CO2 LASER THEORY

by David R. Whitehouse

Only the electronic ground state is of interest for a CO2 laser. Thus, molecular translation, vibration and rotation are the principal forms of energy storage and interchange. The important energy levels of the molecule are shown in Illustration A. Carbon dioxide is a linear, symmetric molecule with the carbon atom balanced against the two oxygen atoms, O-C-O. Therefore, three characteristic vibrational modes exist: the symmetric mode, where the two oxygen atoms vibrate against each other; the bending mode, where the carbon atom moves out of the molecular axis, thus bending the molecule; and the asymmetric mode, where the two oxygen atoms oscillate against the carbon atom. Each of these vibrating modes is quantized, and the particular vibrational state of the molecule is designated by three integral numbers. The first number is the quantum level or excitation number of the symmetric mode; the second number is the excitation number of the bending mode; and the third number is the excitation number of the asymmetric mode. Only the energy levels of the pure modes are given, but all the mixed vibrational modes exist as well (for example: 111).

Each and every vibrational state is degenerate or further subdivided into a whole series of levels brought about by gross rotation of the vibrating molecule. These levels are also quantized and designated by J, the rotational quantum number. They are shown in Illustration B on an expanded energy scale for the 001 and 100 vibrational levels. The lasing transition at 10.6 microns is a simultaneous change of these vibrational and rotational quantum states. However, this may occur on any one of a number of rotational transitions. The selection rules for the vibrational-rotational transition are that J must change by ±1. Those transitions where J increases by 1 are called P-branch transitions ( for example, P10), and those that decrease are called R-branch transitions (for example, R10). The wavelengths associated with the dominant transitions are P18-10.57 microns, P20-10.59 microns, P22-10.61 microns, etc. When the inversion in the total vibrational level occurs because of the discharge, the P branch transitions have more gain than their R counterparts. Since the P and R transitions compete with each other, the medium lases only on the P branches unless special precautions are taken. Those P branches which have the highest gain are most likely to show lasing action.



EXCITATION AND DE-EXCITATION
THE VIBRATIONAL LEVELS

Experiments show that inversion is not a result of direct electronic excitation but is caused by vibrational energy interchange between N2 and CO2. The lowest vibrational level of N2 is essentially metastable since the molecule is symmetric and cannot radiate. In a nitrogen discharge, about 20-30 per cent of the molecules are vibrationally excited. The energy level of n = 1 matches the level of the 001 state of CO2 within 2 mV, as illustrated at A (well within the thermal energy spread of about 25 mV). When the two gases are mixed, effective resonance collisional transfer occurs between the two states. Also, the second level of N2 matches the 002 mode of CO2 and so on up the ladder. The net result is that the asymmetric ladder, 00n, becomes highly excited; or it has a high vibrational temperature.

The lower laser level and associated symmetric ladder is not excited by N2 because there are no energy resonances. However, when the laser is operative, the 100 mode and its chain become heavily populated. This chain cannot radiate to the ground state and can only radiate weakly to the lower bending modes. However, there is a close "Fermi" resonance of the 100 mode with the 020 mode, and collisions can cause this transition.

The addition of helium gas helps to increase laser power. Its low mass makes it effective in trading its kinetic energy with the molecule internal energy modes for small energy changes.

As a result, the helium is effective in:

• Cooling the CO2 rotational temperature (but not the vibrational temperature)
• Increasing the thermal conduction to the wall (thus keeping the translational temperature low, the Doppler width small, and the gain high).
• Increasing the depopulation rate of the 010 level of CO2 which in turn depopulates the 100 lower level. This is because the 010, 020, and 100 levels are all strongly coupled together through resonant collisions.

The important lifetimes in the CO2 laser are practically all determined by collisional phenomena. The radiative lifetimes vary from a few milliseconds to a few seconds, whereas the mean free time between molecular collision is of the order 10 to 100 nanoseconds. These times must then be scaled up by the number of collisions needed to effect a certain energy transfer. The important transfer times are summarized in Illustration C, where the partial pressures normally used in the CO2 laser are assumed.