Chapter 5 Notes

Expanding on the algebra in section chapter 5.6, we begin with an unusual definition of inversion:

From this equation we take the derivative and substitute using equations 5.6.4 and 5.6.5 in the text as follows:

The result is equation 5.9.7. We must now substitute for N₂(t) in order to eliminate this term from the equation as follows:

Indeed, as stated in the text, the "x 2" appears since a decrease of one atom from the N₂ level results in an increase in one atom in the N₁ level - the total change in the inversion is hence two atoms.

Substitution for N₂(t) is now made in equation 5.9.7 to yield 5.9.8 and the solution follows as per the text. Note that in the examples used for the three and four-level laser the definition of inversion is the more conventional one in which:

In section 5.9 we showed the saturation intensity as:

We may now perform a few example calculations to show how saturation intensity may be computed for various transitions. This example also ties together several concepts in the chapter.

First, we need to calculate the (upper lasing) lifetime of the species. Consider the red transition of the HeNe laser with parameters l=632.8nm and transitional probability 0.0339*108 s^-1 (This is the Einstein 'A' coefficient and may be found in the CRC Handbook of Chemistry and Physics). We may approximate tau as the spontaneous lifetime since this is a four-level laser. Exact calculation of tau (i.e., the upper lasing level) may be made by summing the probabilities of all transitions beginning at that level. In this case all of the visible transitions originate from this level plus the IR transition at 730.5nm and a UV transition at 60.0nm (see Chapter 9 for all transitions and then the CRC handbook for transitional probabilities of all of these levels). The eight visible transitions (plus the IR transition at 730.5nm) are all found in the diagram in chapter 9 with the exception of the 60.0nm transition which has a transitional probability of 0.259*10⁸ s^-1 (which is much larger than any other transition originating from that upper level).

The lifetime of this upper level is hence calculated to be 30ns however the 60nm transition is resonance trapped (see section 5.4) and so can essentially be eliminated from the calculation since it's individual lifetime will be very long. The lifetime of the level is then calculated at 1.4*10^-7s.

Second, we need to compute the cross-section of the transition. From section 5.8 we can use equation 5.8.2 in which t_sp is as per calculated above. In the case of the gain lineshape, g(f), we may approximate this by using 1/ Df (since the HeNe laser is primarily broadened by Doppler means). Equation 5.8.2 then becomes:

We have calculated the Doppler linewidth in example 4.7.1 and so we may calculate s = 3.6*10^-17m² then calculate the saturation intensity as:

While this may seem like an enormous figure, consider that the beam is only 1mm is diameter and so the saturation power would be 48mW.

As discussed in this section, this is a naïve estimate since it does not take a number of factors into account, the most important being broadening of the gain medium. It does, however, illustrate how saturation intensity depends on various parameters of the transition.

From chapter 9 it can be seen that gain of the 543.5nm transition is about 0.02 times that of the 632.8nm red transition. We can arrive at this conclusion by (i) finding the 'A' coefficient for each transition from the CRC handbook then (ii) computing the cross-section for each transition using the simplification for s as found above. It is determined, then, that for the 632.8nm transition s = 3.6*10^-17m², for the 611.8nm transition s = 6.0*10^-18m², and for the 543.5nm transition s = 2.2*10^-18m². The ratio, then, for the cross-sections of each transition, and hence gain of each transition, follow that of table 9.2.1.

In order to explain gain saturation quantitatively, consider the basic rate equation for the upper-lasing level (ULL) of any laser with an added term describing the downward path provided out of this level by stimulated emission.

For example, the basic rate equation (5.7.2 in the text):

Where the extra terms represent 'decay' of ULL populations due to stimulated emission in the presence of cavity radiation (where 'r' is the cavity flux) and absorption from the LLL to the ULL which is also a stimulated process. The stimulated emission term (proportional to the population of the ULL N₂) is the rate from section 4.5 of the text. Furthermore, B can be solved as a function of frequency and level lifetime.

As r increases, the amplitude of this stimulated emission term does as well which serves to decrease the population at the ULL (N₂ in this case) so that the magnitude of the term decreases in turn as well ... this is the very nature of saturation as explained in chapter 5.

It may also be noted that the final term, the absorption term, becomes significant as the LLL populates - in a four level laser this may not pose a problem since a depopulation mechanism usually exists to maintain the population of N₁ at a low level however in a three level laser where N₁ is ground, growth of this level will decrease inversion eventually causing lasing to cease!

To further illustrate saturation using the rate equations, consider a very long four-level amplifier pumped at a constant rate. It is assumed that the LLL is essentially empty and so inversion DN is approximately equal to N_ULL. Since gain is proportional to inversion (section 5.8 of the text), gain may also be plotted.

At one end of the amplifier, a probe beam of very small intensity enters. As the probe beam travels down the amplifier, gain increases exponentially according to equation 4.6.1 however the gain is not a constant value but is dependent on the inversion, which constantly decreases as the rate of stimulated emission increases due to increased flux ,r, inside the amplifier. The population of the ULL (and hence gain, in dark blue) is seen to decrease as follows:

Clearly visible are regions where growth of flux (shown in violet) is exponential (unsaturated) and linear growth (saturated).

Experimental results are presented below for gain measured on the orange and red HeNe transitions using the direct method outlined in chapter 4. The unsaturated gain of the red transition falls well within accepted values (as well as values determined using the second method outlined below). In the case of the red transition, it can be seen that the gain decreases as intensity of the probe beam increases however a quick calculation of the saturation intensity shows that a much more powerful HeNe laser than commonly available (> 15mW) would be required to achieve saturation. Such lasers are unusual at best although the author will be attempting to measure gain using a HeNe with a rated output of over 100mW (A large-frame Spectra-Physics SP-125 HeNe) in the near future. Note that error bars are not shown in this graph as they are in the text.

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