Sommerfeld Precursor

We can deform the original path of integration into a large semi-circle of radius in the upper half-plane, plus two segments of the real axis, as shown in Figure 9. Because of the denominator , the integrand tends to zero as on the real axis. We can add the path in the lower half-plane that is shown as a dotted line in the figure, because if the radius of the semi-circular portion of this lower path is increased to infinity then the integrand vanishes exponentially as . Therefore, we can replace our original path of integration by the entire circle . Thus,

in the limit that the radius of the circle tends to infinity.

The dispersion relation (871) yields

(892) |

in the limit . Using the abbreviation

and, henceforth, neglecting with respect to , we obtain

(894) |

from Equation (892). This expression can also be written

Let

(896) |

It follows that

(897) |

giving

(898) |

Substituting the angular variable for in Equation (896), we obtain

Here, we have taken as the radius of the circular integration path in the -plane. This is indeed a large radius because . From symmetry, Equation (900) simplifies to give

The following mathematical identity is fairly well known,^{}

(901) |

where is Bessel function of order . It follows from Equation (901) that

Here, we have made use of the fact that .

The properties of Bessel functions are described in many standard references on mathematical functions (see, for instance, Abramowitz and Stegun). In the small argument limit, , we find that

(903) |

On the other hand, in the large argument limit, , we obtain

(904) |

The behavior of is further illustrated in Figure 10.

We are now in a position to make some quantitative statements regarding
the signal that first arrives at a depth
within the dispersive medium.
This signal propagates at the velocity of light in vacuum, and
is called the *Sommerfeld precursor*. The first important point
to note is that the amplitude of the Sommerfeld precursor is very small
compared to that of the incident wave (whose amplitude is normalized to
unity). We can easily see this because, in deriving Equation (903),
we assumed that
on the circular integration
path
. Because the magnitude of
is always less than, or of order,
unity, it is clear that
. This is a comforting result, because
in a naive treatment of wave propagation through a dielectric medium, the
wave-front propagates at the group velocity
(which is less
than
) and, therefore, no signal should reach a depth
within the medium
before time
. We are finding that there is, in fact, a precursor
that arrives at
, but that this signal is fairly weak. Note
from Equation (894) that
is proportional to
. Consequently, the
amplitude of the Sommerfeld precursor decreases as the inverse of the
distance traveled by the wave-front through the dispersive medium
[because
attains its maximum value when
].
Thus, the Sommerfeld precursor is likely to become undetectable after
the wave has traveled a long distance through the medium.

Equation (903) can be written

(905) |

where , and

(906) |

The normalized Sommerfeld precursor is shown in Figure 11. It can be seen that both the amplitude and the oscillation period of the precursor gradually increase. The roots of [i.e., the solutions of ] are spaced at distances of approximately apart. Thus, the time interval for the th half period of the precursor is approximately given by

(907) |

Note that the initial period of oscillation,

(908) |

is extremely small compared to the incident period . Moreover, the initial period of oscillation is completely independent of the frequency of the incident wave. In fact, depends only on the propagation distance , and the dispersive power of the medium. The period also decreases with increasing distance, , traveled by the wave-front though the medium. So, when visible radiation is incident on a dispersive medium, it is quite possible for the first signal detected well inside the medium to lie in the X-ray region of the electromagnetic spectrum.