Page 83 - High Power Laser Handbook

52 G a s , C h e m i c a l , a n d F r e e - E l e c t r o n L a s e r s Chemical Lasers 53

independent of vibrational level. Then, ignoring higher-order terms,
thermodynamics tells us that the equilibrium rotational fraction for
level J is given by Eq. (3.3a). Assuming the population of vibrational
level v can be described as F(v) and ignoring the v dependence of B e
and Z, the expression in brackets in Eq. (3.8) can be approximated as
follows:

2
e (–J +J)B e /(kT) {F(v + 1) – F(v)[(2J – 1)/(2J + 1)]e –2JB e /(kT) }/Z (3.9)

For P-branch lasing, the exponential factor multiplying the second
term allows one to achieve gain, even when the total population in
vibrational level v + 1—F(v + 1)—is smaller than F(v). This is called a
partial inversion and the effect is substantial for HF and DF due to the
large B values. Note that the analogous factor for R-branch lasing
e
necessitates absolute inversion and increases the difficulty in achiev-
ing threshold.
The normalized line shape g(ν) includes line-width dependence.
It is simple to calculate line width at very low pressures based on
Doppler broadening. At line center, Eq. (3.10) applies.

g(0) = 2[ln(2)/π] 1/2 /∆ν (3.10)
D
∆ν = 2ν [2 ln(2) kT/Mc ] (3.11)
2 1/2
D 0
where ν = centerline frequency
0
T = absolute temperature
M = molecular weight
c = speed of light
At high pressures, which are not typical of CW devices but which
are common to most pulsed devices, pressure broadening becomes
dominant, and line width becomes inversely proportional to pres-
sure. Taking into account pressure broadening for CW (as well as for
pulsed systems) requires the use of Voigt functions, which combine
pressure broadening and Doppler effects. In practice, representative
4
values of small signal gain are typically in the order of a few percent
per centimeter in CW HF devices and are moderately higher in
pulsed devices.

3.3.3 Chemically Excited Species Generation
The chemical reactions used to produce vibrationally excited HF are
given in Eqs. (3.12) and (3.13), which are referred to as the cold and
hot reactions, respectively.

F + H → HF* + H + 31.5 kcal/mol (cold) (3.12)
2
H + F → HF* + F + 98 kcal/mol (hot) (3.13)
2