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4 CO, Isotope Lasers and Their Applications 69

In an analogous fashion, consideration should be given to the energy-level
differences that exist between the (0001) upper levels of rare isotopes of CO, and
the (u = I) levels of rare isotopic N, in order to optimize lasing efficiency ie.g..
13C1601 and 15N,).

3. ROTATIONAL ENERGY-LEVEL SUBSTRUCTURE OF THE CO, MOLECULE

In the CO, laser system eigenstates of the molecule are characterized by the
rotational quantum number J in addition to the vibrational quantum numbers ul’
u,. and u3. Lasing transitions actually occur between rotational levels xhar
-
belong to two different vibrational modes, as illustrated in Fig. 3, which shon s
the detailed vibrational-rotational energy-level structure of the CO, molecule
that is characteristic of the laser transitions in the (0001 j-[ 1000, 0200],,, regular
bands. Laser oscillations occur between two rotational levels belonging to the
two different vibrational modes. The center of the band corresponds to the spac-
ing between the vibrational levels in the absence of any rotational energy (J = 01.
The rotational energies of a given vibrational state, vI. relative to the J = 0 level
are 13 1-36-38]

where B, is the rotational constant of the i’th vibrational state, and D,, H,, Lu, etc..
are spectroscopic constants of the molecule. which are very small compared to B,.
Quantum-mechanical selection rules allow only those transitions between
vibrational-rotational levels of the regular band for which the change in the
rotational quantum number J corresponds to 4J = k 1. Transitions from (4 to
(S+l) are called P(J) lines, whereas those from (J) to (J-1) are named R(s3
lines. According to spectroscopic custom, the rotational part of the transitions
is designated by the rotational quantum number J that is characterizing the
lower level of a lasing transition. This form of designation is illustrated in Fig.
3, which explicitly shows the P(20) and R(20) lasing transitions of the P and R
branches in the (0001)-[1000, 02001, and (OOOl)-[ 1000. 0200],, regular bands
centered about the 10.4- and 9.3-pm wavelengths, respectively. Frequently,
abbreviated forms of laser line designations, such as I-P(20), P,(20) or lOP(20)
are also used. In view of the more recently discovered hot bands. sequence
bands, and sequence hot bands (which are described in a later section of this
chapter). abbreviated forms of line designations should only be used when no
ambiguities exist about the vibrational band affiliations. As indicated in Fig. 3,
the spacings between rotational levels gradually increase toward the higher