*2.1.1. Application example: Laser focalization*

Example: EuristicXOver:

Example: Uniform Mutation:

The mutation take an element Vcj

*Vcj* (*k* ) ={ *Vcj* (*k* -1)

als are selected for the generation of the next population.

*Vcj* (*k* -1)

where w(k) is weight function which decreases with the iteration k.

(k-1) and Vb

(k-1), the children Va

*Vb* (*k* ) =*Vb* (*k* -1)

+ *w*(*k*)(1 - *Vcj*

+ *w*(*k*)*Vcj*

+ *r*(*Vb* (*k* -1) - *Va* (*k* -1) )

*Va* (*k* ) =*Va* (*k* -1)

*Mutations* functions: the genes of the parent are randomly modified.

(k) and Vb

(k-1) and mutate it in a new one by the rule:

) *if rand* >0.5

(*<sup>k</sup>* -1) *if rand* <0.5

(*k* -1)

**Figure 1.** Diagram representing the genetic algorithm principle. The algorithm starts from a random population and then each individual is measured and the population is sorted according to its fitness. Then, some of the best individu‐

(k) are generated by the following

From the parents Va

46 Adaptive Optics Progress

rule:

The intensity of a laser in its focal spot is largely dependent on the quality of the focal point, and this effect is even stronger in nonlinear optics. Often, in laser systems it is not simple to reach an optimal alignment, so that AO devices can be very useful in these cases.

For example, in ref. [24] it was demonstrated how an AO sensorless optimization based on a genetic algorithm can largely enhance the XUV high-order harmonics (HH) generated by the interaction of an ultrafast laser and a gas jet.

The AO system was composed by an electrostatic deformable mirror (Okotech) placed be‐ fore the interaction chamber as illustrated in Fig. 2. The feedback for the genetic algorithm was the photon flux at the shortest wavelengths acquired placing a photomultiplier tube at the XUV spectrometer output.

**Figure 2.** Experimental setup for the optimization of a laser focalization used for high order harmonics generation in ultrafast nonlinear optics. The pulsed laser beam interacts with a gas jet in the interaction chamber. The photomulti‐ plier tube collects the signal from the spectrograph and feeds the genetic algorithm that drives the deformable mirror DM.

The laser pulse was generated by a Ti:S CPA laser system with a hollow-fiber to realize the compression of the pulse duration. The typical values used in the experiment are 6 fs of du‐ ration, 200 µJ of pulse energy, at 1 kHz repetition rate (all the experimental details are de‐ scribed in Villoresi et al. 2004). The focusing of the laser pulses on the gas jet, after the modifications introduced by the Deformable Mirror (DM), is obtained by means of a 250 mm focal length spherical mirror. The spectrometer that analyzes the HHs beam is based on a flat varied-line-spacing grazing-incidence grating with two toroidal mirrors.

The real-time acquisition of the spectral intensity is realized by the combination of a solarblind open microchannel-plate (MCP) with MgF2 photocathode and a phosphor screen placed on the spectrometer focal plane, which converts the HHs XUV spectrum in the visi‐ ble, and by a photomultiplier which acquires a HHs spectral interval selected with a slit. In this way, the single-shot intensity of a single harmonic, or group of harmonics, is used as feedback by the algorithm. A separate optical channel acquires in parallel the image of all at the MCP, from which the HHs spectrum is obtained.

The genetic algorithm used a population of 80 individuals, with a deterministic selection rule that saved the 13 best ones. Both mutations and crossover were used. The results showed an increase of the XUV photons by a factor of 5 when the algorithm was applied. Moreover, the cutoff region moved to shorter wavelengths as reported in Fig 3. The optimi‐ zation process took about 20 iterations to converge.

**Figure 4.** Ants start randomly their search for food, then the shortest path gets the higher content of pheromone.

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49

The main essence of the Ant Colonies optimization algorithm [27] is to simulate the ant be‐ havior for the optimization of a given problem. The algorithm steps necessary for running

As an example we show in Fig. 5 the simulation of the application of the ant colony strategy to a deformable mirror with 32 actuators and 8 bits control. In this example the actuators and their control values are the domain in which the ants can move. In the simulation the

Finally, the ants will follow with larger probability the path having the highest content of pheromone.

the optimization are listed in Table 2.

**Table 2.** Steps of an ant colony algorithm.

**Figure 3.** Result of the experimental optimization of the high order harmonics generation spectra in the case of the flat AO mirror (black line) and in the case of the optimized wavefront (red line).

#### **2.2. Ant colonies**

Ant colonies, in natural world, search the food by walking randomly. After having found it, they return to their colony leaving down a pheromone trail. If other ants cross the same trail they will not walk randomly but they will likely follow it and will reinforce the pheromone trail. The more ants will find food at the end of the trail, the more pheromone will mark it. However, since the pheromone evaporates reducing its strength, the described process will make the shortest path which will be the one with the highest density of pheromone, so pro‐ viding a selection among all the possible paths, as illustrated in Fig. 4.

**Figure 4.** Ants start randomly their search for food, then the shortest path gets the higher content of pheromone. Finally, the ants will follow with larger probability the path having the highest content of pheromone.

The main essence of the Ant Colonies optimization algorithm [27] is to simulate the ant be‐ havior for the optimization of a given problem. The algorithm steps necessary for running the optimization are listed in Table 2.


**Table 2.** Steps of an ant colony algorithm.

showed an increase of the XUV photons by a factor of 5 when the algorithm was applied. Moreover, the cutoff region moved to shorter wavelengths as reported in Fig 3. The optimi‐

**Figure 3.** Result of the experimental optimization of the high order harmonics generation spectra in the case of the

Ant colonies, in natural world, search the food by walking randomly. After having found it, they return to their colony leaving down a pheromone trail. If other ants cross the same trail they will not walk randomly but they will likely follow it and will reinforce the pheromone trail. The more ants will find food at the end of the trail, the more pheromone will mark it. However, since the pheromone evaporates reducing its strength, the described process will make the shortest path which will be the one with the highest density of pheromone, so pro‐

flat AO mirror (black line) and in the case of the optimized wavefront (red line).

viding a selection among all the possible paths, as illustrated in Fig. 4.

**2.2. Ant colonies**

zation process took about 20 iterations to converge.

48 Adaptive Optics Progress

As an example we show in Fig. 5 the simulation of the application of the ant colony strategy to a deformable mirror with 32 actuators and 8 bits control. In this example the actuators and their control values are the domain in which the ants can move. In the simulation the shortest path is a parabolic function, which is represented by the red line. Fig. 5 (top) shows the initial random pheromone distribution, while Fig. 5 (bottom) shows the pheromone dis‐ tribution at the end of the optimization process.

**3. Image based algorithms**

**3.1. Devices for sensorless modal correction**

cus, astigmatism, coma) and of the spherical aberration.

correctors [32].

rithm to converge.

shown in (c).

Although the stochastic optimization algorithms have been demonstrated to represent im‐ portant tools for optical experiments, new techniques, which demonstrated to be more effec‐ tive, have recently been introduced. The use of a modal approach, based on the application of bias aberrations and of a suitable metrics, sorted out some of the limitations of the search algorithms, such as the long convergence time and the need of a training for the determina‐ tion of the algorithm parameters. This new approach demonstrated to be effective both in visual optics and in laser optimization, as described later in this section. The arbitrary gener‐ ation of aberrations can be achieved through the use of deformable mirrors, either thanks to a preliminary calibration of them or through the design of a suitable new class of wavefront

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Electrostatic membrane deformable mirrors rely on the electrostatic pressure between an ac‐ tuator pad array and a thin metalized membrane [33]. Thus, the more the actuators the bet‐ ter the wavefront resolution that the mirror can control. The use of these deformable mirrors is, then, subjected to the acquisition of the deformation generated by each electrode. On the other hand, this kind of DMs can also be used with the optimization algorithms. The draw‐ back, in this case, is that the higher the number of actuators the longer will take to the algo‐

Recently, a new type of deformable mirrors suitable for the direct generation of aberrated wavefronts was designed. The modal membrane deformable mirror, MDM, relies on the use of a graphite layer electrode arrangement (see Fig. 6) for the generation of a continuous dis‐ tribution of the electric field which allows the generation of the low order aberrations (defo‐

**Figure 6.** Electrostatic modal membrane deformable mirror, MDM. (a) Layout of the electrodes of the MDM; (b) volt‐ age and electrostatic pressure distribution which generates the astigmatism shape illustrated in the interferogram

**Figure 5.** Implementation of an ant colony strategy for the optimization of a deformable mirror with 32 actuators and 8 bit control. The red curve represents the shortest (optimized) path. The top panel shows the initial random phero‐ mone distribution while the bottom panel shows the pheromone at the end of the selection process.

#### *2.2.1. Application example: Quantum optics*

The quality of an optical wavefront plays an important role in Spontaneous Down Conver‐ sion (SPDC) process. As demonstrated by [31] the use of a deformable mirror can enhance the generation of photon pairs acting on the wavefront before the generation takes place in the nonlinear crystal. In that system the optimization was carried out by the use of an elec‐ trostatic DM (PAN, Adaptica srl) and the application of the ant colonies algorithm.

In the experiment, the pump beam is reflected by the DM to a BBO type-I nonlinear crystal. Then, the degenerate SPDC photons at 808 nm are selected and measured by a high efficien‐ cy SPADs (Single Photon Avalanche Diode). Since the wavefront has a strong effect on the downconverted light, it can strongly affect the coupling in the fibers of the SPAD detectors. The feedback for the algorithm imposed the condition of photon coincidences. It was dem‐ onstrated in the experiment that the coincidences rate was increased by about 20% when the optimization algorithm was applied. The algorithm used about 80 ants and the convergence took place in about 800 iterations.
