Wavefront control in adaptive microscopy using Shack-Hartmann sensors with arbitrarily shaped pupils

Bing Dong; Martin J. Booth

doi:10.1364/OE.26.001655

1. Introduction

Adaptive optics (AO) systems generally use a wavefront sensor like a Shack-Hartmann sensor (SHS) to measure aberration and a wavefront corrector like a deformable mirror (DM) to perform phase-conjugate compensation. An AO system running in closed-loop continuously converts the slope measurements of the SHS to driving commands of the DM. In most cases, the pupil shape is circular and unchanged during correction. In other words, the amplitude variations at the pupil plane are typically small and have little effect on slope measurements. However, in some applications with severe aberrations the pupil shape can change during correction and the wavefront slopes in some pupil regions will become unmeasurable. Recently, SHS based AO has been used in confocal and two-photon microscopes to correct specimen induced aberrations to improve image quality [1–4]. Strong scattering in specimens can create varying dark regions within the pupil and spots in some subapertures can be totally missing (see Fig. 11 in [2] and Supplementary Fig. 1 in [4]). The wavefront slopes in dark regions were replaced by zero value in [2], which would however misestimate the aberration. In essence, the change of pupil shape leads to a change of wavefront reconstruction or control matrix, which should be updated accordingly during correction.

Wavefront control can be a two-step process that first converts the slope data to optical path difference or phase distribution (i.e. wavefront reconstruction) and then calculates corrective commands for the DM. In [5], the pupil shape is assumed circular and unchanged. Wavefront reconstruction algorithms from gradients within arbitrarily shaped pupils has been reported in previous literature for optical metrology or testing [6–10], however, not yet concerning wavefront control. These algorithms are classified as either zonal or modal, depending on whether the representation of reconstructed wavefront is values at sampling grid points or coefficients of orthogonal polynomials defined over the pupil area. Zou & Zhang proposed a zonal least-squares algorithm using a domain extension technique to establish a normal equation set for irregular-shaped pupils [6]. The accuracy of this algorithm can be improved by introducing Gerchberg-type iterations, which however is very time consuming [7,8]. Mochi & Goldberg developed a modal, non-iterative wavefront reconstruction algorithm based on Gram–Schmidt orthonormalization [9]. Ye et al. presented a modal algorithm based on numerical orthogonal gradient polynomials that can be used to directly fit the slope data within arbitrarily shaped pupil [10]. As speed of operation is important, we will not adopt any iterative zonal reconstruction algorithms and will only focus on modal reconstruction algorithm. After wavefront reconstruction, the control signals of the DM are calculated by simple matrix manipulations, provided that the response matrix of DM is pre-calibrated by an SHS sensor or interferometer. We call this modal control hereafter.

If we do not need to know the phase distribution exactly, the intermediate wavefront reconstruction step can be omitted and the slope data are directly translated to DM commands, namely the slope control in [11]. However, in many applications, only a part of phase aberration is assigned to the DM for correction. It is common in adaptive microscopy to remove tip, tilt and defocus modes from the control matrix, as these modes cannot be detected in an epi-configuration microscope. Furthermore, AO systems consisting of two DMs (namely woofer and tweeter) have to correct low and high order aberrations in a decoupled manner. In this case, it is essential to constrain each DM from generating some aberration modes to realize decoupling control. This is achievable by adding proper constraints on the slope control matrix [12,13].

For our current aim of using SHS based AO in biological microscopy, both modal and slope wavefront control approaches compatible with arbitrarily shaped pupils are investigated in this paper in order to find an improved solution to this issue. It is important to discuss both approaches, as the effects of missing SH spots could be different in each case, as are the mathematical methods to deal with this problem. We also investigate partial wavefront control methods that exclude specific aberration modes from the correction loop. The principles and procedures of slope and modal wavefront control for arbitrary pupils are presented in Section 2. Numerical simulations and results are given in Section 3. Experimental demonstrations are shown in Section 4.

2. Wavefront control with arbitrarily shaped pupils

2.1. Slope Control with arbitrarily shaped pupil

The principle of slope control is expressed as

S = R_{s} A

where

S = {[\begin{matrix} S^{x} \\ S^{y} \end{matrix}]}_{2 M \times 1} A = {[\begin{matrix} a_{1} \\ ⋮ \\ a_{n} \end{matrix}]}_{N \times 1} R_{s} = {[\begin{matrix} S_{1, 1}^{x} & S_{1, 2}^{x} & \dots & S_{1, N}^{x} \\ ⋮ & ⋮ & \dots & ⋮ \\ S_{M, 1}^{x} & S_{M, 2}^{x} & \dots & S_{M, N}^{x} \\ S_{1, 1}^{y} & S_{1, 2}^{y} & \dots & S_{1, N}^{y} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ S_{M, 1}^{y} & S_{M, 2}^{y} & \dots & S_{M, N}^{y} \end{matrix}]}_{2 M \times N}

S is a slope vector measured by the SHS in both x and y directions; A is the DM command vector; R_s is the slope response matrix of the DM; M is the number of subapertures of the SHS and N is the number of actuators of the DM. Each column of R_s is composed of the slope variations after sending a unit driving command to an actuator. R_s typically has many more rows than columns and is a sparse matrix because of the finite spread of each actuator’s influence function. So Eq. (1) is essentially an overdetermined linear equation and A can be solved by a least-squares method through a pseudo-inverse;

A = {(R_{s}^{T} R_{s})}^{- 1} R_{s}^{T} S

A more robust way to calculate A is using singular value decomposition (SVD). The matrix R_s can be represented as the product of three matrices.

R_{s} = U Σ V^{T}

The pseudoinverse of R_s is obtained by

R_{s}^{+} = V Σ^{+} U^{T}

Σ

is a diagonal matrix consisting the singular values of R_s.

Σ^{+}

is formed by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix. The ratio of the maximum singular value to the minimum is called the condition number, which is required to be small to achieve stable closed-loop correction. If any diagonal element of

Σ

is close to zero, the condition number will be very large and R_s is approximately singular. To get around this problem, one can simply zero the elements in

Σ^{+}

that are related to small singular values. Furthermore, to have a small low condition number, the subaperture number across an actuator should be optimized. As shown in [14], at least 1.5 subapertures per actuator are required. The command vector A is given by

A = R_{s}^{+} S

In practice, R_s is measured by the same SHS that is also used for aberration detection and $R_{s}^{+}$ should be calculated in advance before correction. The pupil area should be fully illuminated during calibration in order to obtain complete slope data in R_s. However, the pupil shape may change during correction and the wavefront slopes in some subapertures will become unmeasurable. We have two options to deal with this problem. The first one is simply replacing the unmeasurable slopes with zero in the vector S and while still using the pre-calculated $R_{s}^{+}$ [2]. This is a simple approach but could lead to lower corrective accuracy. The second option is to remove rows related to invalid subapertures from the matrix R_s and recalculate its pseudoinverse. In the second way, Eq. (5) is revised as

A = {\tilde{R}}_{s}^{+} \tilde{S}

where

{\tilde{R}}_{s}

and

\tilde{S}

are composed of available slope data.

2.2. Modal control with arbitrarily shaped pupils

In modal control, the wavefront aberration represented by modal coefficients is reconstructed from discrete slope data measured by the SHS. The wavefront aberration can be decomposed into orthogonal modes defined over the pupil area

W_{E r r} (x, y) = \sum_{i = 1}^{I} c_{i} Q_{i} (x, y)

where

Q_{i} (x, y)

are orthogonal modes in the pupil area and

c_{i}

are modal coefficients.

Using matrix notation, Eq. (7) can be rewritten as

W_{E r r} = Q C

The measured slope vector S is given by

S = \nabla W_{E r r} = (\nabla Q) C

where

\nabla

denotes evaluation of gradient in the x and y directions. The coefficient vector of wavefront aberration is solved from Eq. (9) as

C = {(\nabla Q)}^{+} S

After wavefront reconstruction, the DM command vector A can be solved from

C = R_{c} A

R_c is the modal response matrix of the DM, which is converted from the pre-calibrated slope response matrix R_s as

R_{c} = {(\nabla Q)}^{+} R_{s}

From Eqs. (11) and (12), vector A is calculated by

A = R_{s}^{+} (\nabla Q) C

For a circular pupil,

{Q}

are the well-known Zernike polynomials. For arbitrary pupils, Zernike polynomials are not orthogonal and not suitable for modal wavefront reconstruction. However, we can establish

{Q}

that are orthogonal in arbitrary pupils using linear combinations of Zernike polynomials

{Z}

[15].

Q_{n} = \sum_{k = 1}^{K} D_{n k} Z_{k}

where

D_{n k}

is the conversion coefficient and K is the number of Zernike polynomials used.

Equation (14) can be rewritten in matrix form as

Q = Z D^{T}

Q is an M × N matrix; Z is an M × K matrix; D is an N × K conversion matrix. Q and Z are evaluated at the centers of M subapertures.

Considering the orthogonality of ${Q}$ and Eq. (15), we have

Q^{T} Q = D Z^{T} Q = M I

where

I

is the identity matrix.

Combining Eq. (15) with Eq. (16), we can get

D Z^{T} Z D^{T} = M I

Let

D = {(P^{T})}^{- 1}

and substitute it into Eq. (17), we have

P^{T} P = Z^{T} Z / M

As $Z^{T} Z$ is a symmetric positive-definite matrix, P can be obtained uniquely by computing the Cholesky decomposition of $Z^{T} Z$ . P is an upper triangular matrix and D is a lower triangular matrix. So $Q_{n}$ is formed by a combination of the first n terms of the Zernike polynomials.

For arbitrary pupils, Eqs. (11), (12) and (13) can be rewritten as

C = P {(\nabla \tilde{Z})}^{+} \tilde{S}

R_{c} = P {(\nabla \tilde{Z})}^{+} {\tilde{R}}_{s}

A = {\tilde{R}}_{s}^{+} (\nabla \tilde{Z}) P^{- 1} C

where

\tilde{S}

,

{\tilde{R}}_{s}

and

\nabla \tilde{Z}

are calculated from valid subapertures. The matrix P should be updated by Cholesky decomposition of

{\tilde{Z}}^{T} \tilde{Z}

if the pupil shape varies, as the samples used to calculate Z will change.

2.3. Partial correction with arbitrarily shaped pupils

The wavefront control methods based on Eq. (6) or Eq. (21) try to correct all aberration modes. However, partial correction that keeps some modes uncorrected is required in some cases. This is common, for example, in biological microscopy in an epi-illumination microscope, as tip, tilt and defocus modes cannot be detected, so are usually removed from the control loop. For modal control, partial correction is easy to implement by setting specific modal coefficients in vector C to zero and Eq. (21) can be modified as

A = {\tilde{R}}_{s}^{+} (\nabla \tilde{Z}) P^{- 1} C_{r}

where C_r is the modified coefficient vector.

For slope control, partial correction is not that straightforward. Supposing we have derived a set of orthogonal modes ${Q}$ defined over pupil area using the method provided in Section 2.2, the DM-induced wavefront $W_{D M}$ should be orthogonal to mode $Q_{j}$ , which is not required for correction, which means

〈 W_{D M}, Q_{j} 〉 = 0

where

〈 〉

denotes an inner product.

W_{D M}

is the linear sum of influence functions, which can also be represented by a linear combination of orthogonal modes

{Q}

.

W_{D M} = \sum_{n = 1}^{N} a_{n} F_{n} = \sum_{n = 1}^{N} a_{n} (\sum_{i = 1}^{I} b_{n i} Q_{i})

where

F_{n}

is the influence function of actuator n and

b_{n i} (i = 1 \dots I)

is the modal expansion coefficient. So Eq. (23) can be rewritten as

〈 \sum_{n = 1}^{N} a_{n} (\sum_{i = 1}^{I} b_{n i} Q_{i}), Q_{j} 〉 = 0

Using the orthogonality

〈 Q_{i}, Q_{j} 〉 = δ_{i j}

where

δ_{i j}

is the Kronecker delta, Eq. (25) can be simplified as

\sum_{n = 1}^{N} a_{n} b_{n j} = 0

Incorporating Eq. (26) into Eq. (1) and using matrix notation, we can get

[\begin{matrix} S \\ 0 \end{matrix}] = [\begin{matrix} R_{s} \\ γ R_{d} \end{matrix}] A

where

R_{d} = {b_{n j}, n = 1 \dots N}

is a row vector consisting of each influence function’s modal expansion coefficient of mode

Q_{j}

.

R_{d}

can be easily expanded (i.e. to contain more rows) to inhibit the correction of more aberration modes. Please note that

R_{d}

is actually composed of some selected rows of

R_{c}

in Eq. (12). γ is a gain factor. Since the row number of R_s is much bigger than that of R_d, γ should be large enough to ensure the suppression of modes in

R_{d}

. Here we choose

γ = 1000

empirically in both simulation and experiment. The command vector A can be solved from Eq. (27) still using the SVD method.

2.4 Wavefront control procedures for arbitrarily shaped pupils

The flowcharts of slope control and modal control for arbitrary shape-changing pupils are shown in Fig. 1. The blue boxes in Fig. 1(a) denote extra steps due to requirement of partial correction in slope control. These steps are very similar to those in the blue boxes of Fig. 1(b). So if partial correction for a shape-changing pupil is required, the workload of one cycle in slope control is comparable to that in modal control. Otherwise, slope control is much simpler and faster than modal control.

Fig. 1 Flowchart of (a) slope control and (b) modal control for arbitrarily shaped pupils.

Abstract

1. Introduction

2. Wavefront control with arbitrarily shaped pupils

2.1. Slope Control with arbitrarily shaped pupil

2.2. Modal control with arbitrarily shaped pupils

2.3. Partial correction with arbitrarily shaped pupils

2.4 Wavefront control procedures for arbitrarily shaped pupils

3. Numerical Simulation

3.1. Simulation Model

3.2 Simulation Results for Circular Pupil

3.3. Simulation Results of Arbitrary Shape Pupil

4. Experimental Demonstration

5. Conclusions

Funding

References and links

Cited By

Figures (16)

Equations (31)

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