Two-level model

Laser-induced fluorescence can be divided into three main processes by considering a simple two-level model. Figure 3.1 shows a simple diagram of a two-level model. First, a molecule in energy level 1 absorbs a photon from the laser and moves to level 2; this is the excitation process. Second, the excited molecule may undergo collisions, relaxing back down to energy level 1 without emitting a photon; this will be called the energy transfer process. Third, the molecule may instead relax to the lower energy state by emitting a photon; this is the fluorescence process. The following sections describe each of these processes in more detail.

excited state

2

Q

PXNofB

ground state

1

Figure 3.1: The two-level model. The number of molecules in level 1 depends on p,

the gas density, Xno, the mole fraction of nitric oxide and f B, the Boltzmann fraction. The fraction of molecules that are excited depends on gh the spectral overlap integral between the laser and the molecular transition, E, the energy in a laser pulse, and 5/2,

the Einstein transition probability for stimulated absorption. The amount of

fluorescence depends on the relative rates of spontaneous emission and quenching. These rates are characterized by A, the Einstein transition probability for spontaneous emission, and Q, the collisional quenching rate.

3.2.1.1 Excitation

In order for the excitation process to occur, the laser must be tuned to a particular transition in the resonant species (nitric oxide in these experiments). The amount of light that is actually absorbed will depend on several factors. First, it will depend on the number of absorbers present, which is a function of gas density, species mole fraction, and the population of the probed state (or states, in the case of spectrally coincident transitions). The relative population of the probed state is governed by the Boltzmann fraction, and is a function of the rotational quantum number of the probed state and the temperature of the gas. Second, fluorescence will tend to increase with laser energy unless the laser energy is sufficiently high so as to deplete the probed state, in which case the transition is said to be saturated. Third, it will depend on the amount of overlap

between the spectral profile of the laser and the molecular absorption profile. The absorption profile can be broadened and shifted by several means, some of which are functions of temperature and pressure, as will be discussed in greater detail below. Finally, the amount of absorption depends upon the quantum mechanical probability of the absorption transition occurring, a rate which is characterized by the Einstein B coefficient for stimulated absorption.

3.2.1.2 E nergy transfer

Once a molecule of the absorbing species has been excited to a higher electronic state, collisions may alter the energy of or depopulate the excited state. Collision processes that change the rotational (J or N) or vibrational (v) quantum number of the excited molecule are known as rotational energy transfer (RET) and vibrational energy transfer (VET), respectively. For the conditions encountered in our tests, VET is slow as compared to the fluorescence lifetime (Stephenson 1974). RET rates were calculated using LINUS (Palma 1998) over a range of conditions expected in the present experiments. They were calculated to be between a factor of 5 and 1,600 times faster than the spontaneous emission rate. The fluorescence lifetime expected in these tests is

on the order of 200 ns. Molecules having undergone RET may still be in an

electronically excited state, and so may still fluoresce.

An additional collisional process exists in which the excited molecule transfers energy to another molecule, and returns to the electronic ground state without emitting a photon. This process is known as collisional quenching. It serves as a loss mechanism because the excited state is depopulated non-radiatively, reducing the net amount of

fluorescence. In addition, quenching collisions have a tendency to excite vibrational modes of the NO molecule at the same time that they de-excite electronic modes, populating X FI(v'' # )) states (Paul et al. 1994). In some flows, this could reduce the overall fluorescence because it serves to depopulate the probed X2II(v"=0) state. In the present experiments, this mechanism was not a factor because laser pulses were separated by 0.1 s, and the flow velocities were large. Together, these two factors ensured that the molecules tagged by each laser pulse moved well out of the measurement region prior to the next laser pulse.

The calculations of fluorescence intensity carried out for the present experiments have included the effects of quenching collisions by nitrogen, oxygen, and other nitric oxide molecules. Of these three species, the quenching cross section is largest for nitric oxide self-quenching. However, the low mole-fraction of NO results in self-quenching

having a small effect on the overall fluorescence signal. Nitric oxide is strongly

quenched by molecular oxygen ( 0 2). The effect of quenching by oxygen is readily apparent in the diffusion and mixing regions of these flows. The collisional quenching cross-section for molecular nitrogen (N2) is approximately four orders of magnitude smaller than that for either NO or 0 2, so at the temperatures encountered in the flows of the present investigation, N2-NO collisions have a small effect on the overall fluorescence, despite the high percentage of nitrogen in the flow.

3.2.1.3 Fluorescence

Once molecules have been promoted to an excited state, molecules may dissipate the energy they absorbed through a variety of means. Some will relax radiatively; that is,

they will emit a photon as fluorescence. As mentioned previously, excited molecules may emit photons with the same amount of energy (and therefore, the same wavelength) as those that they absorbed, but they may also emit photons of different energies. The net amount of fluorescence can be calculated by integrating the amount of fluorescence from each allowed radiative relaxation pathway. For each pathway, the amount of fluorescence will depend on the population of the excited state and the probability of the relaxation transition. This probability is captured in the Einstein A coefficient for spontaneous emission. The following sections examine each of these parameters in more detail.