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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 24070–24084
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Model-based aberration correction in a closed-loop wavefront-sensor-less adaptive optics system

H. Song, R. Fraanje, G. Schitter, H. Kroese, G. Vdovin, and M. Verhaegen  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 24070-24084 (2010)
http://dx.doi.org/10.1364/OE.18.024070


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Abstract

In many scientific and medical applications, such as laser systems and microscopes, wavefront-sensor-less (WFSless) adaptive optics (AO) systems are used to improve the laser beam quality or the image resolution by correcting the wavefront aberration in the optical path. The lack of direct wavefront measurement in WFSless AO systems imposes a challenge to achieve efficient aberration correction. This paper presents an aberration correction approach for WFSlss AO systems based on the model of the WFSless AO system and a small number of intensity measurements, where the model is identified from the input-output data of the WFSless AO system by black-box identification. This approach is validated in an experimental setup with 20 static aberrations having Kolmogorov spatial distributions. By correcting N = 9 Zernike modes (N is the number of aberration modes), an intensity improvement from 49% of the maximum value to 89% has been achieved in average based on N + 5 = 14 intensity measurements. With the worst initial intensity, an improvement from 17% of the maximum value to 86% has been achieved based on N + 4 = 13 intensity measurements.

© 2010 Optical Society of America

1. Introduction

In this paper, we further improve the correction speed of the WFSless AO system by wavefront aberration estimation and correction in three steps. First, with the external aberration absent (e.g., the aberration induced by air turbulence, heat or specimen), the WFSless AO system is calibrated such that the system aberration (e.g., initial aberration in the DM, misalignment of the optical components) is removed. Second, still with the external aberration absent, a nonlinear static model of the calibrated WFSless AO system is identified from the measurement data, which describes the transfer from the DM control signal to the intensity measurement. This step is analogue to determining the influence matrix of the DM in WFS-based AO systems; however, in WFSless AO systems, because the transfer from the DM control signal to the intensity measurement is nonlinear, a nonlinear model identification approach is required. Third, when the external aberration is present, the DM is initially excited by N + 2 predefined control signals and the corresponding N + 2 intensity measurements are collected. Aberration is estimated and corrected based on these N + 2 pairs of input-output data and the model of the WFSless AO system, by solving a nonlinear least squares (NLLS) optimization problem online. With new input-output data available, the aberration estimation and correction are refined iteratively. This approach is validated in a WFSless AO experimental setup and the performance of the resulting closed-loop system is evaluated.

The contribution of our work is that a new model-based approach has been proposed and validated for aberration estimation and correction in WFSless AO systems. The paper is organized as follows. Section 2 analyzes the WFSless AO system. Section 3 explains our approach on wavefront aberration estimation and correction. Section 4 describes the experimental setup. Section 5 reports and evaluates the experimental results. Section 6 concludes the work.

2. System analysis

Fig. 1 Schematic of a common closed-loop WFSless AO system. The incident light beam is disturbed in front of the entrance pupil. The control system adapts the control signal u(k) to maximize the intensity measurement y(k).

Because in many cases wavefront aberration is the main factor for intensity measurement reduction at given incident light power [12

12. L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002). [CrossRef] [PubMed]

, 13

13. M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Nat. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002). [CrossRef]

, 17

17. D. Débarre, E. J. Botcherby, M. J. Booth, and T. Wilson, “Adaptive optics for structured illumination microscopy,” Opt. Express 16(13), 9290–9305 (2008). [CrossRef] [PubMed]

], the amplitude variation in the optical field is omitted such that
ai(ξ,η,k)=ai,
(3)
where ai is a constant. Apart from that, if the aberration is corrected within a short time, it is reasonable to consider the wavefront aberration as constant (e.g., when any single point in the specimen is imaged in scanning-type microscopes under normal operational conditions). This simplifies ϕx(ξ, η, k) as
ϕx(ξ,η,k)=ϕx(ξ,η),
(4)
such that ϕx(ξ, η) is time-independent.

Combining Eq. (2)(5), the behavior of the WFSless AO system can be represented as
y(k)=Σ2|Σ1aiexp[j2πλ(ϕx(ξ,η)+D(ξ,η)u(k))]exp[j2πλd(αξ+βη)]dξdη|2dαdβ+w(k).
(6)
Static nonlinearity is visible in Eq. (6) in the wavefront-intensity mapping. Because this mapping is surjective (i.e., different wavefronts can give the same intensity measurement) and not invertible, the wavefront can not be obtained from single intensity measurement. However, with the model of the WFSless AO system describing the transfer from u(k) to y(k) with the aberration ϕx(ξ, η) absent, and at least N + 2 pairs of u(k) and y(k) collected with ϕx(ξ, η) present, the aberration ϕx(ξ, η) can be estimated in the basis defined by D(ξ, η), as will be explained in Section 3.

3. Model-based aberration estimation and correction

3.1. Modeling of the WFSless AO system

Because the DM deformation ϕm(ξ, η) can not be measured in the WFSless AO system and D(ξ, η) can not be obtained with high accuracy, it is difficult to get an accurate model of the real system from Eq. (6). The artifacts in the optical components may also degrade the accuracy of Eq. (6). As will be shown later on, since hundreds of times of intensity calculations are needed by our proposed algorithm to estimate the aberration, the computational complexity in Eq. (6) (e.g., two double integrals for each intensity calculation) will slow down the aberration correction speed. Therefore in our work the AO model is identified directly from u(k) and y(k) by black-box identification [24

24. M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach, (Cambridge University Press, Cambridge, USA, 2007). [CrossRef]

, 25

25. J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: a unified overview,” Automatica31(12), 1691–1724 (1995). [CrossRef]

].

In this sense, the system description in Eq. (6) is represented by
y(k)=g(ϕx(ξ,η)+D(ξ,η)u(k))+w(k),
(7)
where g represents the static nonlinear wavefront-intensity mapping, including the double integral over the coordinate (ξ, η). The wavefront aberration ϕx(ξ, η) can be split into two parts as
ϕx(ξ,η)=D(ξ,η)xϕ1(ξ,η)+Δϕx(ξ,η).
(8)
Here ϕ1(ξ, η) represents the part of ϕx(ξ, η) lying within the range of D(ξ, η) and Δϕx(ξ, η) represents the part of ϕx(ξ, η) which is orthogonal to the range of D(ξ, η). It is assumed that the wavefront aberration can be represented by a finite low-order Zernike aberrations [13

13. M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Nat. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002). [CrossRef]

, 26

26. M. Schwertner, M. J. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high NA adaptive optical microscopy,” Opt. Express 12(26), 6540–6552 (2004). [CrossRef] [PubMed]

, 27

27. G. Vdovin, O. Soloviev, A. Samokhin, and M. Loktev, “Correction of low order aberrations using continuous deformable mirrors,” Opt. Express 16(5), 2859–2866 (2008). [CrossRef] [PubMed]

], then it is possible that the DM can generate these low-order Zernike modes efficiently and Δϕx(ξ, η) can be neglected. As a result, Eq. (8) can be approximated by
ϕx(ξ,η)D(ξ,η)x.
(9)
Substitute Eq. (9) into (7), we have
y(k)g(D(ξ,η)(x+u(k)))+w(k).
(10)
Merging D(ξ, η) and g into one static nonlinear mapping f, we can further simplify the system description as
y(k)f(x+u(k))+w(k).
(11)
Equation (11) considers the aberration as a disturbance directly applied on the input u(k), which allows to identify the model of the WFSless AO system only based on u(k) and y(k) but meanwhile accounting for the influence of the aberration.

With the input-output data u(k) and y(k), the model structure needs to be selected for the nonlinear black-box model. There is a very rich spectrum of possible descriptions for nonlinear black-box models, e.g., neural network [28

28. S. Y. Kung, Digital Neural Networks, (Prentice-Hall, Upper Saddle River, NJ, USA, 1993).

, 29

29. S. Haykin, Neural Networks: a Comprehensive Foundation, (Macmillan, New York, USA, 1994).

], fuzzy models [30

30. M. Brown and C. Harris, Neurofuzzy Adaptive Modeling and Control, (Prentice-Hall, New York, USA, 1994).

], etc. Because a 2-layer neural network is able to model a broad range nonlinearities and, from practical point of view, it can be implemented and trained with the MATLAB Neural Network Toolbox [31

31. H. Demuth, M. Beale, and M. Hagan, Neural Network Toolbox 5 User’s Guide, (The MathWorks, Inc., 2007). [PubMed]

] very conveniently, a 2-layer neural network is built in our work, which has NQ neurons in the first layer and one in the second. The output ŷ(k) of the neural network is determined as
y^(k)=f^(u(k))=W1tanh(W2u(k)+s1)+s2.
(12)
W2 ∈ ℝNQ×N and W1 ∈ ℝNQ contain the input and output weights of the neural network, respectively; s1 ∈ ℝNQ×1 and s2 ∈ ℝ are biases on the input and output neurons, respectively. tanh is the hyperbolic tangent function.

The number of neurons NQ should be defined by the user when constructing the neural network. Parameters W1, W2, s1 and s2 are then optimized by training the neural network with sufficient data points u(k) and y(k). Details on training and validating the neural network can be found, for instance, in [28

28. S. Y. Kung, Digital Neural Networks, (Prentice-Hall, Upper Saddle River, NJ, USA, 1993).

, 29

29. S. Haykin, Neural Networks: a Comprehensive Foundation, (Macmillan, New York, USA, 1994).

].

3.2. Aberration estimation and correction

To obtain an analytical solution of Eq. (13) may be infeasible in practice, for instance, if the nonlinearity has some high-degree components. Alternatively, a numerical solution can be obtained by solving a nonlinear least squares (NLLS) problem as
(a^,x^)=argmina^,x^||Y[1,K]Y^[1,K]||22J(a^,x^),
(14)
with Y[1,K] and Ŷ[1,K] constructed as
Y[1,K]=[y(1)y(2)y(K)],Y^[1,K]=[y^(1)y^(2)y^(K)]=[a^f^(x^+u(1))a^f^(x^+u(2))a^f^(x^+u(K))].
(15)
Here â and are the estimates of a and x, respectively. For given â and , the intensity is estimated by ŷ(k) = âf̂( + u(k)).

Fig. 2 Cost function J() depends on the number of data points for solving the NLLS problem in Eq. (14). For clarity of explanation, the intensity variation a is not considered. In (a), the nonlinearity is represented as y=f(x+u)=(2J1(x+u)x+u)2 to simulate the intensity distribution in the Airy disk [32], with J1 the Bessel function of the first kind. The aberration shift the original system y = f (u) horizontally by x = −1.25. The model uncertainty is neglected, i.e., = f. With single data point P1, the cost function J() has two minima at = −1.25 and = 3.25 as plotted in (b). With points P1 and P2, J() has one unique global minimum at = −1.25 but there is a local minimum at = 2.5. This local minimum vanishes when P3 is added and the domain of convex is increased.

Due to the modeling uncertainty in and the measurement noise in y(k), the accuracy of the aberration estimation may be limited and the intensity may not reach its maximum by the MBAC algorithm. In this situation, other optimization algorithms like simplex algorithm, genetic algorithm, etc., can be used to continue searching for the optimum. Under the assumption that is a close approximation of f, the MBAC algorithm will steer the DM to a point close to its optimum. This point can then be used as a new initial condition for desired nonlinear optimization method, like the simplex algorithm described in [33

33. W. H. Press, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C; the Art of Scientific Computing, 2nd ed. (Cambridge University Press, New York, USA, 1992).

]. The initial simplex of the simplex algorithm is constructed around the control signal which gives the maximum intensity measurement in the MBAC algorithm. The hybrid algorithm (MBAC+Simplex) is described in pseudo code below. The MBAC algorithm stops after a fixed number of intensity measurements P (P is a user-defined number), to distinguish the intensity improvements due to the MBAC algorithm and due to the simplex algorithm. The simplex algorithm stops at time ( is a user-defined number).

MBAC+Simplex algorithm (general description and pseudo code implementation):
  1. Initialization of MBAC, i.e., collecting N + 2 data points
    • Set u(1) = 0.
    • Set u(k) as in Appendix B of [21

      21. M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14(4), 1339–1352 (2006). [CrossRef] [PubMed]

      ], with k
      = 2, ⋯ , N + 2.
    • Set â(k) = 1, with k = 1, ⋯ , N + 2.
    • for k = 1 : N + 2
      • Excite the WFSless AO system with u(k) and collect y(k).
    • end
  2. Aberration estimation and correction by MBAC
    • for k = N + 3 : P
      • p = argmaxp y(p) ;
      • âinit = â(p), init = −u(p);
      • [â(k – 1), (k – 1)] = argminâ, J(â, ) as in Eq.(14), with initial conditions âinit and x̂init.
      • Set u(k) = –(k – 1), excite the system with u(k) and collect y(k).
    • end
  3. Aberration correction by the simplex algorithm
    • p = argmaxp y(p);
    • uc = u(p);
    • Construct simplex around uc as u(k) = u(kP + 1) + uc with k = P + 1, ⋯ , P + N + 1.
    • for k = P + 1 :
      • Run simplex algorithm as in [33

        33. W. H. Press, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C; the Art of Scientific Computing, 2nd ed. (Cambridge University Press, New York, USA, 1992).

        ].
    • end

4. Experimental setup

The closed-loop WFSless AO experimental setup is the same as in Fig. 1. The collimated laser beam is generated by a He-Ne laser with a wavelength of 632 nm. Aberration is generated by a circular glass plate. One side of the glass plate is polished in such a way that the resulting wavefront aberration has a spatial Kolmogorov distribution [20

20. J. W. Hardy, Adaptive Optics for Astronomical Telescopes,(Oxford University Press, New York, USA, 1998).

]. The intensity transmission of the disturbance generator is about 78% as measured by a power meter (PM100, Thorlabs, Germany). During the modeling of the WFSless AO system, this aberration generator is removed. The entrance pupil has a diameter of 6 mm. It is conjugated to the PDM by lenses L1 and L2. The focal distances of L1 and L2 are 6 cm and 20 cm, respectively. The PDM (37-actuator, OKOTech, The Netherlands) has a clear aperture of 30 mm and only the central area with a radius of 20 mm is illuminated to generate Zernike modes efficiently [27

27. G. Vdovin, O. Soloviev, A. Samokhin, and M. Loktev, “Correction of low order aberrations using continuous deformable mirrors,” Opt. Express 16(5), 2859–2866 (2008). [CrossRef] [PubMed]

]. Lens L3 has a focal distance of 400 mm. The pin hole (NT56-282, Edmunds Optics, with a diameter of 50 μm) is placed at the focal point of L3, followed by a photodiode (TSL250R-LF, TAOS, Korea) measuring the light intensity inside the pin hole. The high voltage amplifier (HVA, OKOTech, The Netherlands) has 40 channels, each with an output range of 0∼300 V, a voltage amplification of 80 at low frequencies and a −3dB bandwidth of 1 kHz. The control algorithm is implemented in MATLAB (Version 7.5.0.342, The MathWorks). Signal generation and data acquisition is accomplished by a dSPACE system (DS1006, dSPACE, Germany) with the digital-to-analog card (DS2103) output range of ±10 V, 14-bit and analog-to-digital card (DS2004) input range of ±10 V, 16-bit. Interfacing between MATLAB and the dSPACE system is done via MLIB (dSPACE, Germany).

Fig. 3 Block diagram of the closed-loop WFSless AO system. The physical WFSless AO system has voltage V(k) as input, but conceptually u(k) can be considered as its input because of the hysteresis compensator and the modal transform.

5. Experiments and results

Experiments have been carried out in the setup described in Section 4 to validate the proposed approach for aberration correction, which mainly consist of three steps as follows:
  1. With the aberration generator absent, the WFSless AO system is calibrated using a simplex optimization algorithm. The system aberration is corrected by adapting the shape of the PDM such that the intensity measurement is maximized.
  2. The WFSless AO system is excited by random control signals u(k) and the intensity measurements y(k) are collected. Based on u(k) and y(k), the WFSless AO system is modeled by a neural network as described in Section 3.1.
  3. Aberration is introduced in the WFSless AO system by the aberration generator and corrected by the proposed MBAC+Simplex algorithm as described in Section 3.3. For a comparison, the simplex algorithm alone is also used to correct the aberration. Intensity improvements by these two algorithms are evaluated and compared.

5.1. System calibration

To allow for bi-directional operation of the PDM in later experiments, all the actuators in the PDM are biased by 150 V initially. A simplex optimization algorithm is then used to correct the system aberration, which maximizes the intensity measurement y(k) by adapting the control signal u(k) as in Eq. (1). The sampling rate of the system during the calibration is fs = 50 Hz, which is much less than the resonance frequency of the PDM (about 1 kHz), so that the AO system is considered static. The maximum intensity measurement is denoted as ymax, which is used to normalize intensity measurement in Section 5.3. The control signal which results in the maximal intensity measurement, denoted as u0, is used as a bias in all the following experiments.

5.2. Modeling of the AO system

To collect enough input-output data for modeling the WFSless AO system, the system is excited by 10000 control signals u(k) in open-loop with the aberration generator absent and the intensity measurements y(k) are collected. The control signals u(k) distribute randomly within the operational range of the PDM, to give a persistent excitation. The sampling rate of the system is also 50 Hz.

Among the 10000 collected data points, 6000 are randomly selected for identification of the AO model and the rest 4000 are for validation. The AO system is modeled as a 2-layer feedforward neural network with NQ neurons in its first layer and one neuron in its second layer as in Eq. (12). The neural network is implemented and trained by MATLAB Neural Network Toolbox [31

31. H. Demuth, M. Beale, and M. Hagan, Neural Network Toolbox 5 User’s Guide, (The MathWorks, Inc., 2007). [PubMed]

]. Parameters W1, W2, s1 and s2 in Eq. (12) are optimized by minimizing the mean square of the fitting error, using Levenberg-Marquardt (LM) backpropagation algorithm, i.e.,
(W1,W2,s1,s2)=argmin(W1*,W2*,s1*,s2*)1NtΣk=1Nt(y(k)y^(k))2.
(16)
Nt is the number of data points for identification, in our case, Nt = 6000.

The accuracy of the model is evaluated by calculating the variance accounted for (VAF) of the model, which is defined as
VAF(y^,y)=(1var(y^y)var(y))×100%.
(17)
Here var(y) is the variance of y. Figure 4 shows the VAFs of the AO model with different number of neurons in the first layer. From this plot, it can be seen that VAF already reaches as high as 98.2% at NQ = 20 for the identification set and 97.8% for the validation set, indicating that the neural network can model the AO system very accurately. The difference in VAF is negligible for NQ > 20. Therefore 20 neurons are used in the first layer, to have a good balance between the model accuracy and the model complexity. Experiments show that the the number of neurons NQ needed to accurately model the system is about twice the number of modes in the system, i.e., NQ ≈ 2N.

Fig. 4 Accuracy of the neural network model for different number of neurons NQ. VAF increases with NQ in both identification and validation sets for NQ ≤ 20. The difference in VAF is negligible for NQ > 20. Hence 20 neurons are used.

5.3. Aberration correction

The aberration generator is inserted in the optical path as in Fig. 1. The MBAC+Simplex algorithm is used to correct the aberration. To have a statistics of the performance, experiments have been carried out for 20 static aberrations, which are generated by rotating the circular glass plate such that the beam is disturbed by different regions of the glass plate.

Figure 5 shows the time line of the WFSless AO system. In each experiment, during the initialization, the aberrated system is excited by N + 2 = 11 control signals u(k), k = 1, ⋯ , N + 2, at a rate of 50 Hz. Inputs u(k) are initialized as in Section 3.3. The amplitude of the simplex is selected as half of the operational range of the PDM. After the intensity y(k), k = 1, ⋯ , N + 2, are collected, the aberration is estimated by solving a NLLS optimization problem as in Eq. (14), using the function fmincon in MATLAB Optimization Toolbox. fmincon is used in our work because: (1) it is computationally very efficient and can be called in MATLAB very conveniently; (2) the convexity of J(â, ) improves with more data points so that a local optimization algorithm like fmincon may already be enough to get an accurate estimation â and x̂. â is constrained to be within [0, 1] during the estimation. As time keeps going, more data points are available and the aberration is estimated and corrected iteratively as in the MBAC+Simplex algorithm. After P = 19 data points, the simplex algorithm (named as Simplex 1) is switched on. For a comparison, the intensity is also maximized by the simplex algorithm alone (Simplex 2). Simplex 1 and Simplex 2 are the same except that the initial guess for Simplex 1 comes from the MBAC algorithm, but the initial guess for Simplex 2 is zero. Both simplex algorithms stop after = 200 intensity measurements, when they have converged. The sampling intervals between the 11th and the 19th samples vary because of the computational time of the NLLS algorithm, as will be discussed later. After Simplex 1 is switched on, the sampling rate returns to 50 Hz.

Fig. 5 Time line of the WFSless AO system with the MBAC algorithm, including initialization and aberration correction. The initial sampling interval is ts = 20 ms. The computational time tc,1 for the first aberration estimation takes about 40 ms, while the estimation time tc,2 afterwards takes about 20 ms because a better initial guess is provided for the solving the NLLS problem.

Figure 6 shows the convergence curve for one static aberration which gives the lowest initial intensity. The intensity has been normalized as (k) = y(k)/(ymax * 0.78), where (k) is the normalized intensity and the intensity transmission ratio (78%) of the disturbance generator is accounted for. The initial intensity without correction is 0.17. After N + 2 = 11 samples are collected, the aberration is estimated and corrected by the MBAC algorithm. The intensity increases to 0.38 (about 2.2 times of the initial value) at the 12th time sample. With one more data sample acquired, the intensity jumps to 0.83 at the 13th time sample, which is almost 5 times of the initial value. At the 14th time sample, the intensity already converges to 0.86 and the intensity keeps at about 0.86 from the 15th and 19th samples.

Fig. 6 Aberration correction with the MBAC+Simplex algorithm and with the simplex algorithm alone, for one static aberration. The MBAC algorithm consists of the initialization and the aberration correction. The initial intensity is 0.17. With the MBAC algorithm, the intensity converges to 0.86 at the 14th time sample, which it takes 30 time samples for the simplex algorithm alone to reach 0.8. The simplex algorithm after MBAC also shows faster convergence than the simplex algorithm alone.

The MBAC algorithm stops after 19 time samples and Simplex 1 is switched on thereafter. Simplex 1 is initialized from the 20th to the 29th time samples. The initial simplex of Simplex 1 is constructed around the input point which gave the highest intensity in the past 19 samples, as described at the end of Section 3.2. Since the initialization of the simplex algorithm is only for data collection, intensity fluctuation is observed from the 20th to the 29th time samples as expected. However, after the initialization of Simplex 1 is completed, the intensity is further improved by Simplex 1 as can be seen from the small plot in Fig. 6. This plot shows that Simplex 1 converges faster than Simplex 2 because the MBAC algorithm provides a better initial value for Simplex 1.

Figure 7 shows the convergence curve averaged over 20 experiments and the standard deviation of (k) for k ≥ 12. The initial intensity is 0.49 in average. With the MBAC algorithm, the intensity increases to 0.82 (an improvement of 67%) and 0.87 (an improvement of 78%) at the 12th and 13th time sample, respectively. The intensity converges to 0.89 at the 15th time sample, while it takes Simplex 2 about 45 time samples to reach the same level. Because Simplex 1 starts at a better initial condition provide by MBAC, the intensity reaches 0.95 at the 60th time sample, while Simplex 2 takes 90 time samples to reach the same level. A significant improvement has been achieved in correction speed. The standard deviation of (k) with MBAC is also smaller than with the simplex algorithm. For instance, at the 15th time sample, the standard deviation of (k) with the MBAC algorithm is about 0.02 while that with Simplex 2 is 0.08, about 3 times larger. This indicates that the MBAC algorithm can improve the intensity in a more deterministic manner than simplex.

Fig. 7 Correction of 20 static aberrations. The initial intensity is 0.49 in average. With the MBAC algorithm, the intensity increases to 0.82 at the 12th time sample and to 0.87 at the 13th time sample. The intensity converges to 0.89 by the MBAC at the 15th time sample, while Simplex 2 needs 45 time samples to reach the same level. The standard deviation of (k) is also reduced with the MBAC algorithm, indicating that MBAC can give a more deterministic intensity improvement than simplex.

5.4. Computational complexity

Referring to Fig. 5, the computational time varies from each time when the aberration is estimated. In the first aberration estimation after 11 data samples, the cost function J(â, ) is evaluated for about 578 times by the function fmincon and tc,1 is about 40 ms in average. The sampling interval between the 11th and the 12th time sample is then equal to Ts,1 = tc,1 + ts = 40 + 20 = 60 ms. In the aberration estimations afterwards, because a better initial guess is provided for â and , the number of cost function evaluations is reduced to 251 in average and the computational time tc,2 reduces to about 20 ms. The sampling interval becomes Ts,2 = 20 + 20 = 40 ms.

In applications where the correction speed is the most important, the MBAC algorithm alone can be used and the correction may stop, e.g., after 15 time samples in our experiments where the intensity reaches 0.89. This leads to a total correction time of ts × 11 + Ts,1 + Ts,2 × 3 = 400 ms, while the simplex algorithm alone needs 45 time samples (i.e., ts × 45 = 900 ms) to reach the same intensity level. A reduction of 56% has been achieved in the correction time. If a higher intensity end value is desired, e.g., 0.95, simplex alone needs 90 time samples in average (i.e., ts × 90 = 1.80 s). The hybrid MBAC+Simplex algorithm needs 60 time samples (19 time samples by MBAC and 41 by Simplex 1), which takes ts × 11+ Ts,1 + Ts,2 × 7+ ts × 41 = 1.24 s in average. The time needed by the MBAC+Simplex algorithm is only 70% of that by the simplex algorithm alone.

6. Conclusion

A new approach has been proposed for aberration estimation and correction in WFSless AO systems. The wavefront aberration is estimated by solving a NLLS problem online, based on the model of the WFSless AO system and a minimum number of N + 2 intensity measurements. Experimental results show that in average 82% of the maximum intensity can be achieved at the N + 3 = 12th time sample by the MBAC algorithm and intensity converges to 89% at the 15th time sample. With the better initial condition provided by the MBAC algorithm, the simplex algorithm also shows faster convergence than used alone.

Future work will further improve the correction speed by increasing the sampling rate of the control system and considering the dynamics of the DM.

Acknowledgments

This work is supported by Delft Center for Mechatronics and Microsystems (DCMM). We would like to thank Mr. Arjan van Dijke from TUDelft for his contribution in the implementation of the experimental setup and Dr. Niek Doelman from TNO for the Komolgorov aberration generator.

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G. Vdovin, “Optimization-based operation of micromachined deformable mirrors,” Proc. SPIE 3353, 902–909 (1998). [CrossRef]

3.

M. A. Vorontsov, G. W. Carhart, M. Cohen, and G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17(8), 1440–1453 (2000). [CrossRef]

4.

W. Lubeigt, G. Valentine, J. M. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express 10(13), 550–555 (2002). [PubMed]

5.

U. Wittrock, I. Buske, and H. M. Heuck, “Adaptive aberration control in laser amplifiers and laser resonators,” Proc. SPIE 4969, 122–136 (2003). [CrossRef]

6.

M. de Boer, K. Hinnen, M. Verhaegen, R. Fraanje, G. Vdovin, and N. Doelman, “Control of a thermal deformable mirror: correction of a static disturbance with limited sensor information,” in Proceedings of the 4th International Workshop on Adaptive Optics for Industry and Medicine, pages 61–71, Münster, Germany, 2003.

7.

R. El-Agmy, H. Bulte, A. H. Greenaway, and D. Reid, “Adaptive beam profile control using a simulated annealing algorithm,” Opt. Express 13(16), 6085–6091 (2005). [CrossRef] [PubMed]

8.

A. A. Aleksandrov, A. V. Kudryashov, A. L. Rukosuev, T. Yu. Cherezova, and Yu. V. Sheldakova, “An adaptive optical system for controlling laser radiation,” J. Opt. Technol. 74(8), 550–554 (2007). [CrossRef]

9.

P. Yang, Y. Liu, W. Yang, M. W. Ao, S. J. Hu, B. Xu, and W. H. Jiang, “Adaptive mode optimization of a continuous-wave solid-state laser using an intracavity piezoelectric deformable mirror,” Opt. Commun. 278(2), 377–381 (2007). [CrossRef]

10.

W. Lubeigt, S. P. Poland, G. J. Valentine, A. J. Wright, J. M. Girkin, and D. Burns, “Search-based active optic systems for aberration correction in time-independent applications,” Appl. Opt. 49(3), 307–314 (2010). [CrossRef] [PubMed]

11.

O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy. Opt. Lett. , 25(1):52–54, 2000. [CrossRef]

12.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002). [CrossRef] [PubMed]

13.

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Nat. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002). [CrossRef]

14.

P. Marsh, D. Burns, and J. M. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express 11(10), 1123–1130 (2003). [CrossRef] [PubMed]

15.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67(1), 36–44 (2005). [CrossRef] [PubMed]

16.

S. P. Poland, A. J. Wright, and J. M. Girkin, “Evaluation of fitness parameters used in an iterative approach to aberration correction in optical sectioning microscopy,” Appl. Opt. 47(6), 731–736 (2008). [CrossRef] [PubMed]

17.

D. Débarre, E. J. Botcherby, M. J. Booth, and T. Wilson, “Adaptive optics for structured illumination microscopy,” Opt. Express 16(13), 9290–9305 (2008). [CrossRef] [PubMed]

18.

D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34(16), 2495–2497 (2009). [CrossRef] [PubMed]

19.

F. Roddier, Adaptive Optics in Astronomy, (Cambridge University Press, Cambridge, UK, 1999). [CrossRef]

20.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes,(Oxford University Press, New York, USA, 1998).

21.

M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14(4), 1339–1352 (2006). [CrossRef] [PubMed]

22.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, USA, 1996).

23.

H. Song, G. Vdovin, R. Fraanje, G. Schitter, and M. Verhaegen, “Extracting hysteresis from nonlinear measurement of wavefront-sensorless adaptive optics system,” Opt. Lett. 34(1), 61–63 (2009). [CrossRef]

24.

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach, (Cambridge University Press, Cambridge, USA, 2007). [CrossRef]

25.

J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: a unified overview,” Automatica31(12), 1691–1724 (1995). [CrossRef]

26.

M. Schwertner, M. J. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high NA adaptive optical microscopy,” Opt. Express 12(26), 6540–6552 (2004). [CrossRef] [PubMed]

27.

G. Vdovin, O. Soloviev, A. Samokhin, and M. Loktev, “Correction of low order aberrations using continuous deformable mirrors,” Opt. Express 16(5), 2859–2866 (2008). [CrossRef] [PubMed]

28.

S. Y. Kung, Digital Neural Networks, (Prentice-Hall, Upper Saddle River, NJ, USA, 1993).

29.

S. Haykin, Neural Networks: a Comprehensive Foundation, (Macmillan, New York, USA, 1994).

30.

M. Brown and C. Harris, Neurofuzzy Adaptive Modeling and Control, (Prentice-Hall, New York, USA, 1994).

31.

H. Demuth, M. Beale, and M. Hagan, Neural Network Toolbox 5 User’s Guide, (The MathWorks, Inc., 2007). [PubMed]

32.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, 7th ed. (Cambridge University Press, Cambridge, UK, 1999).

33.

W. H. Press, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C; the Art of Scientific Computing, 2nd ed. (Cambridge University Press, New York, USA, 1992).

34.

M. Loktev, D. Monteiroa, and G. Vdovin, “Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions,” Opt. Commun. 192, 91–99 (2001). [CrossRef]

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(220.1000) Optical design and fabrication : Aberration compensation
(110.0113) Imaging systems : Imaging through turbid media

ToC Category:
Active and Adaptive Optics

History
Original Manuscript: July 15, 2010
Manuscript Accepted: August 23, 2010
Published: November 3, 2010

Virtual Issues
Vol. 6, Iss. 1 Virtual Journal for Biomedical Optics

Citation
H. Song, R. Fraanje, G. Schitter, H. Kroese, G. Vdovin, and M. Verhaegen, "Model-based aberration correction in a closed-loop wavefront-sensor-less adaptive optics system," Opt. Express 18, 24070-24084 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24070


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References

  1. M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, and D. G. Voelz, "Adaptive imaging system for phase-distorted extended source and multiple-distance objects," Appl. Opt. 36(15), 3319-3328 (1997). [CrossRef] [PubMed]
  2. G. Vdovin, "Optimization-based operation of micromachined deformable mirrors," Proc. SPIE 3353, 902-909 (1998). [CrossRef]
  3. M. A. Vorontsov, G. W. Carhart, M. Cohen, and G. Cauwenberghs, "Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration," J. Opt. Soc. Am. A 17(8), 1440-1453 (2000). [CrossRef]
  4. W. Lubeigt, G. Valentine, J. M. Girkin, E. Bente, and D. Burns, "Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror," Opt. Express 10(13), 550-555 (2002). [PubMed]
  5. U. Wittrock, I. Buske, and H. M. Heuck, "Adaptive aberration control in laser amplifiers and laser resonators," Proc. SPIE 4969, 122-136 (2003). [CrossRef]
  6. M. de Boer, K. Hinnen, M. Verhaegen, R. Fraanje, G. Vdovin, and N. Doelman, "Control of a thermal deformable mirror: correction of a static disturbance with limited sensor information," in Proceedings of the 4th International Workshop on Adaptive Optics for Industry and Medicine, pages 61-71, Münster, Germany, 2003.
  7. R. El-Agmy, H. Bulte, A. H. Greenaway, and D. Reid, "Adaptive beam profile control using a simulated annealing algorithm," Opt. Express 13(16), 6085-6091 (2005). [CrossRef] [PubMed]
  8. A. A. Aleksandrov, A. V. Kudryashov, A. L. Rukosuev, T. Yu. Cherezova, and Yu. V. Sheldakova, "An adaptive optical system for controlling laser radiation," J. Opt. Technol. 74(8), 550-554 (2007). [CrossRef]
  9. P. Yang, Y. Liu, W. Yang, M. W. Ao, S. J. Hu, B. Xu, and W. H. Jiang, "Adaptive mode optimization of a continuous-wave solid-state laser using an intracavity piezoelectric deformable mirror," Opt. Commun. 278(2), 377-381 (2007). [CrossRef]
  10. W. Lubeigt, S. P. Poland, G. J. Valentine, A. J. Wright, J. M. Girkin, and D. Burns, "Search-based active optic systems for aberration correction in time-independent applications," Appl. Opt. 49(3), 307-314 (2010). [CrossRef] [PubMed]
  11. O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, "Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy," Opt. Lett. 25(1), 52-54 (2000). [CrossRef]
  12. L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, "Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror," J. Microsc. 206(1), 65-71 (2002). [CrossRef] [PubMed]
  13. M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, "Adaptive aberration correction in a confocal microscope," Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788-5792 (2002). [CrossRef]
  14. P. Marsh, D. Burns, and J. M. Girkin, "Practical implementation of adaptive optics in multiphoton microscopy," Opt. Express 11(10), 1123-1130 (2003). [CrossRef] [PubMed]
  15. A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, "Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy," Microsc. Res. Tech. 67(1), 36-44 (2005). [CrossRef] [PubMed]
  16. S. P. Poland, A. J. Wright, and J. M. Girkin, "Evaluation of fitness parameters used in an iterative approach to aberration correction in optical sectioning microscopy," Appl. Opt. 47(6), 731-736 (2008). [CrossRef] [PubMed]
  17. D. Débarre, E. J. Botcherby, M. J. Booth, and T. Wilson, "Adaptive optics for structured illumination microscopy," Opt. Express 16(13), 9290-9305 (2008). [CrossRef] [PubMed]
  18. D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, "Image-based adaptive optics for two-photon microscopy," Opt. Lett. 34(16), 2495-2497 (2009). [CrossRef] [PubMed]
  19. F. Roddier, Adaptive Optics in Astronomy, (Cambridge University Press, Cambridge, UK, 1999). [CrossRef]
  20. J. W. Hardy, Adaptive Optics for Astronomical Telescopes, (Oxford University Press, New York, USA, 1998).
  21. M. J. Booth, "Wave front sensor-less adaptive optics: a model-based approach using sphere packings," Opt. Express 14(4), 1339-1352 (2006). [CrossRef] [PubMed]
  22. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, USA, 1996).
  23. H. Song, G. Vdovin, R. Fraanje, G. Schitter, and M. Verhaegen, "Extracting hysteresis from nonlinear measurement of wavefront-sensorless adaptive optics system," Opt. Lett. 34(1), 61-63 (2009). [CrossRef]
  24. M. Verhaegen, and V. Verdult, Filtering and System Identification: A Least Squares Approach, (Cambridge University Press, Cambridge, USA, 2007). [CrossRef]
  25. J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, "Nonlinear black-box modeling in system identification: a unified overview," Automatica 31(12), 1691-1724 (1995). [CrossRef]
  26. M. Schwertner, M. J. Booth, and T. Wilson, "Characterizing specimen induced aberrations for high NA adaptive optical microscopy," Opt. Express 12(26), 6540-6552 (2004). [CrossRef] [PubMed]
  27. G. Vdovin, O. Soloviev, A. Samokhin, and M. Loktev, "Correction of low order aberrations using continuous deformable mirrors," Opt. Express 16(5), 2859-2866 (2008). [CrossRef] [PubMed]
  28. S. Y. Kung, Digital Neural Networks, (Prentice-Hall, Upper Saddle River, NJ, USA, 1993).
  29. S. Haykin, Neural Networks: a Comprehensive Foundation, (Macmillan, New York, USA, 1994).
  30. M. Brown, and C. Harris, Neurofuzzy Adaptive Modeling and Control, (Prentice-Hall, New York, USA, 1994).
  31. H. Demuth, M. Beale, and M. Hagan, Neural Network Toolbox 5 User’s Guide, (The MathWorks, Inc., 2007). [PubMed]
  32. M. Born, and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, 7th ed. (Cambridge University Press, Cambridge, UK, 1999).
  33. W. H. Press, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C; the Art of Scientific Computing, 2nd ed. (Cambridge University Press, New York, USA, 1992).
  34. M. Loktev, D. Monteiroa, and G. Vdovin, "Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions," Opt. Commun. 192, 91-99 (2001). [CrossRef]

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