LOS CARNAVALES DEL 2012, UN NUEVO EPISODIO EN RECHAZO AL GOBIERNO

In the small signal regime, a continuous laser signal passing through a medium with uniform small signal gain will grow in power exponentially with dis- tance, z, through the medium. However, for a larger signal intensity a signif- icant number of excited ions decay by stimulated emission and since gain is proportional to the number of excited ions, the gain is therefore reduced and the amplifier is said to be saturated. In the heavily saturated regime the signal intensity grows approximately linearly with distance along the gain medium as all the excited atoms are stimulated. Complete saturation, however, is never achievable since the signal intensity required would be infinite. In the follow- ing sections the small signal gain analysis is extended to account for signal intensities large enough to cause saturation effects in the gain medium. Sieg- man [6] analyzed the behavior of a lossless single pass amplifier with continu- ous pump and signal beams and homogeneous saturation of the gain medium.

The basic differential equation governing the growth rate of a signal with in- tensityIl(z)along a lossless amplifier is given by

1 Il(z) dIl(z) dz = g0(z) 1 +Il(z)/Isat , (4.34) where Isat = hνl σ21τf . (4.35)

Isatis the signal intensity required to reduce the gain to one half its small signal

value. Since the gain coefficient varies with signal intensity and distance z along the amplifier it is not possible to simply express the gain G in terms of G0 and the input intensity Iin. Instead equation (4.34) can be rearranged

and integrated over the length of the amplifier and over an input and output intensityIin andIout:

Z Iout Iin 1 I + 1 Isat dI = Z l 0 g0(z)dz, (4.36)

which can be solved to give

ln Iout Iin + Iout−Iin Isat =g0l= lnG0. (4.37)

By definition, the gain G is given by G = Iout/Iin and so (4.37) can be rear-

ranged to give the useful formula Iin Isat = 1 G−1ln G0 G . (4.38)

Using the knowledge that the maximum gain achievable for an amplifier isG0,

numerical values ofGin the range1 < G < G0 can be inserted into equation

(4.38) to effectively show the variation in gain withIin.

For a practical amplifier it is normally desirable to operate in a saturated regime so that the maximum power is extracted from the amplifier. This however im- plies using a higher signal intensity and consequently the gain is lower. Oscil- lators are generally much better at extracting gain because the laser intensity inside the cavity is much higher due to feedback from the cavity mirrors. The extracted intensity,Iextr, of the amplifier can be calculated using (4.38):

Iextr ≡Iout−Iin = ln G0 G Isat. (4.39)

In the highly saturated regime where the gain approaches one, this equation can be used to show the total available intensity, Iavail, that can be extracted

from the amplifier assuming 100% extraction efficiency: Iavail = lim G→1ln G0 G Isat =IsatlnG0. (4.40)

Substituting equation (4.13), the small signal gain forr ≤ wp with a ‘top-hat’

pump beam, and equation (4.35) into equation (4.46), and multiplying by the pump beam area (πwp2) can yield the simple expression that confirms that,

ignoring the effect of ETU, the maximum power available in an amplifier is the pump power absorbed times one minus the quantum defect.

Pavail = νl νp Ppηabs. (4.41)

Equations (4.39) and (4.46) can be combined to form an expression for the power extraction efficiencyηextr:

ηextr = Iextr Iavail = 1− lnG lnG0 . (4.42)

If we take the simple example of an end-pumped amplifier with a pump power of 10W, Gaussian pump and signal beams of radii 300µm , and we make the simplifying assumptions that all the pump power is absorbed over the length of the gain medium and ETU is negligible, we can make direct comparisons between Nd:YVO4 , Nd;YAG and Nd:YLF. Using the data in Table 2.4, for 1%

doping concentration for each material, and equations (4.35) and (4.22), the small signal gains and saturation intensities were calculated, as shown in Table 4.1.

Material G0 Isat

Nd:YVO4 25.4 8.30 W mm−2

Nd:YAG 2.52 29.0 W mm−2

Nd:YLF 3.42 21.7 W mm−2

Table 4.1: Comparison of small signal gain and saturation intensity for Nd:YVO4 , Nd:YAG and Nd:YLF

This table clearly shows the significantly higher small signal gains achievable in Nd:YVO4compared to Nd:YAG and Nd:YLF under similar conditions. Ad-

ditionally, the lower saturation intensity in Nd:YVO4 means that greater ex-

4.4 further justifies the decision to use Nd:YVO4 , by showing the variation in

gain with input signal power, for the examples given above, found by inserting the values from Table 4.1 into equations (4.38) and (4.42).

Figure 4.4: The variation in gain (solid lines) and extraction ef- ficiency (dashed lines) with input signal power, for Nd:YVO4 ,

Nd:YAG and Nd:YLF amplifiers.

The graph shows that for, all signal powers, the gain and extraction efficiency is significantly higher in Nd:YVO4 , than in Nd:YAG and Nd:YLF, which have

more similar values, although Nd:YLF is shown to have slightly higher gain and extraction efficiency than Nd:YAG. An additional advantage of Nd:YVO4

is its higher absorption coefficient (∼3 times that of Nd:YAG or Nd:YLF), which has been neglected here, which implies a greater fraction of pump power ab- sorbed.