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Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box Sergio M. Dutra Copyright 2005 John Wiley & Sons, Inc.
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Appendix A
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Modes of a perfectly conducting closed cavity: A quick review
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There are many different modes, each of which will have a different resonant frequency corresponding to some particular complicated arrangement of the electric and magnetic fields. Each of these arrangements is called a resonant mode. The resonance frequency of each mode can be calculated by solving Maxwell's equations for the electric and magnetic fields in the cavity. -Richard Phillips Feynman [206]
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In Section 2.2 we showed that Maxwell equations involve only fields at the same point in Fourier space. Electrodynamics then becomes extremely simple. But that was all done in free space. For a cavity, the fields have to satisfy the constraints imposed by the boundary conditions on the walls. Here we show how the Fourier-
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PERFECT CAVITY MODES
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space procedure that simplified electrodynamics in free space can be generalized for a perfectly conducting closed cavity (a perfect cavity for short). We will see that the plane waves appearing in the Fourier transforms in free space can just be replaced by a different sort of wave, whose shape is determined by the boundary conditions of the cavity. These waves are the cavity modes. They are extremely useful, especially for quantizing the electromagnetic field, because Maxwell equations for the electromagnetic field in the cavity involve only fields at the same point in mode space. From this point of view, the plane waves that appeared in the Fourier transforms in Section 2.2 are just modes offree space. Let us consider a closed cavity of any shape without any matter inside it, and made up of perfectly conducting walls. Any electromagnetic radiation inside it can be described by Maxwell equations without sources:
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(AA)
c8t ' 18 VAB=--E c8t '
with the boundary conditions)
nl\ Els =
18 VAE= ---B
(A.5) (A.6)
n Bls =0,
where n is the normal to the surface of the cavity. Taking the curl of (A.3), using (A.4) and (A.I), we find that E obeys the wave equation
1 82 c28t2E.
(A.7)
It is possible to show [458] that any solution of (A.7) can be written as a linear
combination of special solutions that factor out as a product of a function of r alone by a function of t alone, that is,
E(r, t) = x(r)T(t).
So if we can find these special solutions, we can obtain any solution of (A. 7). Substituting (A.8) in (A.7), we find that
(A.8)
V 2 Xi(r) T(t) Xi(r) = c2T(t) '
(A.9)
where the subscript i = 1,2,3 stands for the independent components of X in any arbitrary coordinate system. Because the left-hand side of (A.9) depends only rand
1 If
you are not familiar with these boundary conditions. see Appendix B.
PERFECT CAVITY MODES
the right-hand side only on t, they can only agree for all values of rand t if they are both equal to a constant. Moreover, as the right-hand side of (A.9) is independent of i, this constant should be the same for every value of i. Calling this constant TJ, we obtain V2x(r)
= TJx(r),
(A. 10)
T(t) = c2 TJT(t).
(A.II)
In principle, TJ can be any complex number. The boundary conditions on the cavity walls, however, detennine what kind of complex number TJ can be. First, we show that TJ must be a negative real number in the case of a perfect cavity. For that, we notice that if we take the scalar product of (A. I0) with X and integrate, the right-hand side of (A. 10) will be a positive number times TJ. Now look at the left-hand side of (A.IO). As V . X = 0, we can replace V 2X by - V 1\ (V 1\ X), so that
dV(X)2 = -
dV X . [V 1\ (V 1\ X)]