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All Optical Tutorials on Thorlabs.comAO TutorialIntroduction: Typically, an AO system is comprised from three components: (1) a wavefront sensor, which measures these wavefront deviations, (2) a deformable mirror, which can change shape in order to modify a highly distorted optical wavefront, and (3) realtime control software, which uses the information collected by the wavefront sensor to calculate the appropriate shape that the deformable mirror should assume in order to compensate for the distorted wavefront. Together, these three components operate in a closedloop fashion. By this, we mean that any changes caused by the AO system can also be detected by that system. In principle, this closedloop system is fundamentally simple; it measures the phase as a function of the position of the optical wavefront under consideration, determines its aberration, computes a correction, reshapes the deformable mirror, observes the consequence of that correction, and then repeats this process over and over again as necessary if the phase aberration varies with time. Via this procedure, the AO system is able to improve optical resolution of an image by removing aberrations from the wavefront of the light being imaged. The Wavefront Sensor: A ShackHartmann wavefront sensor uses a lenslet array to divide an incoming beam into a bunch of smaller beams, each of which is imaged onto a CCD camera, which is placed at the focal plane of the lenslet array. If a uniform plane wave is incident on a ShackHartmann wavefront sensor (refer to Fig. 1), a focused spot is formed along the optical axis of each lenslet, yielding a regularly spaced grid of spots in the focal plane. However, if a distorted wavefront (i.e., any nonflat wavefront) is used, the focal spots will be displaced from the optical axis of each lenslet. The amount of shift of each spot’s centroid is proportional to the local slope (i.e., tilt) of the wavefront at the location of that lenslet. The wavefront phase can then be reconstructed (within a constant) from the spot displacement information obtained (see Fig. 2). Figure 1. When a planar wavefront is incident on the ShackHartmann wavefront sensor's microlens array, the light imaged on the CCD sensor will display a regularly spaced grid of spots. If, however, the wavefront is aberrated, individual spots will be displaced from the optical axis of each lenslet; if the displacement is large enough, the image spot may even appear to be missing. This information is used to calculate the shape of the wavefront that was incident on the microlens array. Figure 2. Two ShackHartmann wavefront sensor screen captures are shown: the spot field (lefthand frame) and the calculated wavefront based on that spot field information (righthand frame). Figure 3. Dynamic range and measurement sensitivity are competing properties of a ShackHartmann wavefront sensor. Here, f, Δy, and d represent the focal length of the lenslet, the spot displacement, and the lenslet diameter, respectively. The equations provided for the measurement sensitivity θ_{ min} and the dynamic range θ_{max} are obtained using the small angle approximation. θ_{min} is the minimum wavefront slope that can be measured by the wavefront sensor. The minimum detectable spot displacement Δy_{min} depends on the pixel size of the photodetector, the accuracy of the centroid algorithm, and the signal to noise ratio of the sensor. θ_{max} is the maximum wavefront slope that can be measured by the wavefront sensor and corresponds to a spot displacement of Δy_{max}, which is equal to half of the lenslet diameter. Therefore, increasing the sensitivity will decrease the dynamic range and vice versa. The four parameters that greatly affect the performance of a given ShackHartmann wavefront sensor are the number of lenslets (or lenslet diameter, which typically ranges from ~100 – 600 μm), dynamic range, measurement sensitivity, and the focal length of the lenslet array (typical values range from a few millimeters to about 30 mm). The number of lenslets restricts the maximum number of Zernike coefficients that a reconstruction algorithm can reliably calculate; studies have found that the maximum number of coefficients that can be used to represent the original wavefront is approximately the same as the number of lenslets. When selecting the number of lenslets needed, one must take into account the amount of distortion s/he is trying to model (i.e., how many Zernike coefficients are needed to effectively represent the true wave aberration). When it comes to measurement sensitivity θ_{min} and dynamic range θ_{max}, these are competing specifications (see Fig. 3 to the right). The former determines the minimum phase that can be detected while the latter determines the maximum phase that can be measured. A ShackHartmann sensor’s measurement accuracy (i.e., the minimum wavefront slope that can be measured reliably) depends on its ability to precisely measure the displacement of a focused spot with respect to a reference position, which is located along the optical axis of the lenslet. A conventional algorithm will fail to determine the correct centroid of a spot if it partially overlaps another spot or if the focal spot of a lenslet falls outside of the area of the sensor assigned to detect it (i.e., spot crossover). Special algorithms can be implemented to overcome these problems, but they limit the dynamic range of the sensor (i.e., the maximum wavefront slope that can be measured reliably). The dynamic range of a system can be increased by using a lenslet with either a larger diameter or a shorter focal length. However, the lenslet diameter is tied to the needed number of Zernike coefficients; therefore, the only other way to increase the dynamic range is to shorten the focal length of the lenslet, but this in turn, decreases the measurement sensitivity. Ideally, choose the longest focal length lens that meets both the dynamic range and measurement sensitivity requirements. The ShackHartmann wavefront sensor is capable of providing information about the intensity profile as well as the calculated wavefront. Be careful not to confuse these. The lefthand frame of Fig. 4 shows a sample intensity profile, whereas the righthand frame shows the corresponding wavefront profile. It is possible to obtain the same intensity profile from various wavefunction distributions. Figure 4. Several pieces of information are provided by the ShackHartmann wavefront sensor, including information about the total power at each lenslet and the calculated wavefront distribution present. Here, the lefthand frame shows a sample intensity profile, while the righthand frame shows the corresponding wavefront. The Deformable Mirror: Figure 5. The aberration compensation capabilities of a flat and MEMS deformable mirror are compared. (a) If an unaberrated wavefront is incident on a flat mirror surface, the reflected wavefront will remain unaberrated. (b) A flat mirror is not able to compensate for any deformations in the wavefront; therefore, an incoming highly aberrated wavefront will retain its aberrations upon reflection. (c) A MEMS deformable mirror is able to modify its surface profile to compensate for aberrations; the DM assumes the appropriate conjugate shape to modify the highly aberrated incident wavefront so that it is unaberrated upon reflection. Figure 6. Cross sectional schematics of the main components of BMC's continuous (left) and segmented (right) MEMS deformable mirrors. The range of wavefronts that can be corrected by a particular DM is limited by the actuator stroke and resolution, the number and distribution of actuators, and the model used to determine the appropriate control signals for the DM; the first two are physical limitations of the DM itself, whereas the last one is a limitation of the control software. The actuator stroke is another term for the dynamic range (i.e., the maximum displacement) of the DM actuators and is typically measured in microns. Inadequate actuator stroke leads to poor performance and can prevent the convergence of the control loop. The number of actuators determines the number of degrees of freedom that the mirror can correct for. Although many different actuator arrays have been proposed, including square, triangular, and hexagonal, most DMs are built with square actuator arrays, which are easy to position on a Cartesian coordinate system and map easily to the square detector arrays on the wavefront sensors. To fit the square array on a circular aperture, the corner actuators are sometimes removed (e.g., the deformable mirror included with the AOK1UM01 or AOK1UP01 has a 12 x 12 actuator configuration but only 140 actuators since the corner ones are not used). Although more actuators can be placed within a given area using some of the other configurations, the additional fabrication complexity usually does not warrant that choice. Figure 7. A crosslike pattern is created on the DM surface by applying the voltages necessary for maximum deflection of the 44 actuators that comprise the middle two rows and middle two columns of the array. The frame on the left shows a screen shot of the AO kit software depicting the DM surface, whereas the frame on the right, which was obtained through quasidark field illumination, shows the actual DM surface when programmed to these settings. Note that the white light source used for illumination is visible in the lower righthand corner of the photograph. Figure 7 (left frame) shows a screen shot of a cross formed on the 12 x 12 actuator array of the DM included with the adaptive optics kit. To create this screen shot, the voltages applied to the middle two rows and middle two columns of actuators were set to cause full deflection of the mirror membrane. In addition to the software screen shot depicting the DM surface, quasidark field illumination was used to obtain a photograph of the actual DM surface when programmed to these settings (see Fig. 7, right frame) The Control Software: In essence, the control software uses the spot field deviations to reconstructs the phase of the beam (in this case, using Zernike polynomials) and then sends conjugate commands to the DM. A leastsquares fitting routine is applied to the calculated wavefront phase in order to determine the effective Zernike polynomial data outputted for the end user. Although not the only form possible, Zernike polynomials provide a unique and convenient way to describe the phase of a beam. These polynomials form an orthogonal basis set over a unit circle with different terms representing the amount of focus, tilt, astigmatism, comma, et cetera; the polynomials are normalized so that the maximum of each term (except the piston term) is +1, the minimum is –1, and the average over the surface is always zero. Furthermore, no two aberrations ever add up to a third, thereby leaving no doubt about the type of aberration that is present.  
