Saturation Behavior of Homogeneously and Inhomogeneously Broadened Transitions

4.6 Saturation Behavior of Homogeneously and Inhomogeneously Broadened Transitions 81 the resultant lineshape function has to be evaluated by performing a convolution of the different lineshape functions.

Problem 4.2 Obtain the lineshape function in the presence of both natural and Doppler broadening Solution From Maxwell’s velocity distribution, the fraction of atoms with their center frequency lying betweenωandω+dωis given by

f (ω)dω= M

2πkBT

1 2 c

ω0 exp

−Mc² 2kBT

(ω−ω0)² ω²₀

dω (4.55)

These atoms are characterized by a naturally broadened lineshape function described by h(ω−ω)=2tsp

π

1

1+(ω−ω)²4t²_sp (4.56)

Thus the resultant lineshape function will be given by g(ω)=

f (ω)h(ω−ω)dω (4.57)

which is nothing but the convolution of f (ω) with h(ω)

Example 4.9 Neodymium doped in YAG and in glass are two very important lasers. The host YAG is crystalline while glass is amorphous. Thus the broadening in YAG host is expected to be much smaller than in glass host. In fact the linewidth at 300 K for Nd:YAG is about 120 GHz while that for Nd:glass is about 5400 GHz.

4.6 Saturation Behavior of Homogeneously

at the oscillation frequency again to the value at threshold. It may be mentioned that the gain could exceed the threshold value on a transient basis but not under steady state operation.

Now in a homogenously broadened transition all the atoms have identical line- shapes peaked at the same frequency. Thus all atoms interact with the same oscillating mode and the increase in pumping power cannot increase the gain at other frequencies and thus the laser will oscillate only in a single longitudinal mode (see Fig.4.9). This observation has been verified experimentally on some homo- geneously broadened transitions such as Nd:YAG laser. The fact that a laser with homogeneously broadened transition can oscillate in many modes is due to spatial hole burning. This can be understood from the fact that each mode is a standing wave pattern between the resonator mirrors. Thus there are regions of high popu- lation inversion (at the nodes of the field where the field amplitude is very small) and regions of saturated population inversion (at the antinodes of the field where the field has maximum value). If one considers another mode which has (at least over some portions) antinodes at the nodes corresponding to the central oscillating mode, then this mode can draw energy from the atoms and, if the loss can be compensated by gain, this mode can also oscillate.

In contrast to the case of homogeneous broadening, if the laser medium is inho- mogeneouly broadened then a given mode at a central frequency can interact with only a group of atoms whose response curve contains the mode frequency (see Fig. 4.10). Thus if the pumping is increased beyond threshold, the gain at the oscillating frequency remains fixed but the gain at other frequencies can go on increasing (see Fig.4.10). Thus, in an inhomogeneously broadened line one can have multimode oscillation and as one can see from Fig.4.10. Each oscillating mode

“burns holes in the frequency space” of the gain profile. These general conclusions regarding homogeneously and inhomogeneoulsy broadened lines have been verified experimentally.

ω Gain curve

Oscillating mode

Loss line

Fig. 4.9 In a homogeneously broadened transition, gain can compensate loss at only one oscillating mode leading to single longitudinal mode operation

4.6 Saturation Behavior of Homogeneously and Inhomogeneously Broadened Transitions 83

Gain curve

ω Oscillating

modes

Loss line Fig. 4.10 As the pumping is

increased beyond threshold, under steady-state operation the gain at the various oscillating frequencies cannot increase beyond the threshold value but the gain at other frequencies may be much above the threshold value.

The various frequencies are said to burn holes in the gain curve

Various techniques for single longitudinal mode oscillation of inhomogeneously broadened lasers are discussed inChapter 7.

Let us now consider an inhomogeneously broadened laser medium and let us assume that only a single mode exists within the entire gain profile. Let us also assume to begin with that the frequency of the mode does not coincide with the line center and that we slowly change the frequency of the mode so that it passes through the center of the profile to the other side of the peak in the gain profile. In order to determine the variation of the power output as the frequency is scanned through the line center, we observe that a mode of the laser is actually made up of two traveling waves traveling along opposite directions along the resonator axis. Thus when the mode frequency does not coincide with the line center, the wave travelling from left to right in the resonator will interact with those atoms whose z-directed velocities are near to [see Eq. (4.49)]:

vz= ω−ω21

ω21

c (4.58)

while the wave moving from right to left would interact with those atoms whose z-directed velocity would be

vz= −ω−ω21

ω21

c (4.59)

Thus there are two groups of atoms with equal and opposite z-directed velocities which are strongly interacting with the mode. As the frequency of the mode is tuned to the center these groups of atoms change with the frequency, and at the line center, the mode can interact only with the groups of atoms having a zero value of z-directed velocity. Thus the power output must decrease slightly when the mode frequency is tuned through the line center. In fact, this has been observed experimentally and is referred to as the Lamb dip – the presence of a Lamb dip in a He–Ne laser was shown by McFarlane, Bennet, and Lamb (1963).

4.7 Quantum Theory for the Evaluation of the Transition Rates