Multiple-Stage Amplification

While there are benefits of using multiple passes when the extraction efficiency for a single pass is low, the detrimental thermal effects can be severe. Multiple- stage amplifiers, on the other hand, can add much more power than multiple pass configurations because a greater pump power can be distributed over a larger volume of gain medium and therefore offer higher gain, without neces- sarily causing more severe thermal effects.

Employing a multiple-stage amplifier chain as a route for power scaling, has a number of advantages compared to simply increasing the power of a single- stage amplifier. The most obvious being that thermal effects can be managed more easily because the pump power is distributed over a larger volume of gain medium facilitating more efficient cooling. This reduces the likelihood of thermal fracture and hence allows higher pump powers. Distributing the pump power over many rods with low doping concentrations also reduces the impact of ETU, since the inversion density is reduced. The reduction in ETU leads to an increase in available gain and also reduction in total heat loading in the gain medium. This, in turn, leads to a reduction in thermal lens strength and also a reduction in thermal lens induced aberration of the signal beam. However, the thermal lensing and thermal lens aberrations are still accumu- lated over a number of amplifier stages such that they are dependent on the total heat deposited in all the amplifier stages as well on the material properties and the geometric parameters of each amplifier stage. With regard to thermal lensing, amplifiers have an advantage over oscillators in that dynamic stabil- ity is not such an important issue since they remain stable even for very strong thermal lensing. However, when a number of amplifier stages are combined, the thermal lensing effects become compounded and the resulting change in signal beam radius can become an important issue. One technique that can be used to address this problem is relay imaging of the signal beam, from one rod to another, to ensure that a particular spot size is re-imaged several times,

independently of thermal lens strength. This technique and its limitations are discussed in more detail in Chapter 5.

Leaving aside the issues associated with thermal lens focusing and diffraction of the signal beam, and again assuming constant pump and signal spot sizes over the length of the crystal, the beam quality of a signal after passing sev- eral aberrated thermal lenses can be calculated by applying equations (4.58) to (4.56) in sequence, where the M_f2 for the nth lens becomes the M_i2 for the

(n+ 1)thlens. Or in effect, for N amplifier stages [13]:

M_f2 = v u u t(M2 i)2+ N X n=1 (M2 q,n)2. (4.72)

For a number of amplifier stages with the same beam quality degradation fac- torM2

q, the outputM2 for a diffraction limited input beam is shown in Figure

4.14.

Figure 4.14: Beam quality degradation as a function of the number of amplifier stages

The graph shows that the beam quality degrades to a lesser degree with each successive amplifier stage. Higher values ofM_q2 are shown to lead to more se- vere beam quality degradation. From equation (4.62) for a truncated Gaussian

pump beam, in the presence of ETU, using the data above for a 0.1 at.% doped Nd:YVO4 crystal, and for a signal to pump overlap ratio of 0.7, the absorbed

pump power required to giveM2

q values of 0.5, 1.0 and 1.5, as in the graph, are

approximately 15W, 27W and 37W respectively, which are similar to the pump powers used in the experimental work described in this thesis. As already mentioned, the validity of equation (4.58) is limited whenwlis close towp but,

more importantly, the above analysis neglects the effect of the radial gain pro- file on the beam quality, which is shown in Chapter 5 to play an important role in the degradation in beam quality in a multiple-stage amplifier system. There are a number of factors which can limit the maximum number of am- plifier stages which can be combined, whilst still achieving a net increase in power and brightness. One factor is the transmission losses through each am- plifier stage which become significant when the signal power becomes high because the gain becomes comparable to the loss. For example, assuming that a number of amplifier stages in a chain, have the same pump power, the same pump beam radius and the same signal-pump overlap ratio, the output can be modelled, including a transmission loss for each amplifier stage. For a simple example of a 0.1 at.% doped, Nd:YVO4 amplifier with a pump power Pp of

30W, assumed to be completely absorbed, a pump beam radius wp of 300µm

, assumed to be Gaussian, and a signal-pump overlap ratio wp/wl of 0.7, the

small signal gain G0, neglecting ETU, was calculated using equation (4.22).

Applying equation (4.50) in sequence for an input power Pin for the first am-

plifier of 5W, Figure 4.15 was produced showing the output power and gain as a function of the number of amplifier stages for the lossless case, and assuming a 5% loss per amplifier stage.

The graph shows that, for an amplifier with loss, for a large number of am- plifier stages, the output power reaches a limit at∼250W, after around 40 am- plifier stages, where for the lossless case the power continues to increase lin- early with the number of stages, as the gain becomes saturated for each stage. The number of stages before which the limit is reached depends on the input power, the loss per stage and the amount of stored energy. This means that, in order to keep adding a significant amount of power with each stage, the pump power needs to be increased.

Figure 4.15: Gain (red) and output power (green) as a function of the number of amplifier stages. Dotted line for lossless amplifier, solid line for 5% loss per stage.

Another consideration is the degradation in beam quality for a large number of amplifier stages, which limits the brightness scaling potential of an ampli- fier chain before the power scaling limit is reached. For the amplifier stages described above, the value ofM_q2, from equation (4.62), for a truncated Gaus- sian pump beam, is 0.75. By calculating the brightness gain for each amplifier stage and hence the relative output brightness, Figure 4.16 was produced. For the lossless case, the relative brightness is shown to approach a limit after around 10 amplifier stages due to the cumulative effect of the beam quality degradation. When losses are taken into account (5% loss per stage), the rel- ative brightness rolls over after around 6 amplifier stages, as a result of the brightness gain falling below 1. In this case, increasing the pump power in successive amplifier stage, could be used to increase the maximum brightness, but only up to the limit where the aberrations for a single stage become too severe to cause a net gain in brightness. From Figure 4.12 it can be seen that an increase in pump power should be combined with a reduction inwl/wp in

order to maximise the brightness gain.

By this theory, increasing the pump power to successive amplifier stages can avoid a power scaling limit being reached. Similarly, if increasing the pump

Figure 4.16: Brightness gain (pink) and relative output brightness (green) as a function of the number of amplifier stages. Dotted line for lossless amplifier, solid line for 5% loss per stage.

power is combined with reducing the signal-pump overlap, the brightness scaling limit can be increased. However, this approach is still fundamentally limited by ETU, and thermal damage of the gain medium, which become sig- nificant at high pump powers even for crystals of low doping concentration. For this reason, using higher pump powers needs to be combined with using larger pump beam radii, to reduce the impact of ETU, which results a reduced gain. Additionally, if the signal beam radius is also increased to maintain a good spatial overlap, the signal intensity is therefore reduced leading to lower saturation.

4.6

Summary

By starting with a simple analysis of an end-pumped amplifier and building upon it, this chapter has explored the influence of a range of design parameters on the theoretical performance of laser amplifiers and amplifier chains. The to- tal small signal gain for ‘top-hat’ and Gaussian pump beams was derived for a Gaussian signal beam. Particular emphasis was given to the influence of the signal-pump overlap ratio on the amplifier performance because its opti-

mum value is difficult to determine and depends on a number of parameters. In the small signal limit, the approximation used for the pump beam profile, whether Gaussian or ‘top-hat’ , makes a large difference to the dependence of the gain on the signal-pump overlap for values ofwl/wp < 1. In the ‘top-hat’

case, the small signal gain levels off for low values ofwl/wp where, for Gaus-

sian pump beams, the small signal gain continues to increase as wl/wp → 0.

The effect of ETU on the analytical small signal gain was also taken into ac- count and found to significantly reduce the small signal gain for high pump intensities and high doping concentrations. This supports the design strategy of using low doping concentrations and longer crystal lengths to achieve high gain amplification with minimal loss in inversion density due to ETU as well as reducing thermal loading by distributing the absorbed pump power over a greater length of gain medium. The exponential dependence of small signal gain on the population inversion and the interaction length, makes its absolute value very sensitive to upconversion effects as well as the effects of misalign- ment and loss in population inversion due to amplified spontaneous emission (ASE), which is discussed in the next chapter. These combined effects gener- ally result in a measured small signal gain which is significantly lower than the value predicted from the model. However this analysis still provides useful clues on how to increase the gain for low power signals, such as by controlling the signal-pump overlap.

An analytical model for the saturated gain in a cw amplifier was then pre- sented which accounts for the reduction in gain caused by an intense signal. The gain therefore reaches a steady-state value where the pump rate equals the combined rate of stimulated and spontaneous emission. This model describes the relationship between the input intensity and the gain, in terms of the small signal gain and the saturation intensity, which is a constant for a particular transition in a particular gain medium. This is effectively the signal intensity required to reduce the gain to half its small signal value. The model was used to show how the gain reduces and the extraction efficiency increases as the in- put intensity is increased. To use this model to make predictions of amplifier performance however, requires iterative techniques which fit a given solution to a set of starting conditions. To simplify the analysis and also to extend it to apply to pulse amplification, a model was described for pulse amplification

where the pumping during the pulse is neglected. From this starting point, Franz and Nodvik’s expression for the gain can be derived. This allows the gain or extraction efficiency to be predicted for a set of initial conditions. By considering the change in gain during the pulse, this model was used to pre- dict the output pulse shapes as a function of time, for rectangular input pulses, under a number of operating conditions. The analysis showed the asymme- try of the output pulses, characterised by a sharp peak intensity at the leading edge of the pulse and a decay rate which depends on the initial gain and on the input signal energy. The rate of decay, which corresponds to the rate of gain saturation, was shown to increase with both initial gain and input energy, as one would expect. The model was then extended, using an iterative technique, to account for the build up time of the gain in between pulses which enables the steady-state initial gain after many pulse to be predicted for a known small signal. The main advantage of operating an amplifier in pulsed mode was at- tributed to the large gain in peak power achievable by allowing the gain to build up between pulses to approach its small signal value.

Having described a model for amplifier gain in both cw and pulsed mode of operation, a comparison was made between the two models to test their com- patibility and therefore determine the validity of the more useful pulsed model for making predictions of amplifier performance for cw systems. In general, reasonably good agreement was found between them with slightly higher gain predicted for the pulsed model in the partially saturated regime.

To apply the modelling to practical amplifier systems and to gain insight into the best parameters under which to operate; the model, which so far only considered a local area of the gain medium, was integrated numerically over the gain region to include the whole pump and signal beams. Therefore the influence of the signal-pump overlap and the pump and signal powers on the amplifier performance could be assessed. By first considering only the power gain, the gain as a function of signal-pump overlap was modelled. This showed a peak in gain occurring in the overlap range0.7< wl/wp <0.8which

increased slightly with pump power and showed only small differences be- tween the ‘top-hat’ and Gaussian pump beam approximations. Greater differ- ences between the two approximations were found to occur for small input energies, as was expected from the small signal gain analysis in Section 4.2.1.2.

The model was then modified to account for degradation in beam quality pre- dicted by a quartic approximation of the phase aberration caused by thermal lensing. This led to an analysis of the brightness gain GB = G/(Mx2My2)as a

function of the signal-pump overlap. In contrast to the power gain analysis, the peak brightness gain was shown to reduce with increasing pump power due to the increase thermal aberration. Additionally the optimum overlap, for maximum brightness, was significantly smaller0.48< wl/wp < 0.72than that

for maximum power gain.

Finally, the model for saturated pulse amplification was extended to apply to multiple amplifier stages and multiple passes of an amplifier. A step by step approach was proposed for predicting the gain for multiple amplifier stages by simply using the output from one as the input to the next. A similar approach was described for a double-pass amplifier however, in this case, the gain for the second pass is reduced by the first pass. A modification to this analysis was also necessary when there was overlap between the counter-propagating sig- nals since they then compete for gain. By considering the beam quality degra- dation as well as the diminishing extracted power achievable in multiple-pass configurations, the advantages of single-pass amplifiers were highlighted. For a multiple-stage amplifier, a simple example was given to illustrate the advantages and limitations of combining a large number of amplifiers in series. By introducing a transmission loss for each amplifier stage it was shown that a power scaling limit can be reached where the loss equals the gain. Additionally the brightness of such an amplifier, which is limited by the thermal aberrations for a lossless amplifier, was shown to roll over when a loss was introduced. The use of multiple amplifier stages has been shown, in principle, to be an effective route to power and brightness scaling, provided that system parameters are carefully controlled. However, limitations still ultimately arise due to ETU, thermal lens focusing and thermal damage to the gain medium.

4.7

References

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