Adaptive phase aberration correction based on imperialist competitive algorithm R. Yazdani,1,* M. Hajimahmoodzadeh,1,2 and H. R. Fallah1,2 1
Department of Physics, University of Isfahan, Isfahan, Iran
2
Quantum Optics Research Group, University of Isfahan, Isfahan, Iran *Corresponding author: [email protected]
Received 12 August 2013; revised 30 November 2013; accepted 2 December 2013; posted 4 December 2013 (Doc. ID 195607); published 23 December 2013
We investigate numerically the feasibility of phase aberration correction in a wavefront sensorless adaptive optical system, based on the imperialist competitive algorithm (ICA). Considering a 61-element deformable mirror (DM) and the Strehl ratio as the cost function of ICA, this algorithm is employed to search the optimum surface profile of DM for correcting the phase aberrations in a solid-state laser system. The correction results show that ICA is a powerful correction algorithm for static or slowly changing phase aberrations in optical systems, such as solid-state lasers. The correction capability and the convergence speed of this algorithm are compared with those of the genetic algorithm (GA) and stochastic parallel gradient descent (SPGD) algorithm. The results indicate that these algorithms have almost the same correction capability. Also, ICA and GA are almost the same in convergence speed and SPGD is the fastest of these algorithms. © 2013 Optical Society of America OCIS codes: (010.1080) Active or adaptive optics; (220.1080) Active or adaptive optics; (220.1000) Aberration compensation. http://dx.doi.org/10.1364/AO.53.000132
1. Introduction
Adaptive optics (AO) is a powerful technology to compensate for the dynamical phase aberrations in real time. The basic idea of the dynamical correction was first proposed by Babcock in 1953 [1]. Although AO was initially developed for astronomical viewing and atmosphere propagation limited by turbulence effect, it now has various medical and commercial applications [2–4]. The three critical components of an AO system are a wavefront sensor, which detects the phase aberrations in the incident wavefront, an adaptive element such as a deformable mirror (DM), which can correct the phase aberrations and a proper closed-loop control system, which generates signals to the adaptive element. The basic limitation of an AO system is a large number of expensive and complex elements needed for wavefront sensing. 1559-128X/14/010132-09$15.00/0 © 2014 Optical Society of America 132
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Moreover, in some cases, direct wavefront sensing is not straightforward [5] and even, sometimes, there is no need to have the real time information about the wavefront aberrations. In these cases, the wavefront sensorless AO, based on stochastic optimization methods, can provide an alternative approach to the control problem, in which the wavefront sensor is replaced by a low-cost and simpler system. As a common strategy used by model-free optimization techniques in the wavefront sensorless AO [6], the adaptive element is reconfigured so that the system performance metric related to the image quality is optimized. The system performance metric, J, is defined according to the intensity distribution in the image plane of system and optimized when the system aberrations are compensated. For optimizing J, some algorithms have already been used such as the genetic algorithm (GA) [7–9], simulated annealing [9,10], stochastic parallel gradient descent (SPGD) [11,12], ant colony [13], and hill climbing [14]. In this paper, we introduce the imperialist competitive
algorithm (ICA) and investigate numerically the feasibility of its application in a wavefront sensorless AO system as the optimization control algorithm. The ICA was proposed by Atashpaz-Gargari and Lucas in 2007 [15]. ICA is one of the metaheuristic optimization techniques simulating the social political process of imperialism and imperialistic competition. This algorithm has been widely utilized in various areas of engineering and science, such as designing controllers for industrial systems [16], solving optimization problems in communication systems [17], and solving scheduling and production management problems [18]. In solid-state laser systems, imperfections of optical components and pump-induced thermal distortions degrade the laser beam quality. In this paper, we successfully apply ICA in a wavefront sensorless AO system for improving the Nd:YAG laser beam quality. It has already been shown that the phase aberrations of an Nd:YAG laser change very slowly, so the convergence rate is not very important for such AO systems [19]. We simulate an AO system with a 61-element DM and consider the Strehl ratio as the system performance metric and cost function in ICA and report the simulation results after phase aberration correction with ICA. Finally, we compare the correction capability and the convergence speed of ICA with those of GA and SPGD algorithm. The effects of noise are not included in our simulation analysis. 2. Imperialist Competitive Algorithm (ICA)
ICA is a novel population-based metaheuristic algorithm inspired by the sociopolitical imperialistic competition. This algorithm starts with an initial population known as the countries. Some of the best countries are selected to be the imperialists and the rest form the colonies of the imperialists. Each imperialist along with its colonies generates an empire. In the search space, imperialistic competition among the empires is the basis of ICA which directs the search process toward the powerful imperialist or the optimum point. The procedures of ICA are described as follows. A.
Creating Initial Empires
In ICA, the countries are feasible solutions in the search space. For an optimization problem with N variables, a country is a 1 × N array defined as follows: country C u1 ; u2 ; u3 ; …; uN :
among the imperialists depending on their powers. For this purpose, assuming that J is to be maximized, a normalized cost is defined by: J n J n − minfJ i g;
(3)
where J n is the cost of the nth imperialist and J n is its normalized cost. Having J n , The normalized power of the nth imperialist is defined by: Jn Pn PN i : J
(4)
i1
Then, the initial number of colonies of the nth empire will be: NCn roundfPn · N col g;
(5)
which are selected randomly among the N col N − N colonies. Figure 1 shows the initial empires. As shown in this figure, bigger empires have a greater number of colonies while weaker ones have less. In Fig. 1, imperialist 1 has formed the most powerful empire and consequently has the greatest number of colonies. B. Assimilation: Movement of the Colonies of an Empire toward the Imperialist
To increase their power, the imperialists try to develop their colonies through the assimilation policy, where the colonies are forced to move toward them along different sociopolitical axes, such as culture, language, and religion. This process in ICA is modeled by moving all of the colonies toward the imperialist along different optimization axes. This can be done for each colony C u1 ; u2 ; …; uN by adding the random vector x x1 ; x2 ; …; xN to it, such that the vector u1 x1 ; u2 x2 ; …; uN xN becomes the updated state of the colony C. The elements of x-vector can be defined as: xi ∼ U0; β × di ;
(6)
where U denotes the uniform distribution, β is a number greater than one, and di is the distance
(1)
The cost of a country is defined by evaluating the cost function J at variables u1 ; u2 ; u3 ; …; uN , i.e., J JC Ju1 ; u2 ; u3 ; …; uN :
(2)
In the beginning, the initial countries of size N are produced. N of the most powerful countries, which have the best costs, are selected as the imperialists and the rest will be the colonies which distribute
Fig. 1. Creating the initial empires. The imperialist with more colonies has bigger ⋆ mark. 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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Fig. 4. (a) Exchanging the positions of a colony and the imperialist. (b) The entire empire after position exchange. Fig. 2. Assimilation. Moving colony toward its relevant imperialist.
between the colony and its imperialist state along the ith optimization axis. More precisely, di is the difference between the ith variable value of the colony, ui , and that of its imperialist, di juiimp − uicol j. This movement for a one-dimensional problem is shown in Fig. 2. β > 1 causes the colonies to get closer to the imperialist state from both sides along every dimension. β ≫ 1 gradually results in divergence of colonies from the imperialist state while a β very close to 1 reduces the search ability of the algorithm. β 2 is a proper choice in most of the implementations. There are other methods for modeling the assimilation policy and we refer the interested reader to see [18]. C. Revolution: A Sudden Change in the Socio-Political Characteristics of a Country
Revolution is a fundamental change in the power or organizational structures of a colony that takes place in a relatively short period of time. That is, instead of being assimilated by an imperialist, the colony randomly changes its position in the sociopolitical axis. In ICA, the new position of a colony after revolution can be obtained by randomly changing its variable values in the allowed range. Figure 3 shows the revolution for a two-dimensional optimization problem. The revolution in ICA increases the exploration of the algorithm and prevents the early convergence of countries to a local extremum. The revolution rate, pr in the algorithm, indicates the percentage of colonies in each empire which will randomly change their position. A very high value of revolution
Fig. 3. Revolution. A sudden change in socio-political characteristics of a country. 134
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decreases the exploitation power of the algorithm and can reduce its convergence rate. D.
Exchanging Position of the Imperialist and a Colony
After assimilation for all colonies and revolution for a percentage of them, a colony may reach a position with a better cost than that of the imperialist, i.e., JCcol > JCimp . In this case, the positions of the imperialist and the colony are exchanged. So the colony will be the new imperialist, whereas the old imperialist will become a colony of the same empire. After that, the algorithm will continue by the imperialist in a new position and then the colonies start moving toward this position. Figure 4(a) depicts the position exchange between a colony and the imperialist. In this figure, the best colony of the empire is shown in a darker color. This colony has a higher cost than the imperialist. Figure 4(b) shows the empire after exchanging positions between the imperialist and the colony. E. Imperialistic Competition
The most important process in ICA is imperialistic competition, in which all empires try to take possession of colonies of other empires. Gradually, weaker empires lose their colonies to the stronger ones. This process is modeled by choosing the weakest colony of the weakest empire and making a competition among all empires to possess this colony. The weakest empire is the empire with the lowest total cost, defined later by Eq. (7), and the weakest colony is the colony with the lowest cost. Figure 5 shows a
Fig. 5. Imperialistic competition. The more powerful an empire is, the more likely it possesses the weakest colony of the weakest empire.
big picture of the modeled imperialistic competition. Based on its total power, in this competition, each of the empires will have a likelihood of taking possession of the mentioned colony. In other words, this colony will not be possessed by the most powerful empires, but these empires will be more likely to possess it. To start the competition, first, the possession probability of each empire, based on its total power, is found. The total cost of the nth empire is evaluated by: TCn J n ξ · meanfJ ni g;
(7)
where fJ ni g is the set of colony costs of the nth empire and ξ is a positive number, which is considered to be less than one. A little value for ξ causes the total power of the empire to be determined by just the imperialist and increasing it will increase the role of the colonies in determining the total power of an empire. Then, the normalized total cost is obtained by: NTCn TCn − minfTCi g;
(8)
where TCn and NTCn are the total cost and the normalized total cost of the nth empire, respectively. Having the normalized total cost, the possession probability of the nth empire is calculated by: pp n
NTCn : PN NTC i1
(9)
i
For assigning the mentioned colony to empires based on their possession probability, vector P is formed as: P pp1 ; pp2 ; pp3 ; …; ppN :
(10)
Then, the vector R, with the same size as P, whose elements are uniformly distributed random numbers in the range [0,1], is created, R r1 ; r2 ; r3 ; …; rN :
(11)
Vector D is formed by subtracting R from P:
be reaching a predefined number of iterations, obtaining the desired cost, or only one existing imperialist. 3. Wavefront Sensorless Adaptive Optics System Using ICA
To investigate the phase aberration correction capability of ICA in a wavefront sensorless AO system, we consider the problem of focusing an aberrated laser beam. The assumed setup in simulations is shown in Fig. 7. A wavefront with computergenerated phase aberrations is reflected by a piezoelectric DM and focused on the focal plane of a lens through fast Fourier-transform (FFT). The intensity distribution in this plane is recorded by a CCD camera. It is assumed that DM has a continuous faceplate with 61 actuators, and its stroke is enough to correct the phase aberrations in the simulation. The actuator configuration is shown in Fig. 8. Each black dot stands for an actuator. The circled line in this figure denotes the effective aperture. The phase distribution, φx; y, generated by the 61-element DM can be represented as: φx; y
61 X
uj V j x; y;
(13)
j1
D P − R D1 ; D2 ; D3 ; …; DN pp1 − r1 ; pp2 − r2 ; pp3 − r3 ; …; ppN − rN :
Fig. 6. Eliminating the powerless empires. Empire 4, with no colonies, does not have the power for competition and so it is eliminated from other empires.
(12)
Referring to vector D, the mentioned colony is handed to an empire which in its relevant index, D is maximum. F.
Eliminating the Powerless Empires
An empire will collapse when this empire loses all of its colonies. In this case, this imperialist is considered as a colony and it is assigned to the other empires, as shown in Fig. 6. G.
Termination Criterion
The algorithm is continued until the termination criterion is satisfied. The termination criterion can
Fig. 7. Schematic of wavefront sensorless AO system. 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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Fig. 8. Actuator configuration of 61-element DM.
where uj is the voltage applied onto the jth actuator and V j x; y is the influence function of the jth actuator on the wavefront. From real measurements, we know that the influence function of 61-element DM actuators is approximately a Gaussian distribution [20]: q α ; x − xj 2 y − yj 2 ∕d V j x; y exp lnω
(14) where xj ; yj is the space position of the jth actuator, ω is the coupling coefficient, α is the Gaussian index, and d is the distance between two neighboring actuators. We set ω to 0.08 and α to 2 in our simulations, according to the data from [9]. The ICA is employed to compensate for the input phase aberrations. It is done by adjusting the control voltages of DM for optimizing the performance metric of the system as follows. A.
ICA with Simple Aberrations
We first add defocus and astigmatism aberrations separately to the plane wavefront of the incident
beam. These aberrations are defined over 128 × 128 pixels, which is also the grid of DM (see Fig. 8). The obtained aberrated wavefronts are shown in Figs. 9 and 10. We perform ICA to compensate for these simple aberrations as described below. The cost function J in ICA is considered to be the Strehl ratio (SRctr ) as the system performance metric. SRctr is defined by the ratio of the actual central intensity of the far-field light to that of an ideal wavefront far-field light on the focal plane. The higher the value of SRctr, the better the beam quality. The major purpose of ICA is to acquire the optimum control voltages of DM for maximizing SRctr. For our optimization problem, each country in this algorithm is represented by 61 control voltages, ui , as the variables. Therefore, N 61 is the number of actuators of DM, such that: country C u1 ; u2 ; u3 ; …; u61 :
The initial population includes N 50 countries as candidate solutions, the variable values of which are randomly selected in the range of allowed voltages: 2
3 country1 6 country2 7 6 7 6 7 : 6 7: population 6 7 : 6 7 4 5 : country50
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(16)
According to Eq. (13), for every country (C), a set of the control voltages results in a certain shape of DM and, consequently, a certain value of SRctr (JC). We select 10% of initial countries with the highest SRctr as the imperialists, so N 5 and N col N − N 45. One can choose the small population size but convergence of the system will require more iterations. Also, the large population size takes more computational time. After calculating the
Fig. 9. Wavefront distribution before and after defocus correction with ICA. 136
(15)
Fig. 10. Wavefront distribution before and after astigmatism correction with ICA.
normalized power of the imperialists, Pn , according to Eq. (4), the initial number of colonies of every imperialist, NCn , is determined and the empires are formed. Then, the described steps in Subsections 2.B–2.F are repeated until the termination criterion is satisfied. In our simulations, the termination criterion is reached at the iteration number of 500. This number has been appropriately selected to converge completely. In this case, the best imperialist state, a set of control voltages that results in the highest SRctr , is considered as the best solution of problem. The parameter values of ICA depend on the optimization problem. In our problem, how to choose the best values of these parameters is analyzed through plenty of implementations and, finally, the parameters that result in the most effective convergence process are selected. We set the parameters β in Eq. (6) and ξ in Eq. (7) to 2 and 0.05, respectively, in our simulations. We also set pr , revolution rate, to 0.33 at the first iteration. Then, it decreases when the iteration number increases, such that 0.99pnr , where n is the iteration number of pn1 r the algorithm. A high value of the revolution rate in the beginning of the algorithm causes the whole search space to be explored, the different shapes of DM to be considered, and, consequently, early convergence of solutions to a local extremum to be prevented. Decreasing the revolution rate when the iteration number increases causes the divergence of solutions from optimal (in terms of the wavefront sensorless correction approach this means the optimal shape of DM), to be reduced. It also allows the surrounding space of optimal solutions to be more accurately searched. The wavefront distributions, after defocus and astigmatism correction using ICA, are shown in Figs. 9 and 10, respectively. In Fig. 9, the peak to valley (PV) and root mean square (RMS) of wavefront aberrations are reduced from 0.55λ (λ 1064 nm) and
0.12λ to 0.09λ and 0.007λ, respectively, after defocus correction. In this case, SRctr is increased from 0.32 to 0.99. In Fig. 10, PV and the RMS of wavefront aberrations are reduced from 0.70λ and 0.11λ to 0.10λ and 0.005λ, respectively, after astigmatism correction, and SRctr is raised from 0.42 to 0.99. These optimization results show that ICA has very good capability to correct simple phase aberrations. B. ICA with Complicated Aberrations
We also generate more complicated phase aberrations and add them to the incident beam. The obtained wavefront distribution is shown in Fig. 11. The phase aberrations include 100 Zernike modes with tip and tilt aberrations, and regardless of piston aberration. The Zernike coefficients, which are randomly produced in our method, are shown in Fig. 12(a). As mentioned previously, in our simulations, DM has 61 actuators. Such DM, with 7 actuators across the diameter of the effective aperture, can only appropriately compensate for Zernike modes up to a radial order n 7; that is, up to n 1n 2∕2 36 Zernike modes [21]. Moreover, the main aberration modes in a solid-state laser system are essentially from the first 36 Zernike modes, and the contribution of higher-order modes is less. So we assume that higher-order modes have smaller coefficients. It can also be a way to simulate some kind of noise produced by high-order modes that will affect the efficiency of the low-order mode correction. Under this condition, ICA is used for phase aberration compensation with the same cost function (SRctr ) and initial parameters that were considered in Subsection 3.A. The wavefront distribution, after compensation, is shown in Fig. 11. We find that the PV and RMS of wavefront aberrations are reduced from 1.49λ and 0.14λ to 0.54λ and 0.02λ, respectively. The Zernike coefficients of residual phase aberrations and the Zernike coefficients attenuation, defined as the ratio of residual to input Zernike coefficients, are presented in 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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Fig. 11. Wavefront distribution before and after aberration correction with ICA.
Figs. 12(b) and 12(c), respectively. Figure 12(c) shows clearly which Zernike modes are compensated for (attenuation < 1), which ones are kept invariant (attenuation 1), or even degraded (attenuation > 1). For clarity, the Zernike coefficients attenuation for the first 40 Zernike modes is shown in Fig. 12(d). As expected, the low-order modes, which are controllable by DM, are compensated appropriately. Figure 13(a) shows SRctr during the optimization process for 500 iterations, averaged over 10 corrections. As shown, SRctr increases with the iteration process and reaches 0.92 from 0.36. 0.5
10
Zernike Coefficient
Zernike Coefficient 20
40 60 Zernike Order
80
0
−0.5
100
Coefficients Attenuation
After Correction
0
−0.5
20
(a)
40 60 Zernike Order
80
0
−5
20
0
−0.5
40
40 60 Zernike Order
80
100
(c) 1
GA
Coefficients Attenuation
Coefficients Attenuation
0.5
20 30 Zernike Order
5
−10
100
1
ICA
10
ICA
(b)
1
Coefficients Attenuation
To compare ICA with the standard algorithms, which have already been used for phase aberration compensation, we apply GA and SPGD algorithm to compensate for the same-phase aberrations considered in Subsection 3.B. The procedures of these algorithms are found in [7–9] for GA and [11,12] for SPGD algorithm. We make use of the real-number encoding GA with a nonuniform arithmetical crossover operator [7]. To converge completely, we set the iteration number of the algorithms appropriately. The initial population size and the number of generations of GA are
0.5
Before Correction
−1
4. Comparison of ICA with GA and SPGD Algorithm
0.5
0
−0.5
−1
(d)
10
20 30 Zernike Order
(e)
40
SPGD 0.5
0
−0.5
−1
10
20 30 Zernike Order
40
(f)
Fig. 12. (a) Zernike coefficients of the input phase aberrations. (b) Zernike coefficients of residual phase aberrations, after correction with ICA. (c) Zernike coefficients attenuation, after correction with ICA. Zernike coefficients attenuation for the first 40 Zernike modes, after correction with (d) ICA, (e) GA, and (f) SPGD. 138
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1
SPGD
0.6
0.4
0.8
Strehl Ratio
0.8
Strehl Ratio
0.8
Strehl Ratio
1
GA
ICA
0.6
0.4
0.2 0
100
200
300
400
500
0.6
0.4
0.2 0
200
400
600
0.2 0
1000
2000
Iteration Number
Iteration Number
Iteration Number
(a)
(b)
(c)
3000
Fig. 13. Strehl ratio (SRctr ) during optimization process with (a) ICA, (b) GA, and (c) SPGD, averaged over 10 corrections for each algorithm.
Table 1.
Comparison of Optimization Results of ICA, GA, and SPGD
Algorithm
Averaged SRctr
Final PV (λ)
Final RMS (λ)
0.92 0.92 0.92
0.54 0.58 0.56
0.02 0.02 0.02
ICA GA SPGD
set to 50 and 600, respectively, and the number of iterations of SPGD algorithm is set to 3000. The Zernike coefficients’ attenuation, obtained after phase aberration compensation with GA and SPGD algorithm, are shown in Figs. 12(e) and 12(f), respectively. SRctr , averaged over 10 corrections, during optimization process with GA and SPGD algorithm is shown in Figs. 13(b) and 13(c), respectively. For each of these three algorithms, the final averaged SRctr , along with PV and RMS of residual wavefront aberrations, is given in Table 1. These results, and the comparison of Figs. 12(d)–12(f) and Figs. 13(a)–13(c), indicate that these three algorithms have almost the same correction capability. A.
Analysis of Convergence Speed
An important criterion on which the algorithm can be applied to real-time AO systems is the convergence speed. Because the number of small perturbations sent to the system per iteration is different for different algorithms, we cannot determine convergence speed just from the number of iterations. Therefore, to estimate the convergence speed of the algorithms, we consider the number of mirror changes needed for achieving averaged SRctr of 0.8 in each algorithm. A SRctr of 0.8 is commonly known as the diffraction limit. This number of mirror changes is obtained Table 2.
Comparison of Convergence Speeds of ICA, GA, and SPGD
Algorithm ICA GA SPGD
Averaged Number of Mirror Changes for Achieving SRctr 0.8 74 × 50 3700 75 × 50 3750 185 × 2 370
from multiplying the iteration number in SRctr 0.8 by the number of small perturbations sent to the system per iteration. The number of these perturbations is the number of countries in ICA, the number of generations in GA, and 2 in the SPGD algorithm. The results are presented in Table 2. These data show that ICA and GA have almost the same convergence speed and the SPGD algorithm is the fastest algorithm, such that the number of mirror changes needed by ICA and GA is almost 10 times as many as that of the SPGD algorithm. 5. Conclusion
We have described the basic principle of the ICA. Then, we simulated an AO system with a 61-element DM for phase aberration correction in a solid-state laser beam. The ICA was applied to search the optimum control voltages of DM for maximizing the Strehl ratio. The results obtained from phase aberration correction show that this algorithm has very good correction capability. We compared ICA with GA and the SPGD algorithm and found that they have almost the same correction capability for our simulated phase aberrations. Also, the comparison shows that ICA and GA are almost the same in convergence speed and that the SPGD algorithm is the fastest algorithm, such that the number of mirror changes needed by ICA and GA is almost 10 times as many as that of the SPGD algorithm. The advantage of ICA is that it is far more likely to find the global extremum, and its disadvantage is that its convergence speed is not fast. In conclusion, ICA is a proper correction algorithm for static or slowly changing phase aberrations in optical systems, such as solid-state laser systems, microscopy, and retinal imaging. The aim of this paper was to investigate the feasibility of phase aberration correction with ICA and our future work will be modifying the convergence speed of ICA and using it in a real physical system. References 1. H. W. Babcock, “The possibility of compensating astronomical seeing,” Astron. Soc. Pac. 65, 229–236 (1953). 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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Department of Physics, University of Isfahan, Isfahan, Iran
2
Quantum Optics Research Group, University of Isfahan, Isfahan, Iran *Corresponding author: [email protected]
Received 12 August 2013; revised 30 November 2013; accepted 2 December 2013; posted 4 December 2013 (Doc. ID 195607); published 23 December 2013
We investigate numerically the feasibility of phase aberration correction in a wavefront sensorless adaptive optical system, based on the imperialist competitive algorithm (ICA). Considering a 61-element deformable mirror (DM) and the Strehl ratio as the cost function of ICA, this algorithm is employed to search the optimum surface profile of DM for correcting the phase aberrations in a solid-state laser system. The correction results show that ICA is a powerful correction algorithm for static or slowly changing phase aberrations in optical systems, such as solid-state lasers. The correction capability and the convergence speed of this algorithm are compared with those of the genetic algorithm (GA) and stochastic parallel gradient descent (SPGD) algorithm. The results indicate that these algorithms have almost the same correction capability. Also, ICA and GA are almost the same in convergence speed and SPGD is the fastest of these algorithms. © 2013 Optical Society of America OCIS codes: (010.1080) Active or adaptive optics; (220.1080) Active or adaptive optics; (220.1000) Aberration compensation. http://dx.doi.org/10.1364/AO.53.000132
1. Introduction
Adaptive optics (AO) is a powerful technology to compensate for the dynamical phase aberrations in real time. The basic idea of the dynamical correction was first proposed by Babcock in 1953 [1]. Although AO was initially developed for astronomical viewing and atmosphere propagation limited by turbulence effect, it now has various medical and commercial applications [2–4]. The three critical components of an AO system are a wavefront sensor, which detects the phase aberrations in the incident wavefront, an adaptive element such as a deformable mirror (DM), which can correct the phase aberrations and a proper closed-loop control system, which generates signals to the adaptive element. The basic limitation of an AO system is a large number of expensive and complex elements needed for wavefront sensing. 1559-128X/14/010132-09$15.00/0 © 2014 Optical Society of America 132
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Moreover, in some cases, direct wavefront sensing is not straightforward [5] and even, sometimes, there is no need to have the real time information about the wavefront aberrations. In these cases, the wavefront sensorless AO, based on stochastic optimization methods, can provide an alternative approach to the control problem, in which the wavefront sensor is replaced by a low-cost and simpler system. As a common strategy used by model-free optimization techniques in the wavefront sensorless AO [6], the adaptive element is reconfigured so that the system performance metric related to the image quality is optimized. The system performance metric, J, is defined according to the intensity distribution in the image plane of system and optimized when the system aberrations are compensated. For optimizing J, some algorithms have already been used such as the genetic algorithm (GA) [7–9], simulated annealing [9,10], stochastic parallel gradient descent (SPGD) [11,12], ant colony [13], and hill climbing [14]. In this paper, we introduce the imperialist competitive
algorithm (ICA) and investigate numerically the feasibility of its application in a wavefront sensorless AO system as the optimization control algorithm. The ICA was proposed by Atashpaz-Gargari and Lucas in 2007 [15]. ICA is one of the metaheuristic optimization techniques simulating the social political process of imperialism and imperialistic competition. This algorithm has been widely utilized in various areas of engineering and science, such as designing controllers for industrial systems [16], solving optimization problems in communication systems [17], and solving scheduling and production management problems [18]. In solid-state laser systems, imperfections of optical components and pump-induced thermal distortions degrade the laser beam quality. In this paper, we successfully apply ICA in a wavefront sensorless AO system for improving the Nd:YAG laser beam quality. It has already been shown that the phase aberrations of an Nd:YAG laser change very slowly, so the convergence rate is not very important for such AO systems [19]. We simulate an AO system with a 61-element DM and consider the Strehl ratio as the system performance metric and cost function in ICA and report the simulation results after phase aberration correction with ICA. Finally, we compare the correction capability and the convergence speed of ICA with those of GA and SPGD algorithm. The effects of noise are not included in our simulation analysis. 2. Imperialist Competitive Algorithm (ICA)
ICA is a novel population-based metaheuristic algorithm inspired by the sociopolitical imperialistic competition. This algorithm starts with an initial population known as the countries. Some of the best countries are selected to be the imperialists and the rest form the colonies of the imperialists. Each imperialist along with its colonies generates an empire. In the search space, imperialistic competition among the empires is the basis of ICA which directs the search process toward the powerful imperialist or the optimum point. The procedures of ICA are described as follows. A.
Creating Initial Empires
In ICA, the countries are feasible solutions in the search space. For an optimization problem with N variables, a country is a 1 × N array defined as follows: country C u1 ; u2 ; u3 ; …; uN :
among the imperialists depending on their powers. For this purpose, assuming that J is to be maximized, a normalized cost is defined by: J n J n − minfJ i g;
(3)
where J n is the cost of the nth imperialist and J n is its normalized cost. Having J n , The normalized power of the nth imperialist is defined by: Jn Pn PN i : J
(4)
i1
Then, the initial number of colonies of the nth empire will be: NCn roundfPn · N col g;
(5)
which are selected randomly among the N col N − N colonies. Figure 1 shows the initial empires. As shown in this figure, bigger empires have a greater number of colonies while weaker ones have less. In Fig. 1, imperialist 1 has formed the most powerful empire and consequently has the greatest number of colonies. B. Assimilation: Movement of the Colonies of an Empire toward the Imperialist
To increase their power, the imperialists try to develop their colonies through the assimilation policy, where the colonies are forced to move toward them along different sociopolitical axes, such as culture, language, and religion. This process in ICA is modeled by moving all of the colonies toward the imperialist along different optimization axes. This can be done for each colony C u1 ; u2 ; …; uN by adding the random vector x x1 ; x2 ; …; xN to it, such that the vector u1 x1 ; u2 x2 ; …; uN xN becomes the updated state of the colony C. The elements of x-vector can be defined as: xi ∼ U0; β × di ;
(6)
where U denotes the uniform distribution, β is a number greater than one, and di is the distance
(1)
The cost of a country is defined by evaluating the cost function J at variables u1 ; u2 ; u3 ; …; uN , i.e., J JC Ju1 ; u2 ; u3 ; …; uN :
(2)
In the beginning, the initial countries of size N are produced. N of the most powerful countries, which have the best costs, are selected as the imperialists and the rest will be the colonies which distribute
Fig. 1. Creating the initial empires. The imperialist with more colonies has bigger ⋆ mark. 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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Fig. 4. (a) Exchanging the positions of a colony and the imperialist. (b) The entire empire after position exchange. Fig. 2. Assimilation. Moving colony toward its relevant imperialist.
between the colony and its imperialist state along the ith optimization axis. More precisely, di is the difference between the ith variable value of the colony, ui , and that of its imperialist, di juiimp − uicol j. This movement for a one-dimensional problem is shown in Fig. 2. β > 1 causes the colonies to get closer to the imperialist state from both sides along every dimension. β ≫ 1 gradually results in divergence of colonies from the imperialist state while a β very close to 1 reduces the search ability of the algorithm. β 2 is a proper choice in most of the implementations. There are other methods for modeling the assimilation policy and we refer the interested reader to see [18]. C. Revolution: A Sudden Change in the Socio-Political Characteristics of a Country
Revolution is a fundamental change in the power or organizational structures of a colony that takes place in a relatively short period of time. That is, instead of being assimilated by an imperialist, the colony randomly changes its position in the sociopolitical axis. In ICA, the new position of a colony after revolution can be obtained by randomly changing its variable values in the allowed range. Figure 3 shows the revolution for a two-dimensional optimization problem. The revolution in ICA increases the exploration of the algorithm and prevents the early convergence of countries to a local extremum. The revolution rate, pr in the algorithm, indicates the percentage of colonies in each empire which will randomly change their position. A very high value of revolution
Fig. 3. Revolution. A sudden change in socio-political characteristics of a country. 134
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decreases the exploitation power of the algorithm and can reduce its convergence rate. D.
Exchanging Position of the Imperialist and a Colony
After assimilation for all colonies and revolution for a percentage of them, a colony may reach a position with a better cost than that of the imperialist, i.e., JCcol > JCimp . In this case, the positions of the imperialist and the colony are exchanged. So the colony will be the new imperialist, whereas the old imperialist will become a colony of the same empire. After that, the algorithm will continue by the imperialist in a new position and then the colonies start moving toward this position. Figure 4(a) depicts the position exchange between a colony and the imperialist. In this figure, the best colony of the empire is shown in a darker color. This colony has a higher cost than the imperialist. Figure 4(b) shows the empire after exchanging positions between the imperialist and the colony. E. Imperialistic Competition
The most important process in ICA is imperialistic competition, in which all empires try to take possession of colonies of other empires. Gradually, weaker empires lose their colonies to the stronger ones. This process is modeled by choosing the weakest colony of the weakest empire and making a competition among all empires to possess this colony. The weakest empire is the empire with the lowest total cost, defined later by Eq. (7), and the weakest colony is the colony with the lowest cost. Figure 5 shows a
Fig. 5. Imperialistic competition. The more powerful an empire is, the more likely it possesses the weakest colony of the weakest empire.
big picture of the modeled imperialistic competition. Based on its total power, in this competition, each of the empires will have a likelihood of taking possession of the mentioned colony. In other words, this colony will not be possessed by the most powerful empires, but these empires will be more likely to possess it. To start the competition, first, the possession probability of each empire, based on its total power, is found. The total cost of the nth empire is evaluated by: TCn J n ξ · meanfJ ni g;
(7)
where fJ ni g is the set of colony costs of the nth empire and ξ is a positive number, which is considered to be less than one. A little value for ξ causes the total power of the empire to be determined by just the imperialist and increasing it will increase the role of the colonies in determining the total power of an empire. Then, the normalized total cost is obtained by: NTCn TCn − minfTCi g;
(8)
where TCn and NTCn are the total cost and the normalized total cost of the nth empire, respectively. Having the normalized total cost, the possession probability of the nth empire is calculated by: pp n
NTCn : PN NTC i1
(9)
i
For assigning the mentioned colony to empires based on their possession probability, vector P is formed as: P pp1 ; pp2 ; pp3 ; …; ppN :
(10)
Then, the vector R, with the same size as P, whose elements are uniformly distributed random numbers in the range [0,1], is created, R r1 ; r2 ; r3 ; …; rN :
(11)
Vector D is formed by subtracting R from P:
be reaching a predefined number of iterations, obtaining the desired cost, or only one existing imperialist. 3. Wavefront Sensorless Adaptive Optics System Using ICA
To investigate the phase aberration correction capability of ICA in a wavefront sensorless AO system, we consider the problem of focusing an aberrated laser beam. The assumed setup in simulations is shown in Fig. 7. A wavefront with computergenerated phase aberrations is reflected by a piezoelectric DM and focused on the focal plane of a lens through fast Fourier-transform (FFT). The intensity distribution in this plane is recorded by a CCD camera. It is assumed that DM has a continuous faceplate with 61 actuators, and its stroke is enough to correct the phase aberrations in the simulation. The actuator configuration is shown in Fig. 8. Each black dot stands for an actuator. The circled line in this figure denotes the effective aperture. The phase distribution, φx; y, generated by the 61-element DM can be represented as: φx; y
61 X
uj V j x; y;
(13)
j1
D P − R D1 ; D2 ; D3 ; …; DN pp1 − r1 ; pp2 − r2 ; pp3 − r3 ; …; ppN − rN :
Fig. 6. Eliminating the powerless empires. Empire 4, with no colonies, does not have the power for competition and so it is eliminated from other empires.
(12)
Referring to vector D, the mentioned colony is handed to an empire which in its relevant index, D is maximum. F.
Eliminating the Powerless Empires
An empire will collapse when this empire loses all of its colonies. In this case, this imperialist is considered as a colony and it is assigned to the other empires, as shown in Fig. 6. G.
Termination Criterion
The algorithm is continued until the termination criterion is satisfied. The termination criterion can
Fig. 7. Schematic of wavefront sensorless AO system. 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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Fig. 8. Actuator configuration of 61-element DM.
where uj is the voltage applied onto the jth actuator and V j x; y is the influence function of the jth actuator on the wavefront. From real measurements, we know that the influence function of 61-element DM actuators is approximately a Gaussian distribution [20]: q α ; x − xj 2 y − yj 2 ∕d V j x; y exp lnω
(14) where xj ; yj is the space position of the jth actuator, ω is the coupling coefficient, α is the Gaussian index, and d is the distance between two neighboring actuators. We set ω to 0.08 and α to 2 in our simulations, according to the data from [9]. The ICA is employed to compensate for the input phase aberrations. It is done by adjusting the control voltages of DM for optimizing the performance metric of the system as follows. A.
ICA with Simple Aberrations
We first add defocus and astigmatism aberrations separately to the plane wavefront of the incident
beam. These aberrations are defined over 128 × 128 pixels, which is also the grid of DM (see Fig. 8). The obtained aberrated wavefronts are shown in Figs. 9 and 10. We perform ICA to compensate for these simple aberrations as described below. The cost function J in ICA is considered to be the Strehl ratio (SRctr ) as the system performance metric. SRctr is defined by the ratio of the actual central intensity of the far-field light to that of an ideal wavefront far-field light on the focal plane. The higher the value of SRctr, the better the beam quality. The major purpose of ICA is to acquire the optimum control voltages of DM for maximizing SRctr. For our optimization problem, each country in this algorithm is represented by 61 control voltages, ui , as the variables. Therefore, N 61 is the number of actuators of DM, such that: country C u1 ; u2 ; u3 ; …; u61 :
The initial population includes N 50 countries as candidate solutions, the variable values of which are randomly selected in the range of allowed voltages: 2
3 country1 6 country2 7 6 7 6 7 : 6 7: population 6 7 : 6 7 4 5 : country50
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(16)
According to Eq. (13), for every country (C), a set of the control voltages results in a certain shape of DM and, consequently, a certain value of SRctr (JC). We select 10% of initial countries with the highest SRctr as the imperialists, so N 5 and N col N − N 45. One can choose the small population size but convergence of the system will require more iterations. Also, the large population size takes more computational time. After calculating the
Fig. 9. Wavefront distribution before and after defocus correction with ICA. 136
(15)
Fig. 10. Wavefront distribution before and after astigmatism correction with ICA.
normalized power of the imperialists, Pn , according to Eq. (4), the initial number of colonies of every imperialist, NCn , is determined and the empires are formed. Then, the described steps in Subsections 2.B–2.F are repeated until the termination criterion is satisfied. In our simulations, the termination criterion is reached at the iteration number of 500. This number has been appropriately selected to converge completely. In this case, the best imperialist state, a set of control voltages that results in the highest SRctr , is considered as the best solution of problem. The parameter values of ICA depend on the optimization problem. In our problem, how to choose the best values of these parameters is analyzed through plenty of implementations and, finally, the parameters that result in the most effective convergence process are selected. We set the parameters β in Eq. (6) and ξ in Eq. (7) to 2 and 0.05, respectively, in our simulations. We also set pr , revolution rate, to 0.33 at the first iteration. Then, it decreases when the iteration number increases, such that 0.99pnr , where n is the iteration number of pn1 r the algorithm. A high value of the revolution rate in the beginning of the algorithm causes the whole search space to be explored, the different shapes of DM to be considered, and, consequently, early convergence of solutions to a local extremum to be prevented. Decreasing the revolution rate when the iteration number increases causes the divergence of solutions from optimal (in terms of the wavefront sensorless correction approach this means the optimal shape of DM), to be reduced. It also allows the surrounding space of optimal solutions to be more accurately searched. The wavefront distributions, after defocus and astigmatism correction using ICA, are shown in Figs. 9 and 10, respectively. In Fig. 9, the peak to valley (PV) and root mean square (RMS) of wavefront aberrations are reduced from 0.55λ (λ 1064 nm) and
0.12λ to 0.09λ and 0.007λ, respectively, after defocus correction. In this case, SRctr is increased from 0.32 to 0.99. In Fig. 10, PV and the RMS of wavefront aberrations are reduced from 0.70λ and 0.11λ to 0.10λ and 0.005λ, respectively, after astigmatism correction, and SRctr is raised from 0.42 to 0.99. These optimization results show that ICA has very good capability to correct simple phase aberrations. B. ICA with Complicated Aberrations
We also generate more complicated phase aberrations and add them to the incident beam. The obtained wavefront distribution is shown in Fig. 11. The phase aberrations include 100 Zernike modes with tip and tilt aberrations, and regardless of piston aberration. The Zernike coefficients, which are randomly produced in our method, are shown in Fig. 12(a). As mentioned previously, in our simulations, DM has 61 actuators. Such DM, with 7 actuators across the diameter of the effective aperture, can only appropriately compensate for Zernike modes up to a radial order n 7; that is, up to n 1n 2∕2 36 Zernike modes [21]. Moreover, the main aberration modes in a solid-state laser system are essentially from the first 36 Zernike modes, and the contribution of higher-order modes is less. So we assume that higher-order modes have smaller coefficients. It can also be a way to simulate some kind of noise produced by high-order modes that will affect the efficiency of the low-order mode correction. Under this condition, ICA is used for phase aberration compensation with the same cost function (SRctr ) and initial parameters that were considered in Subsection 3.A. The wavefront distribution, after compensation, is shown in Fig. 11. We find that the PV and RMS of wavefront aberrations are reduced from 1.49λ and 0.14λ to 0.54λ and 0.02λ, respectively. The Zernike coefficients of residual phase aberrations and the Zernike coefficients attenuation, defined as the ratio of residual to input Zernike coefficients, are presented in 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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Fig. 11. Wavefront distribution before and after aberration correction with ICA.
Figs. 12(b) and 12(c), respectively. Figure 12(c) shows clearly which Zernike modes are compensated for (attenuation < 1), which ones are kept invariant (attenuation 1), or even degraded (attenuation > 1). For clarity, the Zernike coefficients attenuation for the first 40 Zernike modes is shown in Fig. 12(d). As expected, the low-order modes, which are controllable by DM, are compensated appropriately. Figure 13(a) shows SRctr during the optimization process for 500 iterations, averaged over 10 corrections. As shown, SRctr increases with the iteration process and reaches 0.92 from 0.36. 0.5
10
Zernike Coefficient
Zernike Coefficient 20
40 60 Zernike Order
80
0
−0.5
100
Coefficients Attenuation
After Correction
0
−0.5
20
(a)
40 60 Zernike Order
80
0
−5
20
0
−0.5
40
40 60 Zernike Order
80
100
(c) 1
GA
Coefficients Attenuation
Coefficients Attenuation
0.5
20 30 Zernike Order
5
−10
100
1
ICA
10
ICA
(b)
1
Coefficients Attenuation
To compare ICA with the standard algorithms, which have already been used for phase aberration compensation, we apply GA and SPGD algorithm to compensate for the same-phase aberrations considered in Subsection 3.B. The procedures of these algorithms are found in [7–9] for GA and [11,12] for SPGD algorithm. We make use of the real-number encoding GA with a nonuniform arithmetical crossover operator [7]. To converge completely, we set the iteration number of the algorithms appropriately. The initial population size and the number of generations of GA are
0.5
Before Correction
−1
4. Comparison of ICA with GA and SPGD Algorithm
0.5
0
−0.5
−1
(d)
10
20 30 Zernike Order
(e)
40
SPGD 0.5
0
−0.5
−1
10
20 30 Zernike Order
40
(f)
Fig. 12. (a) Zernike coefficients of the input phase aberrations. (b) Zernike coefficients of residual phase aberrations, after correction with ICA. (c) Zernike coefficients attenuation, after correction with ICA. Zernike coefficients attenuation for the first 40 Zernike modes, after correction with (d) ICA, (e) GA, and (f) SPGD. 138
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1
SPGD
0.6
0.4
0.8
Strehl Ratio
0.8
Strehl Ratio
0.8
Strehl Ratio
1
GA
ICA
0.6
0.4
0.2 0
100
200
300
400
500
0.6
0.4
0.2 0
200
400
600
0.2 0
1000
2000
Iteration Number
Iteration Number
Iteration Number
(a)
(b)
(c)
3000
Fig. 13. Strehl ratio (SRctr ) during optimization process with (a) ICA, (b) GA, and (c) SPGD, averaged over 10 corrections for each algorithm.
Table 1.
Comparison of Optimization Results of ICA, GA, and SPGD
Algorithm
Averaged SRctr
Final PV (λ)
Final RMS (λ)
0.92 0.92 0.92
0.54 0.58 0.56
0.02 0.02 0.02
ICA GA SPGD
set to 50 and 600, respectively, and the number of iterations of SPGD algorithm is set to 3000. The Zernike coefficients’ attenuation, obtained after phase aberration compensation with GA and SPGD algorithm, are shown in Figs. 12(e) and 12(f), respectively. SRctr , averaged over 10 corrections, during optimization process with GA and SPGD algorithm is shown in Figs. 13(b) and 13(c), respectively. For each of these three algorithms, the final averaged SRctr , along with PV and RMS of residual wavefront aberrations, is given in Table 1. These results, and the comparison of Figs. 12(d)–12(f) and Figs. 13(a)–13(c), indicate that these three algorithms have almost the same correction capability. A.
Analysis of Convergence Speed
An important criterion on which the algorithm can be applied to real-time AO systems is the convergence speed. Because the number of small perturbations sent to the system per iteration is different for different algorithms, we cannot determine convergence speed just from the number of iterations. Therefore, to estimate the convergence speed of the algorithms, we consider the number of mirror changes needed for achieving averaged SRctr of 0.8 in each algorithm. A SRctr of 0.8 is commonly known as the diffraction limit. This number of mirror changes is obtained Table 2.
Comparison of Convergence Speeds of ICA, GA, and SPGD
Algorithm ICA GA SPGD
Averaged Number of Mirror Changes for Achieving SRctr 0.8 74 × 50 3700 75 × 50 3750 185 × 2 370
from multiplying the iteration number in SRctr 0.8 by the number of small perturbations sent to the system per iteration. The number of these perturbations is the number of countries in ICA, the number of generations in GA, and 2 in the SPGD algorithm. The results are presented in Table 2. These data show that ICA and GA have almost the same convergence speed and the SPGD algorithm is the fastest algorithm, such that the number of mirror changes needed by ICA and GA is almost 10 times as many as that of the SPGD algorithm. 5. Conclusion
We have described the basic principle of the ICA. Then, we simulated an AO system with a 61-element DM for phase aberration correction in a solid-state laser beam. The ICA was applied to search the optimum control voltages of DM for maximizing the Strehl ratio. The results obtained from phase aberration correction show that this algorithm has very good correction capability. We compared ICA with GA and the SPGD algorithm and found that they have almost the same correction capability for our simulated phase aberrations. Also, the comparison shows that ICA and GA are almost the same in convergence speed and that the SPGD algorithm is the fastest algorithm, such that the number of mirror changes needed by ICA and GA is almost 10 times as many as that of the SPGD algorithm. The advantage of ICA is that it is far more likely to find the global extremum, and its disadvantage is that its convergence speed is not fast. In conclusion, ICA is a proper correction algorithm for static or slowly changing phase aberrations in optical systems, such as solid-state laser systems, microscopy, and retinal imaging. The aim of this paper was to investigate the feasibility of phase aberration correction with ICA and our future work will be modifying the convergence speed of ICA and using it in a real physical system. References 1. H. W. Babcock, “The possibility of compensating astronomical seeing,” Astron. Soc. Pac. 65, 229–236 (1953). 1 January 2014 / Vol. 53, No. 1 / APPLIED OPTICS
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