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# Resonances

Suppose, now, that a resonance is located at , so that

 (6.29)

in the immediate vicinity of this point, where . Here, is a small real constant. We introduce in our analysis principally as a mathematical artifice to ensure that remains single-valued and finite. However, as will become clear later on, has a physical significance in terms of the damping or the spontaneous excitation of waves.

In the immediate vicinity of the resonance point, , Equations (6.3) and (6.29) yield

 (6.30)

where

 (6.31)

and . This equation is singular at the point . Thus, it is necessary to introduce a branch-cut into the complex- plane, so as to ensure that is single-valued. If then the branch-cut lies in the lower half-plane, whereas if then the branch-cut lies in the upper half-plane. (See Figure 6.1.) Suppose that the argument of is 0 on the positive real -axis. It follows that the argument of on the negative real -axis is when , and when .

Let

 (6.32) (6.33)

In the limit , Equation (6.30) transforms into

 (6.34)

This is a standard equation, known as Bessel's equation of order one (Abramowitz and Stegun 1965c), and possesses two independent solutions, denoted and , respectively. Thus, on the positive real -axis, we can write the most general solution to Equation (6.30) in the form

 (6.35)

where and are two arbitrary constants.

Let

 (6.36) (6.37)

where

 (6.38)

Note that the argument of is zero on the negative real -axis. In the limit , Equation (6.30) transforms into

 (6.39)

This is a standard equation, known as Bessel's modified equation of order one (Abramowitz and Stegun 1965c), and possesses two independent solutions, denoted and , respectively. Thus, on the negative real -axis, we can write the most general solution to Equation (6.30) in the form

 (6.40)

where and are two arbitrary constants.

The Bessel functions , , , and are all perfectly well-defined (i.e., analytic) for complex arguments, so the two expressions (6.35) and (6.40) must, in fact, be identical. In particular, the constants and must somehow be related to the constants and . In order to establish this relationship, it is convenient to investigate the behavior of the expressions (6.35) and (6.40) in the limit of small : that is, . In this limit,

 (6.41) (6.42) (6.43) (6.44)

where is Euler's constant (Abramowitz and Stegun 1965a), and is assumed to lie on the positive real -axis. It follows, by a comparison of Equations (6.35), (6.40), and (6.41)-(6.44), that the choice

 (6.45) (6.46)

ensures that the expressions (6.35) and (6.40) are indeed identical.

In the limit ,

 (6.47) (6.48)

where is assumed to lie on the negative real -axis. It is clear that the solution is unphysical, because it blows up in the evanescent region . Thus, the coefficient in expression (6.40) must be set to zero in order to prevent from blowing up as . According to Equation (6.45), this constraint implies that

 (6.49)

In the limit ,

 (6.50) (6.51)

where is assumed to lie on the positive real -axis. It follows from Equations (6.35), (6.49), (6.50), and (6.51) that, in the non-evanescent region ( ), the most general physical solution takes the form

where is an arbitrary constant.

Suppose that a plane electromagnetic wave, polarized in the -direction, is launched from an antenna, located at large positive , toward the resonance point at . It is assumed that at the launch point. In the non-evanescent region, , the wave can be represented as a linear combination of propagating WKB solutions:

 (6.53)

The first term on the right-hand side of the previous equation represents the incident wave, whereas the second term represents the reflected wave. Here, is the amplitude of the incident wave, and is the amplitude of the reflected wave. In the vicinity of the resonance point (i.e., small and positive, which corresponds to large and positive), the previous expression reduces to

 (6.54)

A comparison of Equations (6.52) and (6.54) shows that if then . In other words, there is a reflected wave, but no incident wave. This corresponds to the spontaneous excitation of waves in the vicinity of the resonance. On the other hand, if then . In other words, there is an incident wave, but no reflected wave. This corresponds to the total absorption of incident waves in the vicinity of the resonance. It is clear that if then represents some sort of spontaneous wave excitation mechanism, whereas if then represents a wave absorption, or damping, mechanism. We would normally expect plasmas to absorb incident wave energy, rather than spontaneously emit waves, so we conclude that, under most circumstances, , and resonances absorb incident waves without reflection.

Next: Resonant Layers Up: Wave Propagation Through Inhomogeneous Previous: Cutoffs
Richard Fitzpatrick 2016-01-23