SPONTANEOUS AND STIMULATED EMISSION

In general, when an electron is in an excited energy state, it must eventually decay to a lower level, giving off a photon of radiation. This event is called “spontaneous emission,” and the photon is emitted in a random direction and a random phase. The average time it takes for the electron to decay is called the time constant for spontaneous emission, and is represented by t. On the other hand, if an electron is in energy state E2, and its decay path is to E1, but, before it has a chance to spontaneously decay, a photon happens to pass by whose energy is approximately

E24E1, there is a probability that the passing photon will cause the electron to decay in such a manner that a photon is emitted at exactly the same wavelength, in exactly the same direction, and with exactly the same phase as the passing photon. This process is called “stimulated emission.” Absorption, spontaneous emission, and stimulated emission are illustrated in figure 36.2.

Now consider the group of atoms shown in figure 36.3: all begin in exactly the same excited state, and most are effectively within the stimulation range of a passing photon. We also will assume that t is very long, and that the probability for stimulated emission is 100 percent. The incoming (stimulating) photon interacts with the first atom, causing stimulated emission of a coherent photon; these two photons then interact with the next two atoms in line, and the result is four coherent photons, on down the line. At the end of the process, we will have eleven coherent photons, all with identical phases and all traveling in the same direction. In other words, the initial photon has been “amplified” by a factor of eleven. Note that the energy to put these atoms in excited states is provided externally by some energy source which is usually referred to as the “pump” source.

Of course, in any real population of atoms, the probability for stimulated emission is quite small. Furthermore, not all of the atoms are usually in an excited state; in fact, the opposite is true. Boltzmann’s principle, a fundamental law of thermodynamics, states that, when a collection of atoms is at thermal equilibrium, the relative population of any two energy levels is given by where N2 and N1 are the populations of the upper and lower energy states, respectively, T is the equilibrium temperature, and k is Boltzmann’s constant. Substituting hn for E24E1 yields For a normal population of atoms, there will always be more atoms in the lower energy levels than in the upper ones. Since the probability for an individual atom to absorb a photon is the same as the probability for an excited atom to emit a photon via stimulated emission, the collection of real atoms will be a net absorber, not a net emitter, and amplification will not be possible. Consequently, to make a laser, we have to create a “population inversion.” POPULATION INVERSION Atomic energy states are much more complex than indicated by the description above. There are many more energy levels, and each one has its own time constants for decay. The four-level energy diagram shown in figure 36.4 is representative of some real lasers. The electron is pumped (excited) into an upper level E4 by some mechanism (for example, a collision with another atom or absorption of high-energy radiation). It then decays to E3, then to E2, and finally to the ground state E1. Let us assume that the time it takes to decay from E2 to E1 is much longer than the time it takes to decay from E2 to E1. In a large population of such atoms, at equilibrium and with a continuous pumping process, a population inversion will occur between the E3 and E2 energy states, and a photon entering the population will be amplified coherently.