## Fast optimal wavefront reconstruction for multi-conjugate adaptive optics using the Fourier domain preconditioned conjugate gradient algorithm

Optics Express, Vol. 14, Issue 17, pp. 7487-7498 (2006)

http://dx.doi.org/10.1364/OE.14.007487

Acrobat PDF (178 KB)

### Abstract

We present two different implementations of the Fourier domain preconditioned conjugate gradient algorithm (FD-PCG) to efficiently solve the large structured linear systems that arise in optimal volume turbulence estimation, or tomography, for multi-conjugate adaptive optics (MCAO). We describe how to deal with several critical technical issues, including the cone coordinate transformation problem and sensor subaperture grid spacing. We also extend the FD-PCG approach to handle the deformable mirror fitting problem for MCAO.

© 2006 Optical Society of America

## 1. Introduction

1. J. M. Beckers, “Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics,” in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M. H. Ulrich, ed., Vol. 30 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 693–703.

2. D. C. Johnston and B. M. Welsh, “Analysis of multi-conjugate adaptive optics,” J. Opt. Soc. Am. A **11**, 394–408 (1994). [CrossRef]

*wavefront reconstruction*we mean the determination of DM actuator commands, given wavefront sensor signals. In this paper we assume a linear model for sensor signals as a function of the turbulence profile, we assume DM displacements depend linearly on actuator commands, and we employ an optimal, or minimum variance, approach to wavefront reconstruction [3

3. T. Fusco, J. M. Conan, G. Rousset, L. M. Mugnier, and V. Michau, “Optimal wave-front reconstruction strategies for multi-conjugate adaptive optics,” J. Opt. Soc. Am. A **18**, 2527–2538 (2001). [CrossRef]

4. R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J-P. Veran, “Adaptive optics requirements definition for TMT,” Advancements in Adaptive Optics, edited by
D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, Proc. SPIE **5490**, 879–890 (2004). [CrossRef]

5. B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A , **19**, 1803–1816 (2002). [CrossRef]

7. L. Gilles, C. R. Vogel, and B. L. Ellerbroek, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A , **19**, 1817–1822 (2002). [CrossRef]

8. L. Gilles, B. L. Ellerbroek, and C. R. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multi-conjugate adaptive optics,” Appl. Opt. **42**, 5233–5250 (2003). [CrossRef] [PubMed]

9. B. L. Ellerbroek, L. Gilles, and C. R. Vogel, “Numerical simulations of multi-conjugate adaptive optics wavefront reconstruction on giant telescopes,” Appl. Opt. **42** (2003), pp. 4811–4818. [CrossRef] [PubMed]

10. Q. Yang, C.R. Vogel, and B.L. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. **45**, No. 21 (2006). [CrossRef] [PubMed]

9. B. L. Ellerbroek, L. Gilles, and C. R. Vogel, “Numerical simulations of multi-conjugate adaptive optics wavefront reconstruction on giant telescopes,” Appl. Opt. **42** (2003), pp. 4811–4818. [CrossRef] [PubMed]

10. Q. Yang, C.R. Vogel, and B.L. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. **45**, No. 21 (2006). [CrossRef] [PubMed]

## 2. Basic Concepts and Notation

*ψ*(

**x**,

*z*

_{l}),

*l*= 1,…,

*n*

_{layer}, where the discrete layer heights

*z*

_{l}are known, and

**x**= (

*x*,

*y*) denotes lateral position. Since rapid refractive index variations are concentrated in regions of high turbulence,

*ψ*is often referred to as the

*atmospheric volume turbulence profile*. (Note that one can obtain the

*z*, by integrating the square of

*ψ*.) We further assume that at any instant in time,

*ψ*is a realization of a wide-sense stationary stochastic process characterized by a Kolmogorov power spectral density function. See [11] for details.

*θ*denotes direction of the guidestar and

**x**denotes location in the pupil plane. Using a geometric optics approximation, we model the propagator

*P*as

*ψ*and

*ϕ*represent optical path differences and have units of length.

*θ*, the output of the (

*i,j*)th element of a Shack-Hartmann wavefront sensor can be modelled [12

12. D. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. , Vol. **67** No. 3, (1977), pp.370–375 [CrossRef]

*h*represents the sensor subaperture width and

*ϕ*to sensor measurements

**s**= (

*s*

^{x},

*s*

^{y}).

*ψ*arises naturally because of the finite number of guidestars in a practical MCAO system. We impose a nodal discretization on

*ψ*in the lateral variable

**x**at each layer

*z*

_{l}and nodal discretization of

*ϕ*in each (discrete) direction

*θ*. Hence the operators

*P*and Γ have discrete matrix (approximate) representations. In addition, the Kolmogorov spectrum for (discretized) volume turbulence has a block diagonal matrix representation in the Fourier domain, with one diagonal block for each layer, due to independence of the layers.

**x**= (

*x*,

*y*) at each altitude

*z*

_{l}. However, telescope apertures are typically circular or annular. Hence it is necessary to introduce an additional masking operator

*M*. We take [

*Ms*](

*x*

_{j},

*y*

_{j}) to equal

*s*(

*x*

_{i},

*y*

_{i}) if (

*x*

_{i},

*y*

_{j}) lies within the aperture and we take it to equal zero for (

*x*

_{i},

*y*

_{i}) outside the telescope aperture. In an abuse of notation, we will also refer to the masking function whose value

*M*(

*x*

_{j},

*y*

_{j}) = 1 for (

*x*

_{j},

*y*

_{j}) inside the aperture and

*M*(

*x*

_{j},

*y*

_{j}) = 0 for (

*x*

_{j},

*y*

_{j}) outside the aperture.

*atmospheric turbulence tomography*, or volume turbulence estimation, we mean the task of estimating the (discretized) volume turbulence profile

*ψ*from noisy sensor measurements

*ϕ*

_{est}. The sampling must be dense enough to well-represent the phase, but not so dense as to make the computations overly expensive. The estimation plus this first stage of the fitting step can together be viewed as a glorified interpolation scheme to extend the phase to directions that have not been sampled by the guidestars.

*m*(

**x**,

*z*

_{k}) denote the displacement of the DM at conjugate altitude

*z*

_{k}. The correction for pupil-plane phase aberrations due to the DMs can be represented as

*P*is again a geometric optics propagation operator with a representation as in Eq. (1); the DM displacements are analogous to turbulence layers placed at the DM conjugate altitudes. We assume the DM displacements can be represented as

**a**

_{k}represents the actuator commands for the kth DM, and

*H*

_{k}is the actuator-to-DM influence operator. We represent the combination of Eqs. (4) and (5) as

**a**= (

**a**

_{1},…,

**a**

_{nDM}) is the concatenation of the actuator commands. As in the estimation step, nodal discretization of the DM displacements gives rise to a discrete matrix to represent

*H*. The second stage of fitting can then be formulated as follows: Find the actuator command vector

**a**for which

*H*

**a**best matches (in a sense that we will precisely define in section 5) the estimated phase

*ϕ*

_{est}.

## 3. Turbulence Estimation for an LGS-NGS MCAO System

5. B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A , **19**, 1803–1816 (2002). [CrossRef]

*ψ*represent a nodal discretization of the volume turbulence profile. We separate wavefront sensor measurements into high-order components

*s*

_{h}(from the LGSs) and low-order components

*s*

_{t}(from the NGSs),

*G*

_{h},

*G*

_{t}has form similar to

*G*in Eq. (3). From [3

3. T. Fusco, J. M. Conan, G. Rousset, L. M. Mugnier, and V. Michau, “Optimal wave-front reconstruction strategies for multi-conjugate adaptive optics,” J. Opt. Soc. Am. A **18**, 2527–2538 (2001). [CrossRef]

*C*

_{t}is the covariance matrix for low order sensor noise, and

*C*

_{ψ}is the covariance matrix for volume turbulence. The first two terms in (8) correspond to the pair of entries in (7), while the third is a

*prior*, or regularization term, whose role is to incorporate prior statistical information about the volume turbulence.

5. B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A , **19**, 1803–1816 (2002). [CrossRef]

10. Q. Yang, C.R. Vogel, and B.L. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. **45**, No. 21 (2006). [CrossRef] [PubMed]

*N*

_{h}describes the noise within the high-order sensors, and

*T*is a low-rank block matrix with 2

*n*

_{LGS}columns (

*n*

_{LGS}denotes number of LGSs). The product

*s*

_{h}can be interpreted as a noise-weighted, tip-tilt-removed version of the high-order sensor signals sh. We then rewrite Eq. (29) as

*n*

_{LGS}and

*n*

_{t}, respectively, where

*n*

_{t}is the total number of low-order sensor measurements. Hence we can decompose

*U*

_{1}has 2

*n*

_{LGS}columns and

*U*

_{2}has

*n*

_{t}columns. On the other hand, the matrix

*A*

_{h}in Eq. (13) has full rank. The Sherman-Morrison formula [6] allows to write

*b*=

*b*

_{h}+

*b*

_{lr}, we obtain from (12) and (17)

*Lb*can be computed by (i) taking a few dot products to obtain

*b*and -

*W*

^{2}

^{T}

*b*; (ii) applying the inverse of a small (2

*n*

_{LGS}+

*n*

_{t}× 2

*n*

_{LGS}+

*n*

_{t}) matrix to the result of (i); and (iii) adding together scalar multiples of the columns of

*W*

_{1},

*W*

_{2}, where the scalars come from (ii). The product

*b*is computed iteratively using FD-PCG.

### 3.1. The Cone-Coordinate Transformation

**x**due to light that has passed through a turbulence layer at altitude

*z*

_{l}from a guidestar at height

*H*with orientation

*θ*is proportional to

*c*= 1 -

*z*

_{l}/

*H*and

**s**=

*z*

_{l}

*θ*are the cone compression factor and the shift, respectively, and the mapping

*cone coordinate transformation*.

*P*

_{t}then has a more complicated representation. To overcome this problem, we interpolate back to standard coordinates before applying

*P*

_{t}. In cone coordinates, Eq. (8) takes the form

*G̃*

_{h}in the first term on the right hand side denotes the cone-coordinate representation of

*G*

_{h}in Eqn. (7). The

*I*

_{c}in the second term denotes the transformation from cone coordinates back to standard coordinates. In practice all operators have discrete matrix representations and

*I*

_{c}is a 2-D interpolation matrix, which is very sparse. In cone coordinates, Eqs. (12)-(15) must be slightly modified, with

*G̃*

_{h}taking the place of

*G*

_{h}and

*G*

_{t}

*I*

_{c}taking the place of

*G*

_{t}. Given

*, one computes*ψ ˜

_{est}## 4. FD-PCG Implementation of the Estimation Step

*A*

_{h}in Eq. (13) nearly all have nice Fourier domain representations. In particular, the discrete gradient operator Γ corresponding to the Fried model (2) has a diagonal Fourier representation due to the Fourier shift theorem. The Fourier representer for the inverse Kolmogorov turbulence covariance matrix

*P*

_{h}has a Fourier representer which has a block decomposition with diagonal blocks, again as a consequence of the shift theorem. The mask

*M*unfortunately does not have a compact representation, but it can be well-approximated by a scalar multiple of a diagonal matrix, as we demonstrated in [10].

*P*

_{h}into blocks

*P*

_{kj}, where index

*k*corresponds to LGS direction and index

*j*corresponds to turbulence layer. Then from Eqs. (13) and (3), Ah also has a block decomposition,

*δ*

_{ij}is the Kronacker delta, the regularization term

*B*

_{j}corresponds to the inverse covariance matrix for the jth layer of turbulence, and Γ

_{x}, Γ

_{y}are the x- and y-components of the discrete gradient Γ.

*P*

_{kj}is a simple shift operator. Hence it has a Fourier representation

*F*represents the 2-D discrete Fourier transform on an

*n*

_{x}×

*n*

_{x}grid and

*P̂*

_{kj}is a simple filter (component-wise scalar multiplication) operator. We assume the components Γ

_{x}, Γ

_{y}of the gradient operator have analogous Fourier-domain filter representations

_{x}, Γ̂

_{y}. In addition, we assume the tip-tilt removed noise covariance is scalar,

*I*. Then

*B̂*

_{j}=

*FB*

_{j}

*F*

^{-1}, ∗ denotes complex conjugate transpose, and

**45**, No. 21 (2006). [CrossRef] [PubMed]

*M̂*by a scalar multiple of the identity to obtain (29) from (28). For simplicity of presentation we have taken the scalar to be 1 in Eq. (29). We will revisit the issue of masking in Section 4.3 below.

### 4.1. Direct Implementation of FD-PCG

*A*

_{h}

*x*=

*b*requires storage of 4 vectors, each the size of the right-hand-side

*b*; see [6] for details. The dominant costs at each iteration are typically from the operator multiplications

*h*=

*A*

_{h}

*d*and the preconditioner applications

*z*=

*C*

^{-1}

*r*.

*x*=

*b*in Eq. (19). In this case, motivated by Eq. (27) and approximation (29), we take the preconditioner to be the matrix

*C*having block components

*C*denote the matrix comprised of blocks

*Ĉ*

_{ij}. We demonstrated in [10] that there exists a permutation, or reordering of rows and columns, for which

*Ĉ*is block diagonal and the diagonal block size is quite small. (Diagonal block size will be addressed in detail in Section 4.3 below.) Hence

*Ĉ*can be inverted directly and efficiently stored. The preconditioning step

*z*=

*C*

^{-1}

*r*is then implemented as follows:

- 2-D Fourier transforms are applied to the blocks of
*r*(note that blocks correspond to turbulence layers; there are*n*_{layer}blocks), yielding a vector*f*.The entries of*r̂*are permuted, yielding a vector*r̃*. *r̃*is multiplied by the block diagonal inverse of the reordered*Ĉ*, yielding*ẑ*.The inverse permutation is applied to*ẑ*, yielding*z*.- Inverse Fourier transforms are applied to the blocks of
*ẑ*to obtain*z*.

*h*=

*A*

_{h}

*d*are carried out as sparse matrix-vector multiplications using the block decomposition (24). The propagators

*P*

_{kj}correspond to simple shift-and-adds and need not be stored, and the matrices

*S*

_{k}are very sparse. We employ Ellerbroek’s biharmonic approximation to the inverse turbulence covariance [5

**19**, 1803–1816 (2002). [CrossRef]

*B*

_{j}are also very sparse. Computations

*z*=

*C*

^{-1}

*r*are dominated by the layer-wise 2-D Fourier transforms in steps (i) and (v).

### 4.2. Transformed Implementation of FD-PCG and Comparison with Direct Implementation

*Âx̂*=

*b*, where

*Â*has blocks given in (27) and b has blocks

*b̂*

_{j}=

*F*[

*b*]

_{j}. In this case the preconditioning step

*ẑ*=

*C*

^{-1}

*r̂*is carried out as in the direct PCG approach, but the Fourier transforms in steps (i) and (v) are omitted.

*ĥ*=

*Âd̂*, we see from (27) that Fourier domain propagations can be carried out as component-wise product, or filter operations, involving (complex-valued) scalar multiplications. Similarly, the regularization terms

*B̂*

_{i}give rise to filtering operations. However, we see from Eq. (28) that since the transformed pupil mask

*M̂*has a full matrix representation, a Fourier transform / inverse Fourier transform pair is required for multiplication by each

*Ŝ*

_{k}.

*n*

_{layer}=

*n*

_{LGS}, then the number of Fourier transforms applied during each PCG iteration is the same for the direct implementation and the transformed implementation. The cost of propagation and regularization is very similar using both approaches. However, the transformed implementation requires

*n*

_{layer}additional Fourier transforms to obtain

*b̂*from

*b*and

*n*

_{layer}additional inverse Fourier transforms to obtain

*x*from

*x̂*. Moreover, the transformed approach requires 4 complex-valued PCG vectors, while the direct approach requires 4 real-valued vectors, which need half as much storage.

### 4.3. Grid Masking for High Order Sensor Subapertures

*C*to closely approximate the operator

*A*

_{h}, we must incorporate information about this grid structure in the Fourier domain. To this end we extend the subaperture vertex grid to the entire computational domain, as shown in Fig. 2, and we define the subaperture mask

*C*from Eq. (30). As in [10

**45**, No. 21 (2006). [CrossRef] [PubMed]

*Ĉ*according to spatial frequency, so the permuted

*Ĉ*is block diagonal with diagonal blocks that are

*n*

_{b}×

*n*

_{b}, where

*n*

_{b}=

*n*

_{layer}(Δ

*s*/Δ

*x*)

^{2}, with Δ

*s*equal to the sensor subaperture spacing and Δ

*x*equal to the spacing of the computational grid. For example, for a 6-layer atmospheric model with the grid shown in Fig. 2, Δ

*s*/Δ

*x*= 2 and the diagonal blocks of the permuted

*Ĉ*are 24 × 24.

## 5. FD-PCG for the Fitting Step

*ψ*

_{est}, one propagates to the pupil plane to obtain an estimated phase,

*ϕ*

_{est}. Slightly modifying the notation introduced at the end of section 2, we model the masked DM corrections to the phase as

*M*is a pupil mask, a is the actuator command vector,

*H*

**a**represents PDM figures at the conjugate altitudes, and

*P*

_{DM}represents propagation from the nsample virtual guidestars, through “layers” that correspond to DM displacements at the conjugate altitudes, to the pupil plane.

*P*

_{DM}has an

*n*

_{sample}×

*n*

_{DM}block decomposition, the actuator influence operator

*H*is block diagonal with

*n*

_{DM}blocks, and

*a*can be decomposed into

*n*

_{DM}blocks.

*W*is to allow selective weighting in certain sampling directions, and the role of

*R*is to stabilize the actuator commands. One can also accommodate fast tip-tilt mirrors or other low-order corrective elements. These give rise to low-rank terms and can be handled as in the estimation step using the Sherman-Morrison formula.

*A*

_{fit}takes the form of

*A*

_{h}in Eq. (13) provided we identify

*MP*

_{DM}

*H*,

*W*, and

*R*in (34) with

*G*

_{h},

*A*

_{fit}must be well-approximated by matrices with sparse Fourier domain representations. For this reason, we select

*W*and

*R*to have blocks with translation invariant structure (the matrix representers are then block circulant with circulant blocks, or BCCB). The propagator

*P*

_{DM}should automatically be translation invariant, and many DM models assume that the blocks of

*H*are translation invariant. As was the case in the estimation step, only the pupil mask does not have a sparse Fourier representer, but it can be approximated as before. We can again employ either of the two FD-PCG solutions strategies described in Sections 4.1 and 4.2.

## 5.1. An Illustrative Example

**45**, No. 21 (2006). [CrossRef] [PubMed]

**45**, No. 21 (2006). [CrossRef] [PubMed]

**45**, No. 21 (2006). [CrossRef] [PubMed]

*H*in Eq. (4), are taken to be bilinear splines with control points that coincide with the high-order sensor subaperture vertices (see Fig. 2). The matrix

*H*has translation invariant structure and is quite sparse. FD-PCG performance is summarized in Fig. 3.

**a**

_{k}denote the approximation to

**a**

_{opt}in Eq. (33) after k FD-PCG iterations, and let

**a**

_{k}for a in Eq. (32). By the

*residual phase error*we mean

*ϕ*

^{true}-

**a**

_{k}and

**a**

_{opt}is negligible and produces negligible change in

**45**, No. 21 (2006). [CrossRef] [PubMed]

## 5.2. Cost of Estimation and Fitting for the Example

*ψ*in the estimation step. Standard implementations of PCG require 4 copies of

*ψ*. There is some additional overhead to store the components of

*A*

_{h}, but by taking advantage of special structure (sparsity and translation invariance), this can be kept relatively small. However, the

*W*

_{1}and

*W*

_{2}matrices in the low-rank matrix

*L*in Eq. (17) require a great deal more storage. Each column of each

*W*

_{i}has

*N*

_{est}elements, and there are 2

*n*

_{LGS}+

*n*

_{t}= 2×5 + 8= 18 columns in

*W*

_{1},

*W*

_{2}for the simulation. Hence the storage costs in the estimation step are dominated by the low-rank terms.

*L*

_{b}(which is O(

*N*

_{est})) is much cheaper than the cost of computing

*b*via FD-PCG iteration. The FD-PCG costs are the cost per iteration multiplied by the total number of iterations. Dominant costs per iteration are multiplications by

*A*

_{h}and by the inverse preconditioner

*C*

^{-1}. Whether we use the direct approach in section 4.1 or the transformed approach in section 4.2 we must apply forward and inverse discrete Fourier transforms to block vectors with

*N*

_{est}entries (each block is 256 × 256 in our simulation). Using the fast Fourier transform, the asymptotic cost for this is then O(

*N*

_{est}log

*N*

_{est}). There are some additional order

*N*

_{est}costs, e.g., from the multiplication by the relatively small blocks of

*Ĉ*

^{-1}.

*ψ*and the block structure of the components of

*A*

_{h}and of

*Ĉ*

^{-1}provide ample opportunities to dramatically reduce computational costs in a parallel computing environment.

*N*

_{fit}=

*N*

_{est}/3 (due to 6 layers in the estimation vs 2 DMs in the fitting). In addition, the 2 or 3 FD-PCG iterations required for the fitting step are much fewer than the 6 to 8 iteration needed for estimation.

## 6. Discussion and Conclusions

**45**, No. 21 (2006). [CrossRef] [PubMed]

- A demonstration that the FD-PCG algorithm can be adapted to efficiently solve the fitting step in optimal MCAO wavefront reconstruction. We show that for a simulated MCAO system for a 30-meter telescope, at most 3 FD-PCG iterations are needed for the fitting step.
- Solution to the cone coordinate transformation problem. This problem arises with mixed LGS-NGS systems and can lead to field distortion if it is not handled correctly. Our solution involves a cone-coordinate-to-standard coordinate interpolation and is efficiently implemented with sparse matrix techniques.
- Solution of problems related to computational grids that have higher resolution than the high-order sensor subaperture grids. These problems are solved with a subaperture mask. This leads to a somewhat more complex preconditioner, but the additional computational overhead is small relative to the cost of the 2-D Fourier transforms.
- Two separate implementations of the FD-PCG algorithm. In the direct approach, PCG vectors lie in the spatial domain and are real-valued, while with the transformed approach the PCG vectors lie in the Fourier domain and are complex valued. The direct approach requires slightly fewer Fourier transforms and slightly less storage than does the transformed approach. Perhaps the only advantage of the transformed approach is that allows easy implementation of wave optics propagators. These have a sparse Fourier domain representation but have a convolution representation in the spatial domain.

## Acknowledgments

## References and links

1. | J. M. Beckers, “Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics,” in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M. H. Ulrich, ed., Vol. 30 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 693–703. |

2. | D. C. Johnston and B. M. Welsh, “Analysis of multi-conjugate adaptive optics,” J. Opt. Soc. Am. A |

3. | T. Fusco, J. M. Conan, G. Rousset, L. M. Mugnier, and V. Michau, “Optimal wave-front reconstruction strategies for multi-conjugate adaptive optics,” J. Opt. Soc. Am. A |

4. | R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J-P. Veran, “Adaptive optics requirements definition for TMT,” Advancements in Adaptive Optics, edited by
D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni, Proc. SPIE |

5. | B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A , |

6. | G. Golub and C. VanLoan, |

7. | L. Gilles, C. R. Vogel, and B. L. Ellerbroek, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A , |

8. | L. Gilles, B. L. Ellerbroek, and C. R. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multi-conjugate adaptive optics,” Appl. Opt. |

9. | B. L. Ellerbroek, L. Gilles, and C. R. Vogel, “Numerical simulations of multi-conjugate adaptive optics wavefront reconstruction on giant telescopes,” Appl. Opt. |

10. | Q. Yang, C.R. Vogel, and B.L. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. |

11. | J. W. Hardy, |

12. | D. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. , Vol. |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

**ToC Category:**

Focus Issue: Adaptive Optics

**History**

Original Manuscript: March 13, 2006

Revised Manuscript: May 10, 2006

Manuscript Accepted: June 2, 2006

Published: August 21, 2006

**Citation**

Curtis R. Vogel and Qiang Yang, "Fast optimal wavefront reconstruction for multi-conjugate adaptive optics using the Fourier domain preconditioned conjugate gradient algorithm," Opt. Express **14**, 7487-7498 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7487

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### References

- J. M. Beckers, "Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics," in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M. H. Ulrich, ed., Vol. 30 of ESO Conference andWorkshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 693-703.
- D. C. Johnston and B. M. Welsh, "Analysis of multi-conjugate adaptive optics," J. Opt. Soc. Am. A 11, 394-408 (1994). [CrossRef]
- T. Fusco, J. M. Conan, G. Rousset, L. M. Mugnier, and V. Michau, "Optimal wave-front reconstruction strategies for multi-conjugate adaptive optics," J. Opt. Soc. Am. A 18, 2527-2538 (2001). [CrossRef]
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