For several of the cavity ringdown spectroscopy (CRDS) experiments described in Chapter 3 (3.3.1 and 3.3.3), kinetic processes occurred with time constants comparable to the lifetime of a photon within the cavity. To determine rate constants for these reactions, data was collected and analyzed using the Simultaneous Kinetics and Ringdown (SKaR) technique published by Brown et al.1 The original SKaR paper assumes that the delay time between the photolysis (initiates radical chemistry) and probe (initiates a cavity ringdown decay) laser pulses is fixed at zero. In this Appendix, we show that a ringdown ratio constructed using many arbitrary delay times is equivalent to a ringdown ratio acquired with a delay time fixed at zero. This modified approach has a variety of advantages that will be discussed and was used in all SKaR experiments described in Chapter 3.
The SKaR technique is typically used with vacuum cavity ringdown times that exceed kinetic lifetimes to ensure that an entire kinetic trace can be captured within a single ringdown event. The probe laser beam is injected into the optical cavity when photolysis occurs so that all kinetic information can be observed during a single cavity decay. After many ringdown times, the signal to noise decreases substantially as intracavity power drops. When the ratio of the ringdown with kinetics is taken to the reference ringdown, the noise propagates as two noisy signals are divided. This sets a limit on the usable time range in SKaR. If the empty cavity ringdown time exceeds the kinetic lifetime, then the usable time range is often sufficient to extract a precise rate coefficient so long as one properly weights the fit to the ringdown ratio. However, achieving long ringdown times requires either a long cavity length or very high reflectivity mirrors that may be expensive or impractical to implement. One can envision an experiment in which the empty cavity ringdown time is shorter than the kinetic lifetime making the ringdown ratio too noisy to precisely extract a rate coefficient, but the ringdown with kinetics is still nonexponential.
Here, we show that by acquiring ringdown ratios at multiple delay times between the probe and photolysis lasers, one can construct a single ratio function that exhibits high signal to noise across an arbitrary number of empty cavity ringdown times. This ratio can
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be fit to extract an extremely precise rate coefficient, alleviating any concerns about whether the empty cavity ringdown time is long enough to implement the SKaR technique.
Let 𝑡 represent the time since photolysis was initiated (the start of kinetics) and let 𝑡𝑑 represent the time at which the probe beam is injected into the optical cavity. Then the intensity of the ringdown acquired during a kinetic event is defined for 𝑡 ≥ 𝑡𝑑 by:
𝐼𝑜𝑛(𝑡) = 𝐼0,𝑜𝑛exp [−𝑐
𝑑 ∫ 𝐴(𝑡′)𝑑𝑡′−(𝑡 − 𝑡𝑑) 𝜏0
𝑡 𝑡𝑑
] (Eqn. 1)
where 𝐼0,𝑜𝑛 is the intensity of the ringdown at 𝑡 = 𝑡𝑑, 𝑐 is the speed of light, 𝑑 is the distance between the cavity mirrors, 𝜏0 is the ringdown time in the absence of photolysis, and 𝐴(𝑡) is a function defined for 𝑡 ≥ 0 that describes absorbance at the probe wavelength after the start of kinetics. If 𝐴(𝑡) is approximately constant between 𝑡𝑑 and a value of 𝑡 by which 𝐼𝑜𝑛(𝑡) is dominated by noise, then Eqn. 1 reduces to the traditional exponential decay for a cavity with static absorbance 𝐴:
𝐼𝑜𝑛(𝑡) = 𝐼0,𝑜𝑛exp [− (𝑐
𝑑𝐴 + 1
𝜏0) (𝑡 − 𝑡𝑑)] (Eqn. 2) The intensity of the ringdown acquired without photolysis is defined for 𝑡 ≥ 𝑡𝑑 by:
𝐼𝑜𝑓𝑓(𝑡) = 𝐼0,𝑜𝑓𝑓exp [−(𝑡 − 𝑡𝑑)
𝜏0 ] (Eqn. 3)
where 𝐼0,𝑜𝑓𝑓 is the intensity of the ringdown at 𝑡 = 𝑡𝑑. Taking the ratio of Eqn. 1 to Eqn.
3 gives the following ratio function defined for 𝑡 ≥ 𝑡𝑑: 𝑅(𝑡) = 𝐼𝑜𝑛(𝑡)
𝐼𝑜𝑓𝑓(𝑡)= 𝐼0,𝑜𝑛
𝐼0,𝑜𝑓𝑓exp [−𝑐
𝑑 ∫ 𝐴(𝑡′)𝑑𝑡′
𝑡 𝑡𝑑
] (Eqn. 4)
For the case that 𝑡𝑑 = 0, the derivation shown thus far reduces to that presented in the original SKaR paper.1
Now consider that one has measured a set of ringdown ratios {𝑅𝑖(𝑡)} with corresponding delay times {𝑡𝑑𝑖}. 𝑅𝑖(𝑡) is defined for 𝑡 ≥ 𝑡𝑑𝑖 by:
𝑅𝑖(𝑡) = (𝐼0,𝑜𝑛 𝐼0,𝑜𝑓𝑓)
𝑖
exp [−𝑐
𝑑 ∫ 𝐴(𝑡′)𝑑𝑡′
𝑡 𝑡𝑑𝑖
] (Eqn. 5)
Without loss of generality, the sets can be reordered after collection such that {𝑡𝑑𝑖} is
173 delay time to the ringdown ratio at the earliest delay time is a constant defined for 𝑡 ≥ 𝑡𝑑𝑖:
𝑅𝑖(𝑡) 𝑅1(𝑡) =
(𝐼0,𝑜𝑛 𝐼0,𝑜𝑓𝑓)
𝑖
exp [−𝑐
𝑑 ∫𝑡𝑡 𝐴(𝑡′)𝑑𝑡′
𝑑𝑖 ]
(𝐼0,𝑜𝑛 𝐼0,𝑜𝑓𝑓)
1
exp [−𝑐
𝑑 ∫𝑡𝑡 𝐴(𝑡′)𝑑𝑡′
𝑑1 ]
(Eqn. 6)
= [ (𝐼0,𝑜𝑛
𝐼0,𝑜𝑓𝑓)
𝑖
(𝐼0,𝑜𝑛 𝐼0,𝑜𝑓𝑓)
1]
exp [−𝑐
𝑑 (∫ 𝐴(𝑡′)𝑑𝑡′
𝑡 𝑡𝑑𝑖
− ∫ 𝐴(𝑡′)𝑑𝑡′
𝑡 𝑡𝑑1
)]
= [ (𝐼0,𝑜𝑛
𝐼0,𝑜𝑓𝑓)
𝑖
(𝐼0,𝑜𝑛 𝐼0,𝑜𝑓𝑓)
1]
exp [−𝑐
𝑑 (∫ 𝐴(𝑡′)𝑑𝑡′
𝑡 𝑡𝑑𝑖
− ∫ 𝐴(𝑡′)𝑑𝑡′
𝑡𝑑𝑖 𝑡𝑑1
− ∫ 𝐴(𝑡′)𝑑𝑡′
𝑡 𝑡𝑑𝑖
)]
= [ (𝐼0,𝑜𝑛
𝐼0,𝑜𝑓𝑓)
𝑖
(𝐼0,𝑜𝑛 𝐼0,𝑜𝑓𝑓)
1] exp [𝑐
𝑑∫ 𝐴(𝑡′)𝑑𝑡′
𝑡𝑑𝑖 𝑡𝑑1
]
= 𝐶𝑖
since 𝑡𝑑𝑖 ≥ 𝑡𝑑1. 𝐶𝑖 is a constant independent of time. Therefore, a ringdown ratio measured at any arbitrary delay time differs from the ringdown ratio measured at the earliest delay time by two constant factors: (1) the initial intensities of the ringdowns and (2) how the absorbance has evolved between the two delay times.
The function 𝑅1(𝑡) can be determined not only by direct measurement at 𝑡𝑑1 but can also be constructed iteratively via measurements of {𝑅𝑖(𝑡)}:
𝑅1(𝑡) =
{
𝑅1𝑚(𝑡) for 𝑡 ≥ 𝑡𝑑1 then avg (𝑅1(𝑡),𝑅2𝑚(𝑡)
𝐶2 ) for 𝑡 ≥ 𝑡𝑑2 …
then avg (𝑅1(𝑡),𝑅𝑛𝑚(𝑡)
𝐶𝑛 ) for 𝑡 ≥ 𝑡𝑑𝑛 }
(Eqn. 7)
where 𝑅𝑖𝑚(𝑡) denotes the measurement of the ringdown ratio at 𝑡𝑑𝑖 (as opposed to its analytical expression) and 𝑛 is the total number of ringdown ratios measured. The values
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of 𝐶𝑖 are determined sequentially as 𝑅1(𝑡) is constructed, through a linear least squares optimization of
𝑅𝑖𝑚(𝑡) = 𝐶𝑖𝑅1(𝑡) (Eqn. 8) over 𝑡 ≥ 𝑡𝑑𝑖. That is, we first construct 𝑅1(𝑡) for 𝑡 ≥ 𝑡𝑑1. 𝐶2 is then computed through a fit to Eqn. 8. Next, 𝑅1(𝑡) for 𝑡 ≥ 𝑡𝑑2 is updated using 𝐶2 in Eqn. 7. This process is repeated for computing 𝐶3 and updating 𝑅1(𝑡) for 𝑡 ≥ 𝑡𝑑3 and so on until all ringdown ratios have been integrated into the construction of 𝑅1(𝑡). At each step the averaging is variance- weighted to ensure that every point in 𝑅1(𝑡) is biased toward the ringdown ratio with the least uncertainty.
A variance-weighted fit of the constructed 𝑅1(𝑡) to Eqn. 5 yields a precise rate coefficient for the kinetic process under study. The proposed method improves upon the original SKaR technique by taking advantage of the high signal to noise exhibited by 𝑅𝑖(𝑡) immediately after 𝑡𝑑𝑖. By measuring ringdown ratios across delay times that span several kinetic lifetimes, and which have sufficiently narrow spacing between the delays, a ringdown ratio is constructed that has excellent signal to noise across the entire kinetic timescale. This approach has two main advantages over simply averaging a large number of ringdown ratios at a single delay time: (1) Transient digitizers have a fixed number of bits which dictate the maximum precision of the A/D conversion. After a certain amount of averaging, there will be no improvements to the signal to noise of a ringdown because the limit of the transient digitizer has been reached. (2) Averaging at a single delay time results in a ringdown ratio that has the least uncertainty at early times and the greatest uncertainty at long times. Fits will therefore be biased toward changes in absorbance that occur near the beginning of a kinetic event, immediately after photolysis. The modified method creates a ringdown ratio with nearly uniform signal to noise at all times, and therefore the fit will unbiasedly represent the entire kinetic timescale. In conclusion, the modified method allows SKaR to be utilized in experiments where the ringdown time is shorter than the kinetic lifetime, but the ringdown with kinetics is still nonexponential.
(1) Brown, S. S.; Ravishankara, A. R.; Stark, H. Simultaneous Kinetics and Ring-down:
Rate Coefficients from Single Cavity Loss Temporal Profiles. J. Phys. Chem. A
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