Relation Between Linewidth and Lifetime

The radiant power of the damped oscillator can be obtained from (3.3) if both sides of the equation are multiplied by mx, which yields after rearranging 

The left-hand side of (3.12) is the time derivative of the total energy W (sum of kinetic energy mx and potential energy Dx /2 = moi x jT), and can therefore be written as 

Inserting x(0 from (3.5) and neglecting terms with yields 

Because the time average sin cot = 1/2, the time-averaged radiant power P = dW/dt is 

Equation (3.15) shows that P and with it the intensity I t) of the spectral line decreases to 1/e of its initial value = 0) after the decay time r = l/y. 

Inserting x(t) from (3.5) yields when neglecting terms with 7 dW 

In Sect.2.7 we saw that the mean lifetime Tj of a molecular level Ej, which decays exponentially by spontaneous emission, is related to the Einstein coefficient Aj by Tj = 1/Aj. Replacing the classical damping constant 7 by the spontaneous transition probability A we can use the classical formulas (3.9-11) as a correct description of the frequency distribution of spontaneous emission and its linewidth. The natural halfwidth of a spectral line spontaneously emitted from the level E is, according to (3.11), 

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Time-independent picture. The opposite extreme from short-pulse excitation involves the use of nearly monochromatic radiation. Practically, this means that the interaction between molecule and radiation field is of longer duration than Tnr. In this limit, the quantity measured is the absorption lineshape. It will be shown below that the linewidth observed in an energy-resolved experiment is related in a very simple way to the predissociation lifetime in the time-resolved experiment. 

We consider in the following the case of a single isolated resonance, far removed from other resonances. If an absorption line has a Lorentzian profile, the linewidth/ T, is related to the total lifetime, r, of the state by the usual relation P oc r 1 [cf. Eq. (7.4.11)]. This width includes contributions from each of these three different types of decay. If interactions between continua of the electron, nuclei, and radiation field are neglected, one can write... 

It is important to realize that the relaxation times might depend on some factors that are properties of the atom or molecule itself and on others that are related to its environment. Thus rotational spectra of gases have linewidths (related to the rotational relaxation times) that depend on the mean times between coUisions for the molecules, which in turn depend on the gas pressure. In liquids, the collision lifetimes are much shorter, and so rotational energy is effectively non-quantized. On the other hand, if the probability of collisions is reduced, as in a molecular beam, we can increase the relaxation time, reduce linewidths, and so improve resolution. Of course, the relaxation time only defines a minimum width of spectral lines, which may be broadened by other experimental factors. 

The homogeneous component includes all mechanisms that involve every atom in both the upper and lower laser levels of the gain medium in identically the same way. These mechanisms include the natural radiative linewidth due to spontaneous emission, broadening due to collisions with other particles such as free electrons, protons, neutrons, or other atoms or ions, or power broadening in which a high-intensity laser beam rapidly cycles an atom between the upper and lower levels at a rate faster than the normal lifetime of the state. In each of these effects, the rate at which the process occurs determines the number of frequency components or the bandwidth (linewidth) required to describe the process, with faster processes corresponding to broader linewidths. The natural radiative linewidth is associated with the radiative decay of individual atoms. It is related to the radiative decay rates of both the upper and lower energy levels involved in the laser transition. 

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