SRLs usually operate in the sealed-off pulsed mode (as do ours, to be described later). This means that energy is delivered to the active volume in the form of a train of pulses. The helium buffer gas temperature will consequently exhibit a time dependency.
Consider a volume (the volume of a laser’s discharge channel) containing m moles of gaseous helium, which is subjected to an input energy pulse. The increase in gas temperature, A7, accompanying the
Chapter 2
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4^
energy pulse is related to the input pulse energy, AE, through the gas
molar heat capacity at constant volume, according to:
AE = mcy AT. [2.4.1]
The equation of state for an ideal gas, such as helium, which is used in the SRL as buffer gas, is:
pV = mRT. [2.4.2]
Hence, eliminating the amount of gas, m, from Equation 2.4.1 gives.
Now, the kinetic theory prediction of the molar heat capacity at constant volume for an ideal gas is.
3
Cv,m = 2 ^ ’ [2.4.4]
SO that Equation 2.4.3 can be rewritten as:
AE = ^ ^ A T . [2.4.5]
2T
The fractional change in gas temperature, expressed in terms of the energy added to a system under isovolumetric conditions, is therefore
Chapter 2
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49given by:
AT 2AE
As the temperature of a fixed amount of gas at constant volume increases so does its pressure. Here the pressure, p, exerted by the gas is the pressure to which the gas relaxes during the inter-pulse period (i.e., the external reservoir pressure).
Notice that the above discussion assumes that all of the input energy is used to raise the internal energy of the gas. However, in practical laser systems, the discharge channel volume will not account for the whole sealed-off volume (due to the finite volume of the laser heads). Hence, in practice, the input pulse energy will not be used
entirely to raise the internal energy of the discharge channel gas, as some will be used to perform work when the gas contained in the discharge channel expands (to equalize the gas pressure in the entire laser tube’s sealed-off volume). This will result in the true fractional change in gas temperature being somewhat lower than that obtained using Equation 2.4.6. However, for the purposes of this discussion, a simplification which leads to an overestimate of the gas temperature rise accompanying an input energy pulse is justified.
Taking 5 pJ cm'^ as a typical value for the specific laser pulse energy of an SRL, operating with a typical efficiency of 0.1%, if the
helium buffer gas pressure is 50 kPa, the fractional change in gas temperature due to an input energy pulse will be less than 0.07. An
Chapter 2
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50increase in gas temperature of 7% is sufficiently low to enable us to treat the laser as operating under steady-state conditions of temperature.
It is interesting to seek the specific input energy at which the gas temperature changes by say 50% during a pulse, as a means of evaluating the limits under which the laser can he treated as a steady- state system. At a pressure of 50 kPa, from Equation 3.4.6, this would occur for a specific input energy of 0.0375 J cm"^. On the basis of a laser efficiency of 0.1%, the expected specific laser pulse energy would then
be 37.5 pJ cm“^. In practice such a laser would have to be operated at low (sub 1 kHz) PRFs to avoid overheating the discharge channel wall or the gas (see Chapter 4). It should be noted that the preceding example is based on a hypothetical laser. No longitudinal-discharge SRL has yet been built which is capable of delivering this sort of specific laser pulse energy under these pressure conditions.
The fractional change in gas temperature is related to the input power density, Py, and PRF,/, through the expression:
Equation 2.4.7 shows us that if the fractional change in gas temperature is to be kept low during a pulse, the repetition rate must be increased with the input power density, P y . The minimum PRF, at a pressure of
50 kPa, which must be used to keep the change in temperature during a pulse below 10% of the starting value, is shown in Figure 2.5 as a function of input power density.
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51Input energy pulses will raise the gas temperature of a cold SRL to a
level after which the temperatures associated with the system show no significant short term or long term time dependences. However, attempting to raise the specific output pulse energy of a given laser tube by increasing the specific input energy and lowering the PRF will result
in an increase in the fractional change in gas temperature, so that the steady-state approximation may not be valid.
Our lasers, to be described, all operate at low specific input energies (<5 mJ cm'^) and high PRFs (ahove 1 kHz), so that the fractional change in gas temperature during an input pulse is expected to be less than 10%. The steady state approximation is therefore applicable in our laser systems and, indeed, to all longitudinally excited SRLs cited in the literature at this time.
2.5 A steady-state thermal loading model for free-convection