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U(t) = Uo exp − ωt
QL
Ith − Io
Ith
123
(5.4)
It follows that the voltage, which is proportional to the square root of the stored
energy, is given by
V (t) = Vo exp − ωt
2QL
Ith − Io
Ith
(5.5)
From Eq. (5.5) one can extract an exceedingly useful quantity, defined as the
effective quality factor
Qeff =
Ith
Ith − Io
QL
(5.6)
This simple relation states that by measuring the effective Q as a function of the
average beam current, in principle, the threshold is easily extracted. With zero
beam current, the effective Q is the QL of the HOM. When Io = Ith, the effective Q
becomes infinite and the HOM voltage does not decay. If the beam current exceeds
the threshold, the amplitude of the voltage oscillations grow exponentially, and is
measured by the Schottky diodes (see Fig. 5.4). Note that Eq. (5.6) is valid both
above and below the threshold current.
The beam-transfer function (BTF) measurement is the second method used to
measure the threshold current and amounts to using a network analyzer to make an
S21 measurement of a particular mode as a function of average beam current. By
measuring the effective Q, that is, the quality factor of the combined HOM-beam
system measured from the −3 dB points of the frequency curve, as a function of
current, Eq. (5.6) can be used to extract the threshold current.
The third and final measure of the threshold is achieved by measuring the
growth rate of the HOM power. The growth rate is described by the time constant