Several AOS have been invented for different microscopes. To optimize specif-ically for their application great importance has to be attached to the selection of the AOS components. In this chapter, the implemented system is explained in detail, especially focusing on the compensation theory of the implemented AOS and the specific implementation. Afterwards a simulation of the com-bined system is outlined to determine the system’s compensation limitations.
Then the combined system is characterized and calibration enhancements are tested. The last section of the chapter will show first results of correcting manually introduced aberrations.
aberration to the DM.
The interaction between the SHWFS and the DM can be realized in several ways. The two basic types of wavefront information are either zonal, where the wavefront is expressed over a small spatial area, or modal, when the wave-front is expressed in terms of mode coefficients of a polynomial expansion over the entire pupil [43]. A common approach in microscopy is a modal method, where the deflections of the SHWFS’ spot diagram are transformed into Zernike polynomials [65]. The linear combination of the Zernike poly-nomials producible by the DM are then applied with a negative sign. This optical path length adjustment is canceling out whilst the excitation wave travels through the aberrated specimen resulting in a diffraction limited PSF [66]. In theory, however, any number of basis sets with the conditions of linear independence and completeness can be used for wavefront representation in the modal approach [66]. Therefore, in our system’s implementation the eigenmodes of the adaptive optics element, the DM, are taken as the basis, hence, limiting our AOS to the actual hardware limitations.
4.1.1. Theory of the Modes from Wavefront Slopes
The theory of an AOS is based on a linear equation system, because most of the adaptive optics operate in a linear regime. Thus, straightforward methods
of linear algebra are sufficient.
A wavefront phaseFat a position (x,y) in the pupil plane is described by a polynomial expansion
F(x,y) =
Â
K k=1akZk(x,y) (4.1)
with the coefficientsak, the polynomial basis functionsZk, also called modes, andKthe total number of modes.
A SHWFS measures the wavefront slopes in two directions at m lenslet posi-tions. Assume that the wavefront slopes in x and in y direction are measured at position m. Hence, the number of total measurements M is 2xm, with m in x direction and m in y direction. The corresponding set of linear equation is given by
∂F(x,y)
∂x m =
Â
K k=1ak∂Zk(x,y)
∂x (4.2)
∂F(x,y)
∂y m =
Â
K k=1ak∂Zk(x,y)
∂y (4.3)
where m implies the evaluation at the coordinates (xm,ym). These two equa-tions, eq. (4.2) and eq. (4.3), can be rewritten as the matrix equation
s= [B]·a (4.4)
with
[B] = 2 66 66 66 66 66 66 66 66 66 4
∂Z(x,y)1
∂x 1
∂Z(x,y)2
∂x 1 · · · ∂Z(∂xx,y)K
∂Z(x,y)1 1
∂x 2
∂Z(x,y)2
∂x 2 · · · ∂Z(∂xx,y)K
... ... 2
∂Z(x,y)1
∂x M
2
∂Z(x,y)2
∂x M
2
· · · ∂Z(∂xx,y)K M
∂Z(x,y)1 2
∂y 1
∂Z(x,y)2
∂y 1 · · · ∂Z(∂yx,y)K
... ... 1
∂Z(x,y)1
∂y M
2
∂Z(x,y)2
∂y M
2
· · · ∂Z(∂yx,y)K M
2
3 77 77 77 77 77 77 77 77 77 5
(4.5)
as the so called control matrix. Bcontains the influence of every modeZk on every lenslet at position m.
As mentioned earlier, in theory, a number of basis sets can be used for wavefront representation with linear independence and completeness as their only requirement [66]. In the implemented control system the eigenmodes of the DM, called poke modes, are used. Because the eigenmodes of the DM are spanned by the set of actuators, the poke modes are equivalent to the actuators and the mode coefficients are equivalent to the actuator heights from the reference plane. This poke mode approach allows direct transformation from deflections on the SHWFS to actuator heights on the DM and thus bypassing transformations in other basis as is necessary with Zernike modes.
During a calibration, the control matrixBis generated. In that process a certain height is applied to every actuator iteratively, the resulting spot diagram is detected by the SHWFS and the deflections for each lenslet are calculated and stored in the control matrix column-wise. After calibration, when an
aberrated wavefront is measured with a SHWFS the deflections s are obtained.
Thus, the poke mode coefficients can be determined by solving the linear equation system eq. (4.4). Due to our system’s composition, consisting of144 actuators - where approximately44are discarded because these are located out of the pupil- and145illuminated lenslets, it has to be emphasized that the control matrix is a rectangular matrix. Hence, a pseudo inversion [67]
is necessary for matrix inversion to solve for the mode coefficients. For our overdetermined system singular value decomposition (SVD) is implemented as pseudo inversion.
4.1.2. Implementation of the Adaptive Optics System
In the implemented AOS a SHWFS is used for wavefront sensing of a two-photon-induced guide star that can either average the aberrations over the FOV by scanning it during wavefront determination or measures the aberration just in a single focal volume where the beam is parked. The control method for wavefront correction is based on slope measurements, which will be discussed briefly.
Before aberration correction can be conducted, the control system has to be calibrated by creating the control matrix B. Prior to the calibration, the system aberrations are corrected using a sensor-less AOS to minimize initial aberrations.
Sensor-less AOS The implemented sensor-less AOS is based on Zernike polynomials with pixel intensity as their metric. This sensor-less AOS is in this thesis mainly used for the elimination of aberrations introduced by the system, called “flatten the system”. The “flatten the system” process iterates through a user-specified amount of Zernike polynmials. Every Zernike polynomial is applied to the DM iteratively in a user-specified coefficient range. An image of the ROI for every Zernike polynomial with a specific coefficient is captured and the pixel with highest intensity is stored. When iterating through the given coefficient range for a Zernike polynomial a parabola-shaped curve, see Figure4.1, for the corresponding pixel intensities will result. The discrete intensity values M for every coefficient a are then fitted with a parabola and the coefficient value for the maximum intensity of the fit is extracted. This procedure is conducted for all inserted Zernike polynomials leading to a vector of Zernike coefficients with the enhanced image intensity. Therefore, a linear combination of the Zernike polynomials with the extracted coefficients gives an optimized system flat pattern minimizing the initial aberrations when applied to the DM. This procedure is conducted prior to every AOS calibration to ensure minimized initial system aberrations and to optimize the PSF before imaging.
Figure4.1.: Example of a sensor-less AO correction for one Zernike polynomial. When iterating through a given coefficient range{0,+x, x},see first three graphs from the left, a parabola-shaped curve for the corresponding pixel intensities{M0, M+, M } will result. The discrete intensity values for every coefficient are then fitted with a parabola and the coefficient valueacorrfor the maximum intensityxof the fit is extracted, see right graph. Figure from [68].
Calibration After minimizing the initial system aberrations, the AOS calibra-tion is conducted. For the AOS calibracalibra-tion a spot diagram is captured and defined as the reference or non-aberrated spot diagram and the center position of the focal spot from every lenslet is determined and stored in a vector. This will be the reference vector for aberration correction. After the center positions of the focal spots are determined, the calibration of the control matrix can be carried out. As described previously, for control matrix determination an actuator is poked with a certain height and the corresponding SHWFS spot diagram is recorded. This spot diagram is then analyzed immediately by determining the center position of the focal spot for each lenslet. Spots corresponding to lenslets that are conjugate to the poked actuator will be deflected by the applied poke. Thus, these spots will be displaced depending on the applied poke height. When subtracting the reference spot positions by
the poke aberrated spot position, deflections will result for lenslets, which were affected by the aberrations. These deflections are then stored in the column of the control matrix, which the actuator corresponds to. This is done for every actuator resulting in matrixBof eq. (4.4). After calibration, artificial-or specimen-induced aberrations can be measured as described below.
Closed-loop control To maximize aberration correction, closed-loop con-trolled aberration correction is implemented. The closed-loop control for aberration measurement is shown in Figure4.2. A SHWFS spot diagram is recorded and then analyzed as mentioned above to determine the x- and y-deflection of each focal spot. The resulting vector is s in eq. (4.4). Thus, the actuator compensation can be easily calculated by solving eq. (4.4) with the use of a pseudo inversion for B. In theory, simply the negative of the recon-structed actuator heights can be applied to the deformable mirror to correct for the measured aberration. However, pseudo inversion introduces large ac-tuator heights for non-contributing acac-tuators, the acac-tuators, which lie outside the DM aperture. Thus, compensation is adjusted by an influence vector. As mentioned above, the control matrix also contains information regarding the influence of every actuator on all the lenslets. Hence, the control matrix is used to determine the influence vector. The influence vector is calculated from the control matrix as following
niv(k) =
qÂmM=1B2m,k
max (4.6)
with
max=maximum( vu ut
Â
Mm=1
Bm,k2 ) (4.7)
Figure4.2.: Control loop for closed-loop aberration correction. In a closed-loop aberration correction, first the spot diagram of the aberrated wavefront is captured. Then the correction is calculated and applied to the DM using the control computer.
Then another spot diagram of the previously corrected wavefront is captured and analyzed again. When the calculated correction factors are smaller than the user-set threshold an aberration corrected image is acquired. Else another aberration correction iteration is conducted.
Before applying the reconstructed actuator heights, the normalized influence vector (NIV) is multiplied weighing the contribution of each actuator. Hence,
actuators on the outside of the pupil with less influence on the spot diagram will have less contribution on the correction. Additionally, the reconstructed actuator heights for outer rim actuators are set to 0, because these have, according to experimental observation, no significant contribution to the aberration correction. Finally, the negative of the recalculated actuator heights, multiplied with the NIV and after setting the outer rim actuators to 0, are added to the deformable mirror for compensation. Because a closed-loop approach is implemented, aberration correction is done iteratively. However, if the maximum closed-loop compensation iteration is set to one, aberration correction terminates here. Otherwise it starts again until the actuator heights for the current iteration step is lower than an expected user-set value, or the maximum compensation iteration is exceeded.