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then gives us the interaction time that would yield that particular trapping state. Now, to fix N ex and vary e, as in Figure 8.6, amounts to fixing the number ofexcited atoms in the gain medium and varying the interaction time. In a laser, however, it is the (effective) interaction time that is fixed by the short lifetime of the atomic energy levels involved in optical transitions, as we mentioned above. Only the number of excited atoms in the gain medium (pump rate) can be varied in a laser. So to make a comparison with the typical laser features discussed in Section 8.3, we will break

Netherlands (22 percent), Canada (133 percent), Russia (133 percent), Spain (73 percent), and China (62 percent) Support for research and development from nonpro ts and foreign sources is virtually nonexistent in Brazil and Cuba There is a small amount of nonpro t funding for Argentina (17 percent) and Mexico (11 percent), and Argentina, Chile, and Mexico have signi cant foreign sup port for research and development Mexico s foreign support is most prominent, relative to total research and development funding, with 67 percent of total funding from foreign sources Chile follows at 64 percent of total funding, and Argentina receives 37 percent of total research and development funding from foreign sources.

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As they obey Sommerfeld's radiation condition, natural modes on their own do not form a complete set. So we cannot expand the quantized electromagnetic radiation field in terms of them alone. In this section we formulate this problem more precisely and suggest how they can be supplemented by a different sort of mode in a way that will make the total set complete. But before starting this discussion, it is convenient to introduce first a fascinating result that is not widely taught in electrodynamics lecture courses today: Although we normally use one scalar potential and a vector potential to describe the electromagnetic field, the same job can be done using only two real scalar potentials or, equivalently, a single complex scalar potential.

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13He was later to become the discoverer of the first elementary particle, the electron. 14See page 95, "Researches, 1880-84" in m of [596].

We know from elementary electrodynamics that a description in terms of electromagnetic potentials introduces redundant degrees of freedom, whose main consequence is giving us the possibility of choosing the gauge that suits us best. Can we use this redundancy to reduce the number of electromagnetic potentials needed to describe the fields Yes, we can. We will now show that this redundancy can be exploited to reduce the usual description in terms of a scalar potential and a vector (with three scalar components) potential to one involving only two scalar potentials. This is not only useful to simplify calculations, but also seems to have some fundamental significance, as we know that light has only two independent polarizations (see s 2 and 3). One can derive this representation in a general way, in the presence of matter [466] and without making specific assumptions about the cavity shape and so on. As we just wish to develop the basic ideas of such representation, however, we will assume vacuum and take a specific cavity geometry that will lead to a simple presentation of these ideas. Consider a cylindrical cavity geometry, where the surfaces of the cavity walls are everywhere parallel to the z-axis. Assuming that the cavity is just sitting in the vacuum, the frequency w = ck components of the fields satisfy the following Maxwell equations:

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