in a high sensitivity to very weak absorption.

Cavity ring down spectroscopy (CRDS) is a very different, but well-known technique for very high sensitivity absorption spectroscopy. Its initial development was described by O'Keefe and Deacon in a 1988 paper [7]. The rapidly developing interest in CRDS was demonstrated by the publication of a steadily increasing number of reviews and applications in subsequent years [8–11], as well as by an excellent perspective on CRDS developments by Vallance in 2005 [12].

Briefly, in CRDS the absorbing sample is placed between two high reflecting mirrors and a pulse of light is reflected back and forth through the sample many times. Sample absorption reduces the pulse energy and the long effective path length (due to the multiple reflections passing through the sample) makes this an extremely sensitive technique for weak absorption. But scattering also reduces the pulse energy and can lead to a significant systematic absorption error. Nevertheless, this is a widely used and powerful technique when scattering is negligible.

One might expect that a combination of these two absorption spectroscopy techniques would provide an extremely powerful and useful new technology—i.e. integrating CRDS (ICRDS). But, ICRDS has not previously been exploited because the diffuse reflectivity of all known materials was simply not high enough to do ring-down spectroscopy in an integrating cavity. But, we have now developed a new material (basically compressed quartz powder) that does have the high diffuse reflectivity required for ring-down spectroscopy in an integrating cavity. Reflectivities of 0.9992 at 532 nm and 0.9969 at 266 nm have been obtained [13]. In a particularly exciting demonstration, the first successful spectral absorption measurements were made on retinal pigmented epithelium (RPE) cells [14]; this is a biological sample whose absorption coefficient has proved difficult to measure due to its large scattering cross section.

ICRDS will open new research vistas by providing very sensitive and accurate direct spectral absorption measurements of both a sample and any particulates suspended in it while being unaffected by the scattering in the sample. As part of the celebration of quantum optics in the International Year of Light, this paper will review the ICRDS concept and will also introduce previously unpublished theoretical analyses of the considerations involved in some of the measurement procedures.

Figure 1 shows a diagram for a typical ICRDS implementation. The cavity has diffuse reflecting walls with a very high reflectivity (i.e. our compressed quartz powder) and is filled with a sample/suspension whose absorption is to be measured. The input laser pulse enters the cavity via a fiber optic and bounces around the cavity, reflecting off the walls and being scattered by any particles in the sample. A second fiber optic samples the laser light in the cavity and sends it to a detector that shows an exponential decrease in the intensity as the light is absorbed while bouncing around the cavity (through the sample).

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The ring-down time in an empty cavity is

$$\begin{eqnarray}&&{\tau }_{{\rm{empty}}}=\displaystyle \frac{1}{-\quad \mathrm{ln}\quad \rho }\displaystyle \frac{\bar{d}}{c},\end{eqnarray} \tag{ 1 }$$

where ρ is the wall reflectivity, $\bar{d}$ is the mean path length between reflections from the cavity wall, and c is the speed of light in the medium that fills the cavity [15]. For an arbitrarily shaped cavity whose volume is V and whose total surface area is S, the mean path $\bar{d}$ is a very simple relation [15]

$$\begin{eqnarray}&&\bar{d}=4\displaystyle \frac{V}{S}.\end{eqnarray} \tag{ 2 }$$

There is one further slight complication. For the diffuse reflector, light does not reflect right at the surface; rather, it slightly penetrates the surface and bounces around before finally exiting the surface in a Lambertian distribution. This is illustrated in figure 2.

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Consequently, at each reflection from the cavity wall, the light spends some average time δt within the wall. To include the fact that photons spend an average time δt within the wall of the cavity at each reflection, the ring-down time, equation (1), becomes

$$\begin{eqnarray}&&{\tau }_{{\rm{empty}}}=\displaystyle \frac{1}{-\quad \mathrm{ln}\quad \rho }\left(\displaystyle \frac{\bar{d}}{c}+\delta t\right).\end{eqnarray} \tag{ 3 }$$

Basically, the total time between reflections is the average time $\bar{d}$/c for a light ray to transit the cavity plus the time spent in the wall. In practice, δt is small; we have measured δt = 3.5 × 10⁻¹² s for our compressed quartz powder [13]. This is a typical value, but it does vary with the compression of the quartz powder. In any event, for a $\bar{d}$ of 3 cm and an empty cavity, the average transit time $\bar{d}$/c of the cavity is 100 ps and the time δt in the wall is then only a 3.5% effect. Larger cavities further decrease the effect of δt.

Now consider the case of a cavity filled with a medium of absorption coefficient α. If the zero of time is chosen to be at the instant of the first reflection, then at the time τ of the (n + 1)th reflection, τ = n$\bar{t},$ where $\bar{t}$ = $\bar{d}$/c is the mean time between reflections at the cavity wall. Hence, at time t = τ, the irradiance E_I incident on the wall is

$$\begin{eqnarray}&&{E}_{{\rm{I}}}(\tau )={\rho }^{n}{({{\rm{e}}}^{-\alpha \bar{d}})}^{n}{E}_{0}={{\rm{e}}}^{n\mathrm{ln}(\rho )-n\alpha \bar{d}}{E}_{0}.\end{eqnarray} \tag{ 4 }$$

Here, ${\rho }^{n}$ is a factor due to the n reflections with reflectivity ρ at each reflection; e^−αd is the average transmission through the medium during each transit of the cavity and is raised to the power n for the n transits of the cavity.

In general, the irradiance E_I incident on the wall at time t is

$$\begin{eqnarray}&&{E}_{{\rm{I}}}(t)={E}_{{\rm{0}}}{{\rm{e}}}^{-t/\tau },\end{eqnarray} \tag{ 5 }$$

and at time t = τ, the irradiance is

$$\begin{eqnarray}&&{E}_{{\rm{I}}}(\tau )={E}_{{\rm{0}}}{{\rm{e}}}^{-\tau /\tau }={E}_{{\rm{0}}}{{\rm{e}}}^{-1}.\end{eqnarray} \tag{ 6 }$$

Thus, from equations (4) and (6)

$$\begin{eqnarray}&&n\quad \mathrm{ln}(\rho )-n\alpha \bar{d}=-1,\end{eqnarray} \tag{ 7 }$$

and the number n of reflections in a time τ_α after the first reflection in a cavity filled with a medium of absorption coefficient α is

$$\begin{eqnarray}&&n=\displaystyle \frac{1}{-\quad \mathrm{ln}(\rho )+\alpha \bar{d}}.\end{eqnarray} \tag{ 8 }$$

The corresponding decay time τ_α is

$$\begin{eqnarray}&&{\tau }_{\alpha }=n\left(\displaystyle \frac{\bar{d}}{c}+\delta t\right)=\displaystyle \frac{1}{-\quad \mathrm{ln}(\rho )+\alpha \bar{d}}\left(\displaystyle \frac{\bar{d}}{c}+\delta t\right).\end{eqnarray} \tag{ 9 }$$

Solving equation (9) for the absorption coefficient gives

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{1}{c{\tau }_{\alpha }}\left(1+\displaystyle \frac{c\delta t}{\bar{d}}\right)+\displaystyle \frac{\mathrm{ln}(\rho )}{\bar{d}}.\end{eqnarray} \tag{ 10 }$$

If a cavity is filled with pure water (absorption coefficient α_pw), then from equation (9)

$$\begin{eqnarray}&&{\tau }_{{\rm{pw}}}=\displaystyle \frac{1}{[-\quad \mathrm{ln}\quad \rho +{\alpha }_{{\rm{pw}}}\bar{d}]}\left(\displaystyle \frac{\bar{d}}{{c}_{{\rm{w}}}}+\delta t\right),\end{eqnarray} \tag{ 11 }$$

where c_w is the speed of light in water. Adding a dye, α_dye, to the water gives

$$\begin{eqnarray}&&{\tau }_{{\rm{dye}}}=\displaystyle \frac{1}{[-\quad \mathrm{ln}\quad \rho +({\alpha }_{{\rm{pw}}}+{\alpha }_{{\rm{dye}}})\bar{d}]}\left(\displaystyle \frac{\bar{d}}{{c}_{{\rm{w}}}}+\delta t\right),\end{eqnarray} \tag{ 12 }$$

and the total absorption coefficient corresponding to equation (10) is

$$\begin{eqnarray}&&{\alpha }_{{\rm{pw}}}+{\alpha }_{{\rm{dye}}}=\displaystyle \frac{1}{{c}_{{\rm{w}}}{\tau }_{{\rm{dye}}}}\left(1+\displaystyle \frac{{c}_{{\rm{w}}}\delta t}{\bar{d}}\right)+\displaystyle \frac{\mathrm{ln}(\rho )}{\bar{d}}.\end{eqnarray} \tag{ 13 }$$

The objective of ring-down spectroscopy is the measurement of absorption coefficients; the measurement of the requisite decay times such as those for τ_empty in equation (3), τ_pw in equation (11), and τ_dye in equation (12) is straightforward. However, there are two parameters, ρ and δt, in these equations that must be considered in any measurements of absorption coefficients. There are several possible approaches.

• (i) Measuring α₀ when a dye, other dissolved substance, or suspended particulates are added to a cavity filled with pure water.

Equations (11) and (12) give

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{0}}-\displaystyle \frac{1}{{\tau }_{{\rm{pw}}}}=\displaystyle \frac{{c}_{{\rm{w}}}}{1+{c}_{{\rm{w}}}\delta t/\bar{d}}{\alpha }_{0}\end{eqnarray} \tag{ 14 }$$

and hence the absorption coefficient for the dye or other dissolved/suspended substance in the water is

$$\begin{eqnarray}&&{\alpha }_{0}=\displaystyle \frac{1}{{c}_{{\rm{w}}}}\left(\displaystyle \frac{1}{{\tau }_{0}}-\displaystyle \frac{1}{{\tau }_{{\rm{pw}}}}\right)\left(1+\displaystyle \frac{{c}_{{\rm{w}}}\delta t}{\bar{d}}\right).\end{eqnarray} \tag{ 15 }$$

Perhaps the outstanding feature of equation (15) is that if the quantity ${c}_{{\rm{w}}}\delta t/\bar{d}$ is known or is negligible, then the absorption coefficient for the dye is given in absolute terms (without any need to compare with an absorption standard) by simple measurement of the two ring-down times. Basically, ICRDS is providing an absorption standard. As an example of the magnitude of ${c}_{{\rm{w}}}\delta t/\bar{d},$ consider a spherical cavity that is 15 cm diameter; this gives $\bar{d}\ =\ 10\; cm$ [15], and also take δt = 0.004 ns [13]. Since the speed of light in water is c_w = 2.25 × 10¹⁰ cm s⁻¹, the correction to α₀ in equation (15) is then ${c}_{{\rm{w}}}\delta t/\bar{d}\ =\ 0.009,$ i.e. 0.9%. This correction term can be made even smaller by using larger cavities; it can also be measured to significantly improve the accuracy.

Specifically, ${c}_{{\rm{w}}}\delta t/\bar{d}$ can be determined if one measures τ_ref for a sample whose absorption coefficient α_ref is known via calibration against some standard reference, then equation (15) (with α₀ and τ₀ replaced by α_ref and τ_ref) can be used to determine (1 + ${c}_{{\rm{w}}}\delta t/\bar{d})$ in terms of the known absorption coefficient α_ref

$$\begin{eqnarray}&&1+\displaystyle \frac{{c}_{{\rm{w}}}\delta t}{\bar{d}}=\displaystyle \frac{{c}_{{\rm{w}}}{\alpha }_{{\rm{ref}}}}{1/{\tau }_{{\rm{ref}}}-1/{\tau }_{{\rm{pw}}}}={c}_{{\rm{w}}}{\alpha }_{{\rm{ref}}}\displaystyle \frac{{\tau }_{{\rm{pw}}}{\tau }_{{\rm{ref}}}}{{\tau }_{{\rm{pw}}}-{\tau }_{{\rm{ref}}}}.\end{eqnarray} \tag{ 16 }$$

Then, α₀ for any dye solution (or particulate suspension) is obtained using equation (15) together with the known quantity from equation (16)

$$\begin{eqnarray}&&{\alpha }_{0}={\alpha }_{{\rm{ref}}}\displaystyle \frac{1/{\tau }_{{\rm{dye}}}-1/{\tau }_{{\rm{pw}}}}{1/{\tau }_{{\rm{ref}}}-1/{\tau }_{{\rm{pw}}}}={\alpha }_{{\rm{ref}}}\displaystyle \frac{{\tau }_{{\rm{ref}}}}{{\tau }_{{\rm{dye}}}}\left(\displaystyle \frac{{\tau }_{{\rm{pw}}}-{\tau }_{{\rm{dye}}}}{{\tau }_{{\rm{pw}}}-{\tau }_{{\rm{ref}}}}\right).\end{eqnarray} \tag{ 17 }$$

It should again be emphasized that the quantity on the left side of equation (16) is only slightly different from unity; consequently, errors in α_ref would be expected to have an insignificant effect on the correction factor. Furthermore, the right hand side of equation (16) can be evaluated to check the accuracy (i.e. how close it is to unity). It should also be noted that δt can be determined from equation (16)

$$\begin{eqnarray}&&\delta t=\displaystyle \frac{\bar{d}}{{c}_{{\rm{w}}}}\left(\displaystyle \frac{{c}_{{\rm{w}}}{\alpha }_{{\rm{ref}}}{\tau }_{{\rm{pw}}}{\tau }_{{\rm{ref}}}}{{\tau }_{{\rm{pw}}}-{\tau }_{{\rm{ref}}}}-1\right).\end{eqnarray} \tag{ 18 }$$

Consider this a little further. Rewriting equation (15) for some reference dye α_ref

$$\begin{eqnarray}&&\displaystyle \frac{{\tau }_{{\rm{pw}}}{\tau }_{{\rm{ref}}}}{{\tau }_{{\rm{pw}}}-{\tau }_{{\rm{ref}}}}{\alpha }_{{\rm{ref}}}{c}_{{\rm{w}}}=1+\displaystyle \frac{{c}_{{\rm{w}}}\delta t}{\bar{d}}.\end{eqnarray} \tag{ 19 }$$

Now, for any specific cavity, $\bar{d}$ and δt are constants and the right hand side of equation (19) must be a constant. Hence, for any given cavity, the left hand side of equation (19) will be a dimensionless parameter that is a constant regardless of the value of α_ref ≠ 0; call it K($\bar{d},$ δt)

$$\begin{eqnarray}&&K(\bar{d},\delta t)=\displaystyle \frac{{\tau }_{{\rm{pw}}}{\tau }_{{\rm{ref}}}}{{\tau }_{{\rm{pw}}}-{\tau }_{{\rm{ref}}}}{\alpha }_{{\rm{ref}}}{c}_{{\rm{w}}}=\displaystyle \frac{1}{1/{\tau }_{{\rm{ref}}}\;-1/{\tau }_{{\rm{pw}}}}{\alpha }_{{\rm{ref}}}{c}_{{\rm{w}}}.\end{eqnarray} \tag{ 20 }$$

To summarize, regardless of the amount of dye dissolved, the measured decay time τ_ref must be such that K is independent of α_ref. Take a cavity filled with water, then add some dye and evaluate K. Add some more dye and evaluate K again; one should get the same value. If one has an absorption standard α_ref, then τ_ref can be measured with some cavity and K($\bar{d},$ δt) for that specific cavity can be accurately determined from equation (20). Then, equation (19) gives

$$\begin{eqnarray}&&1+\displaystyle \frac{{c}_{{\rm{w}}}\delta t}{\bar{d}}=K(\bar{d},\delta t).\end{eqnarray} \tag{ 21 }$$

The absorption coefficient α₀ for any other solution/suspension is obtained by measuring τ₀ for that sample and using equation (20)

$$\begin{eqnarray}&&{\alpha }_{0}=\displaystyle \frac{K(\bar{d},\delta t)}{{c}_{{\rm{w}}}}\left(\displaystyle \frac{1}{{\tau }_{0}}-\displaystyle \frac{1}{{\tau }_{{\rm{pw}}}}\right).\end{eqnarray} \tag{ 22 }$$

This is, in fact, identical to equation (17). Finally, the wall time δt is given by

$$\begin{eqnarray}&&\delta t=\displaystyle \frac{\bar{d}}{{c}_{{\rm{w}}}}[K(\bar{d},\delta t)-1].\end{eqnarray} \tag{ 23 }$$

• (ii) Evaluating parameters by varying the size of the integrating cavity.

Consider first the determination of δt using empty cavities. Equation (3) gives

$$\begin{eqnarray}&&{\tau }_{{\rm{empty}}}=\displaystyle \frac{1}{-c\;\mathrm{ln}\;\rho }\bar{d}+\displaystyle \frac{\delta t}{-\quad \mathrm{ln}\;\rho }.\end{eqnarray} \tag{ 24 }$$

Now measure τ_empty for five different size cavities with identical diffuse reflecting walls and with $\bar{d}$ = d, 2d, 3d, 4d, and 5d. A plot y = τ_empty versus x = $\bar{d}$ is a straight line with slope m and y-intercept b

$$\begin{eqnarray}&&m=\displaystyle \frac{1}{-c\;\mathrm{ln}\;\rho },\quad \quad b=\displaystyle \frac{\delta t}{-\quad \mathrm{ln}\;\rho }.\end{eqnarray} \tag{ 25 }$$

Hence, ρ is given by ln ρ = −1/mc, or explicitly as

$$\begin{eqnarray}&&\rho ={{\rm{e}}}^{-1/mc}=1-\displaystyle \frac{1}{mc}+\displaystyle \frac{1}{{\rm{2}}{m}^{{\rm{2}}}\quad {c}^{{\rm{2}}}}-\ldots \;\approx 1-\displaystyle \frac{1}{mc},\end{eqnarray} \tag{ 26 }$$

and δt is given by

$$\begin{eqnarray}&&\delta t=-b\;\mathrm{ln}\;\rho =\displaystyle \frac{b}{mc}.\end{eqnarray} \tag{ 27 }$$

The approach of using cavities of different sizes and plotting results as a function of $\bar{d}$ can also be used to obtain absorption coefficient measurements. Consider, for example, the measurement of α_o when the cavity is filled with a medium, e.g. a water solution, some other medium, a particle suspension, etc. From equation (10), the absorption coefficient α_o is

$$\begin{eqnarray}&&{\alpha }_{{\rm{o}}}=\displaystyle \frac{1}{{c}_{{\rm{o}}}{\tau }_{{\rm{o}}}}\left(1+\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)+\displaystyle \frac{\mathrm{ln}\;\rho }{\bar{d}},\end{eqnarray} \tag{ 28 }$$

where c_o is the speed of light in the medium. Rearranging gives

$$\begin{eqnarray}&&\displaystyle \frac{1}{{c}_{{\rm{o}}}{\tau }_{{\rm{o}}}}=\left(\displaystyle \frac{-\quad \mathrm{ln}\;\rho }{\bar{d}}+{\alpha }_{{\rm{o}}}\right)\displaystyle \frac{1}{1+{c}_{{\rm{o}}}\delta t/\bar{d}}.\end{eqnarray} \tag{ 29 }$$

The power series expansion

$$\begin{eqnarray}&&\displaystyle \frac{1}{1+{c}_{{\rm{o}}}\delta t/\bar{d}}\approx 1-\left(\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)+{\left(\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)}^{2}-\ldots ,\end{eqnarray} \tag{ 30 }$$

can be inserted into equation (29) to obtain

$$\begin{eqnarray}\displaystyle \frac{1}{{c}_{{\rm{o}}}{\tau }_{{\rm{o}}}} & = & {\alpha }_{{\rm{o}}}+\left[\displaystyle \frac{-\quad \mathrm{ln}\;\rho }{\bar{d}}-{\alpha }_{{\rm{o}}}\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right]\\ & & \times \left\{1-\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}+{\left(\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)}^{2}-{\left(\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)}^{3}+\ldots \right\},\end{eqnarray} \tag{ 31 }$$

or moving a factor ${c}_{{\rm{o}}}\delta t/\bar{d}$ from the square brackets to the curly brackets gives

$$\begin{eqnarray}\displaystyle \frac{1}{{c}_{{\rm{o}}}{\tau }_{{\rm{o}}}} & = & {\alpha }_{m}-\left(\displaystyle \frac{\mathrm{ln}\;\rho }{{c}_{{\rm{o}}}\delta t}+{\alpha }_{{\rm{o}}}\right)\left\{\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}-{\left(\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)}^{2}\right.\\ & & \left.+{\left(\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)}^{3}-{\left(\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)}^{4}+\ldots \right\}.\end{eqnarray} \tag{ 32 }$$

Since ${c}_{{\rm{o}}}\delta t/\bar{d}$ is very small, drop the higher order terms

$$\begin{eqnarray}&&\displaystyle \frac{1}{{c}_{{\rm{o}}}{\tau }_{{\rm{o}}}}={\alpha }_{{\rm{o}}}-\left(\displaystyle \frac{\mathrm{ln}\;\rho }{{c}_{{\rm{o}}}\delta t}+{\alpha }_{{\rm{o}}}\right)\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}.\end{eqnarray} \tag{ 33 }$$

Now, measure τ_o for several $\bar{d}$ and plot y = 1/c_oτ_o versus x = 1/$\bar{d}.$ Fit this straight line data to the function

$$\begin{eqnarray}&&y(x)=A+Bx,\end{eqnarray} \tag{ 34 }$$

and determine the fitting parameters A and B. The parameter A is just the absorption coefficient α_o of the medium in the cavity.

• (iii) Evaluating parameters by varying the absorption of the calibration dyes.

Consider a medium whose absorption coefficient is α_o. Then add carefully measured drops of a calibrated dye solution to obtain an accurately known and steadily increasing α_dye. The decay times τ_dye are measured after each drop of dye is added. From equation (10), we have both

$$\begin{eqnarray}&&{\alpha }_{{\rm{o}}}=\displaystyle \frac{1}{{c}_{{\rm{o}}}{\tau }_{{\rm{o}}}}\left(1+\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)+\displaystyle \frac{\mathrm{ln}\;\rho }{\bar{d}},\end{eqnarray} \tag{ 35 }$$

and

$$\begin{eqnarray}&&{\alpha }_{{\rm{dye}}}+{\alpha }_{{\rm{o}}}=\displaystyle \frac{1}{{c}_{{\rm{o}}}{\tau }_{{\rm{dye}}}}\left(1+\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)+\displaystyle \frac{\mathrm{ln}\;\rho }{\bar{d}}.\end{eqnarray} \tag{ 36 }$$

From equation (35) we obtain

$$\begin{eqnarray}&&\displaystyle \frac{1}{{c}_{{\rm{o}}}}\left(1+\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right)={\tau }_{{\rm{o}}}\left({\alpha }_{{\rm{o}}}-\displaystyle \frac{\mathrm{ln}\;\rho }{\bar{d}}\right).\end{eqnarray} \tag{ 37 }$$

Using this result in equation (36) gives

$$\begin{eqnarray}&&{\alpha }_{{\rm{dye}}}=\displaystyle \frac{{\tau }_{{\rm{o}}}}{{\tau }_{{\rm{dye}}}}\left({\alpha }_{{\rm{o}}}-\displaystyle \frac{\mathrm{ln}\;\rho }{\bar{d}}\right)+\displaystyle \frac{\mathrm{ln}\;\rho }{\bar{d}}-{\alpha }_{{\rm{o}}}.\end{eqnarray} \tag{ 38 }$$

From this expression, a plot of α_dye as a function of 1/τ_dye is a straight line with slope m and y-intercept b where

$$\begin{eqnarray}&&m={\tau }_{{\rm{o}}}\left({\alpha }_{{\rm{o}}}-\displaystyle \frac{\mathrm{ln}\;\rho }{\bar{d}}\right)\end{eqnarray} \tag{ 39 }$$

and

$$\begin{eqnarray}&&b=\displaystyle \frac{\mathrm{ln}\;\rho }{\bar{d}}-{\alpha }_{{\rm{o}}}.\end{eqnarray} \tag{ 40 }$$

Using equation (37) in (39) gives

$$\begin{eqnarray}&&m=\displaystyle \frac{1}{{c}_{{\rm{o}}}}\left(1+\displaystyle \frac{{c}_{{\rm{o}}}\delta t}{\bar{d}}\right),\end{eqnarray} \tag{ 41 }$$

which is clearly a constant for a given cavity (i.e. independent of the absorbing medium in the cavity). The measured y-intercept b and slope m together with the measured decay time τ_o can be used in equation (39) to obtain the absorption coefficient α_o

$$\begin{eqnarray}&&{\alpha }_{{\rm{o}}}=\displaystyle \frac{m}{{\tau }_{{\rm{o}}}}+b{\rm{.}}\end{eqnarray} \tag{ 42 }$$

This result for α_o together with the y-intercept b could be used in equation (40) to obtain the wall reflectivity ρ

$$\begin{eqnarray}&&\mathrm{ln}\;\rho =\bar{d}(b+{\alpha }_{{\rm{o}}}).\end{eqnarray} \tag{ 43 }$$

• (iv) Determining α when the sample occupies only a part of the cavity volume.

Consider the measurement of the absorption coefficient α_s for an unknown absorber which is suspended in a liquid sample that occupies only a part of the integrating cavity volume. This is a brief review of previously published work [14]. Basically, the problem being addressed is that a biological cell sample is small and requires a habitable environment to maintain viability. Specifically, the measurement of the absorption coefficient of living cells must be performed with the cells suspended in a buffer solution, e.g. a phosphate buffered saline (PBS) solution. Consider a cavity in which only part of the volume is occupied by a PBS solution that contains a sample (e.g. cells); then equation (11) is modified to become

$$\begin{eqnarray}&&{\tau }_{{\rm{s}}}=\displaystyle \frac{1}{[-\quad \mathrm{ln}\;\rho +({\alpha }_{{\rm{PBS}}}+{\alpha }_{{\rm{s}}})\bar{d}]}\left(\displaystyle \frac{\bar{d}}{c}+\displaystyle \frac{{\bar{d}}_{{\rm{s}}}}{{c}_{{\rm{s}}}}+\delta t\right),\end{eqnarray} \tag{ 44 }$$

where α_PBS is the absorption coefficient of the PBS buffer solution, α_s is the absorption coefficient of the sample, $\bar{d}$ is the average distance that light travels through the empty volume of the cavity between reflections from the cavity wall, ${\bar{d}}_{{\rm{s}}}$ is the average distance that light travels through the sample volume of the cavity between reflections from the cavity wall, c is the speed of light in the empty cavity, and c_s is the speed of light in the sample. If the PBS solution contains no sample then the decay time τ_PBS is

$$\begin{eqnarray}&&{\tau }_{{\rm{PBS}}}=\displaystyle \frac{1}{[-\quad \mathrm{ln}\;\rho +{\alpha }_{{\rm{PBS}}}\bar{d}]}\left(\displaystyle \frac{\bar{d}}{c}+\displaystyle \frac{{\bar{d}}_{{\rm{s}}}}{{c}_{{\rm{s}}}}+\delta t\right),\end{eqnarray} \tag{ 45 }$$

where the position, shape, and size of the volume of the buffer solution are identical to those of the cell/buffer combination. Subtracting the reciprocals of equations (44) and (45) gives

$$\begin{eqnarray}&&{\tau }_{{\rm{s}}}^{-1}-{\tau }_{{\rm{PBS}}}^{-1}=\displaystyle \frac{{\alpha }_{{\rm{s}}}{\bar{d}}_{{\rm{s}}}}{\left(\displaystyle \frac{\bar{d}}{c}+\displaystyle \frac{{\bar{d}}_{{\rm{s}}}}{{c}_{{\rm{s}}}}+\delta t\right)}.\end{eqnarray} \tag{ 46 }$$

If a dye is added to the buffer solution to obtain a solution with a known absorption coefficient α_dye, then the decay time is

$$\begin{eqnarray}&&{\tau }_{{\rm{dye}}}=\displaystyle \frac{1}{[-\quad \mathrm{ln}\;\rho +{\alpha }_{{\rm{dye}}}\bar{d}]}\left(\displaystyle \frac{\bar{d}}{c}+\displaystyle \frac{{\bar{d}}_{{\rm{s}}}}{{c}_{{\rm{s}}}}+\delta t\right).\end{eqnarray} \tag{ 47 }$$

Now measure τ for two different dye dilutions with known absorption coefficients (α_dye1 and α_dye2) and subtract their inverses, to obtain

$$\begin{eqnarray}&&{\tau }_{{\rm{dye1}}}^{-1}-{\tau }_{{\rm{dye2}}}^{-1}=\displaystyle \frac{({\alpha }_{{\rm{dye1}}}-{\alpha }_{{\rm{dye2}}}){\bar{d}}_{{\rm{s}}}}{\left(\displaystyle \frac{\bar{d}}{c}+\displaystyle \frac{{\bar{d}}_{{\rm{s}}}}{{c}_{{\rm{s}}}}+\delta t\right)}.\end{eqnarray} \tag{ 48 }$$

Divide equation (46) by equation (48) to obtain

$$\begin{eqnarray}&&{\alpha }_{{\rm{s}}}=\displaystyle \frac{{\tau }_{{\rm{s}}}^{-1}-{\tau }_{{\rm{PBS}}}^{-1}}{{\tau }_{{\rm{dye1}}}^{-1}-{\tau }_{{\rm{dye2}}}^{-1}}({\alpha }_{{\rm{dye1}}}-{\alpha }_{{\rm{dye2}}}).\end{eqnarray} \tag{ 49 }$$

Therefore, the absorption coefficient for the dissolved sample can be found based on the values of the absorption coefficients for two known dye solutions and the measurements of the relevant ICRDS decay times.

To demonstrate the capability for high sensitivity absorption measurements in the presence of severe scattering, ICRDS was used to measure the base absorption (i.e. with the pigment removed) of 60 million human RPE cells in a 3 ml PBS solution [14]. The results are shown in figure 3. Method iv described in the previous section was used for the ICRDS measurements. A spectrophotometer was used for the total attenuation (scattering plus absorption) measurements. As a demonstration of the dominance of scattering in this transmission measurement, the spectrophotometer attenuation coefficient from figure 3 for these RPE cells at 500 nm was ∼4.6 cm⁻¹ and the ICRDS measurement of absorption was ∼0.02 cm⁻¹. This is a factor of 230, i.e. the absorption is less than one-half of one percent of the attenuation.

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Figure 4 is another example of the sensitivity of ICRDS; it shows the absorption spectrum of a sample of pig brain tissue. Figures 3 and 4 are examples of biological absorption spectra that were previously inaccessible for study and analysis.

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ICRDS has provided a major new advance in high sensitivity measurements of light absorption even in the presence of severe scattering. The discussions in the Measurement procedures section are an indication of the extraordinary flexibility that the integrating cavity brings to ring-down spectroscopy studies.

A very important application is the measurement of optical absorption in biological tissues. Previously, some attempts have used transmission-style experiments in which an attenuation coefficient is measured (for example, in a spectrophotometer) by observing the decrease in the intensity of a light source as it passes through a sample [16]. This is really an extinction measurement and when scattering is severe (typical in biological samples) the absorption characteristics may be completely obscured. ICRDS can be expected to have major impacts on the understanding and modeling of processes in highly scattering cellular media, e.g. tissues, membranes, cells, organelles, etc. A vast number of other techniques and procedures will be impacted—e.g. monitoring and measuring the optical absorption of ocean waters, or monitoring the environment for pathogens and cytotoxins.

This work received support from the National Science Foundation (NSF) OCE1333425, the George P Mitchell Chair in Experimental Physics, and the Robert A Welch Foundation under grant A-1218.