**3. Brief theoretical background**

#### **3.1. AO calibration procedure**

The AO concept requires a procedure for calculating actuator commands based on WFS sig‐ nals relative to a defined set of zero points, so-called calibration. Both the DCAO demonstra‐ tor and the PoC prototype are calibrated using the same direct slope algorithm. The purpose is to construct an interaction matrix *G* by calculating the sensor response *s* = [s1, s2,..., sm] <sup>T</sup> to a sequence of DM actuator commands *c* = [*c*1, *c*2,..., *c*n] T. Here *s* is a vector of measured wave‐ front slopes, m/2 is the number of subapertures, and *n* is the number of DM actuators. This relation is defined by

$$\mathbf{s} = \mathbf{G} \mathbf{c}\_{\prime} \tag{1}$$

actuator on DM1 and ending with the last actuator on DM2. In the case of five Hartmann patterns with 129 subapertures each and two DMs with a total of 149 actuators we obtain an interaction matrix dimension of 1290×149. The reconstructor matrix *G*<sup>+</sup> is calculated using

where *U* is an m×m unitary matrix, *Λ* is an m×n diagonal matrix with nonzero diagonal ele‐ ments and all other elements equal to zero, and *VT* is the transpose of *V*, an n×n unitary ma‐

*G* <sup>+</sup> =*V Λ* <sup>+</sup>

( *c*1 *c*2 ) =*G* <sup>+</sup> (

which is also the least squares solution to Eq. (1). The diagonal values of *Λ*<sup>+</sup> are set to λ*<sup>i</sup>*

zero if λ*<sup>i</sup>* is less than a defined threshold value. Non-zero singular values correspond to cor‐ rectable modes of the system. Noise sensitivity can be reduced by removing modes with very small singular values. DM actuator commands can then be calculated by matrix multi‐

> *s*1 *s*2 *s*3 *s*4 *s*5

However, even the most meticulous calibration of DM and WFS interaction will not yield optimal imaging performance due to non-common path errors between the wavefront sen‐ sor and the final focal plane of the imaging channel. The reduction of these effects by proper zero point calibration is therefore crucial to achieve optimal performance of an AO system. Several methods have been proposed to improve imaging performance [26-33]. The method implemented in our system is similar to the imaging sharpening method [29, 30], but a novel figure of merit is used, and the inherent singular modes of the AO system are optimized

In SCAO a single GS is used to measure wavefront aberrations and a single DM is used to correct the aberrations in the pupil plane. This will result in a small corrected FOV due to field dependent aberrations in the eye. However, the corrected FOV in the eye can be in‐ creased by using several GS distributed across the FOV and two or more DMs [3, 19-21]. A larger FOV than in SCAO can actually be obtained by using several GS and a single DM in

*G* =*UΛV <sup>T</sup>* , (5)

http://dx.doi.org/10.5772/53640

Dual Conjugate Adaptive Optics Prototype for Wide Field High Resolution Retinal Imaging

of *Λ* are the singular values of *G*. The pseudoinverse

*U <sup>T</sup>* , (6)

). (7)

*-1*, or

7

singular value decomposition (SVD) [25] since

trix. The non-zero diagonal elements λ*<sup>i</sup>*

of *G* can now be computed as

plication:

(patent pending).

**3.2. Corrected field of view**

and the interaction matrix is given by

$$\mathbf{G} = \begin{bmatrix} \partial s\_1 / \partial c\_1 & \partial s\_1 / \partial c\_2 & \cdots & \partial s\_1 / \partial c\_n \\ \partial s\_2 / \partial c\_1 & \partial s\_2 / \partial c\_2 & \cdots & \partial s\_2 / \partial c\_n \\ \vdots & \vdots & & \vdots \\ \partial s\_m / \partial c\_1 & \partial s\_m / \partial c\_2 & \cdots & \partial s\_m / \partial c\_n \end{bmatrix} \tag{2}$$

The relation above has to be modified to allow for multiple GSs and DMs by concatenating multiple *s* and *c* vectors. In the case of five GSs and two DMs we obtain

$$
\begin{pmatrix} s\_1 \\ s\_2 \\ s\_3 \\ s\_4 \\ s\_5 \end{pmatrix} = G \begin{pmatrix} c\_1 \\ c\_2 \end{pmatrix} \\ \tag{3}
$$

where

$$G = \begin{bmatrix} \left\| s\_1 / \left\| c\_1 \right\| \left\| s\_1 / \left\| c\_2 \right\| \\ \left\| s\_2 / \left\| c\_1 \right\| \left\| s\_2 / \left\| c\_2 \right\| \\ \left\| s\_3 / \left\| c\_1 \right\| \left\| s\_3 / \left\| c\_2 \right\| \\ \left\| s\_4 / \left\| c\_1 \right\| \left\| s\_4 / \left\| c\_2 \right\| \\ \left\| s\_5 / \left\| c\_1 \right\| \left\| s\_5 / \left\| c\_2 \right\| \end{bmatrix} \right\| \end{bmatrix} \tag{4}$$

The interaction matrix *G* is constructed by poking each DM actuator in sequence with a pos‐ itive and a negative unit poke and calculating an average response, starting with the first actuator on DM1 and ending with the last actuator on DM2. In the case of five Hartmann patterns with 129 subapertures each and two DMs with a total of 149 actuators we obtain an interaction matrix dimension of 1290×149. The reconstructor matrix *G*<sup>+</sup> is calculated using singular value decomposition (SVD) [25] since

$$\mathbf{G} = \mathbf{U}\boldsymbol{\Lambda}\boldsymbol{\Lambda}\boldsymbol{V}^T \boldsymbol{\Lambda} \tag{5}$$

where *U* is an m×m unitary matrix, *Λ* is an m×n diagonal matrix with nonzero diagonal ele‐ ments and all other elements equal to zero, and *VT* is the transpose of *V*, an n×n unitary ma‐ trix. The non-zero diagonal elements λ*<sup>i</sup>* of *Λ* are the singular values of *G*. The pseudoinverse of *G* can now be computed as

$$\mathbf{G}^\* = \mathbf{V} \boldsymbol{\Lambda}^\* \mathbf{U} \boldsymbol{\Lambda}^T \tag{6}$$

which is also the least squares solution to Eq. (1). The diagonal values of *Λ*<sup>+</sup> are set to λ*<sup>i</sup> -1*, or zero if λ*<sup>i</sup>* is less than a defined threshold value. Non-zero singular values correspond to cor‐ rectable modes of the system. Noise sensitivity can be reduced by removing modes with very small singular values. DM actuator commands can then be calculated by matrix multi‐ plication:

$$\begin{pmatrix} c\_1\\ c\_2\\ c\_2 \end{pmatrix} = G \stackrel{\ast}{\ast} \begin{pmatrix} s\_1\\ s\_2\\ s\_3\\ s\_4\\ s\_5 \end{pmatrix} . \tag{7}$$

However, even the most meticulous calibration of DM and WFS interaction will not yield optimal imaging performance due to non-common path errors between the wavefront sen‐ sor and the final focal plane of the imaging channel. The reduction of these effects by proper zero point calibration is therefore crucial to achieve optimal performance of an AO system. Several methods have been proposed to improve imaging performance [26-33]. The method implemented in our system is similar to the imaging sharpening method [29, 30], but a novel figure of merit is used, and the inherent singular modes of the AO system are optimized (patent pending).

#### **3.2. Corrected field of view**

**3. Brief theoretical background**

and the interaction matrix is given by

a sequence of DM actuator commands *c* = [*c*1, *c*2,..., *c*n]

*G* =

The AO concept requires a procedure for calculating actuator commands based on WFS sig‐ nals relative to a defined set of zero points, so-called calibration. Both the DCAO demonstra‐ tor and the PoC prototype are calibrated using the same direct slope algorithm. The purpose is to construct an interaction matrix *G* by calculating the sensor response *s* = [s1, s2,..., sm]

front slopes, m/2 is the number of subapertures, and *n* is the number of DM actuators. This

∂*s*<sup>1</sup> / ∂*c*<sup>1</sup> ∂*s*<sup>1</sup> / ∂*c*<sup>2</sup> ⋯ ∂*s*<sup>1</sup> / ∂*cn* ∂*s*<sup>2</sup> / ∂*c*<sup>1</sup> ∂*s*<sup>2</sup> / ∂*c*<sup>2</sup> ⋯ ∂*s*<sup>2</sup> / ∂*cn* ⋮ ⋮ ⋮ ∂*sm* / ∂*c*<sup>1</sup> ∂*sm* / ∂*c*<sup>2</sup> ⋯ ∂*sm* / ∂*cn*

The relation above has to be modified to allow for multiple GSs and DMs by concatenating

multiple *s* and *c* vectors. In the case of five GSs and two DMs we obtain

(

*G* =

) <sup>=</sup>*G*( *c*1 *c*2

∂*s*<sup>1</sup> / ∂*c*<sup>1</sup> ∂*s*<sup>1</sup> / ∂*c*<sup>2</sup> ∂*s*<sup>2</sup> / ∂*c*<sup>1</sup> ∂*s*<sup>2</sup> / ∂*c*<sup>2</sup> ∂*s*<sup>3</sup> / ∂*c*<sup>1</sup> ∂*s*<sup>3</sup> / ∂*c*<sup>2</sup> ∂*s*<sup>4</sup> / ∂*c*<sup>1</sup> ∂*s*<sup>4</sup> / ∂*c*<sup>2</sup> ∂*s*<sup>5</sup> / ∂*c*<sup>1</sup> ∂*s*<sup>5</sup> / ∂*c*<sup>2</sup>

The interaction matrix *G* is constructed by poking each DM actuator in sequence with a pos‐ itive and a negative unit poke and calculating an average response, starting with the first

*s*1 *s*2 *s*3 *s*4 *s*5 <sup>T</sup> to

T. Here *s* is a vector of measured wave‐

. (2)

*s* =*Gc*, (1)

), (3)

. (4)

**3.1. AO calibration procedure**

6 Adaptive Optics Progress

relation is defined by

where

In SCAO a single GS is used to measure wavefront aberrations and a single DM is used to correct the aberrations in the pupil plane. This will result in a small corrected FOV due to field dependent aberrations in the eye. However, the corrected FOV in the eye can be in‐ creased by using several GS distributed across the FOV and two or more DMs [3, 19-21]. A larger FOV than in SCAO can actually be obtained by using several GS and a single DM in the pupil plane, analogous to ground layer AO (GLAO) in astronomy [34], but the increase in FOV size and the magnitude of correction will be less than when using multiple DMs.

DMs the light passes through a collimating lens array (CLA) consisting of five identical lens‐ es, one for each GS. The five beams are focused by a lens (L7) to a common focal point (c.f. Fig. 8), collimated by a lens (L8) and individually sampled by the WFS, an arrangement con‐ sequently termed multi-reference WFS. In addition to separating the WFS Hartmann pat‐ terns as in [36] this arrangement makes it possible to filter light from all five GSs using a

Dual Conjugate Adaptive Optics Prototype for Wide Field High Resolution Retinal Imaging

http://dx.doi.org/10.5772/53640

9

Custom written AO software for control of one or two DM and one to five GS was devel‐ oped, tested, and implemented by Landell [37]. The pupil DM (DM1) will apply an identical correction for all field-points in the FOV. The second DM (DM2), positioned in a plane con‐ jugated to a plane approximately 3 mm in front of the retina, will contribute with partially individual corrections for the five angular directions and thus compensate for non-uniform (anisoplanatic) or field-dependent aberrations. The location of DM2 was chosen to ensure an smooth correction over the FOV by allowing sufficient overlap of GS beam footprints.

**Figure 3.** Basic layout of the DCAO demonstrator. Abbreviations: BPF – band-pass filter, BS – beamsplitter, CLA – colli‐ mating lens array, CM – cold mirror, DM1 – pupil DM, DM2 – field DM, FF – fiber ferrules, FS – field stop, FT – flash tube, LA – lenslet array, M – mirror, P – pupil conjugate plane, PL – photographic lens, PM – pupil mask, R – retinal conjugate

For imaging purposes, the retina is illuminated with a flash from a Xenon flash lamp, fil‐ tered by a 575±10 nm wavelength bandpass filter (BP). The narrow bandwidth of the BP is

plane, SF – spatial filter, SLD – superluminescent diode, WBS – wedge beamsplitter.

*4.1.2. DCAO demonstrator retinal imaging*

single pinhole (US Patent 7,639,369).

A relative comparison of simulated corrected FOV for the three cases of SCAO, GLAO, and DCAO in our setup is shown in Fig. 2. The simulated FOV is approximately 7×7 degrees, with a centrally positioned GS in the SCAO simulation and five GS positioned in an 'X' for‐ mation with the four peripheral GSs displaced from the central GS by a visual angle of 3.1 deg in the GLAO and DCAO simulations.

**Figure 2.** Zemax simulation of a corrected 7×7 deg FOV in our setup using the Liou-Brennan eye model [35] for SCAO (left), multiple GS and single DM (middle), and DCAO (right). Color bar represents simulated Strehl ratio.
