To practically maximize correction of the Zernike modes across the field of view in a scanning system, it is necessary for a deformable element to rapidly respond to the changing tissue aberrations. We therefore use the transition rate of the modes to estimate the additional residual wavefront error that occurs due to a lag in the frequencies of the correction element (such as a DM) behind the fast axis scanner and the gradient of Zernike modes in the sample. To assess the refresh rate of a desirable correction element, we calculate the gradient of modes in horizontal direction (the direction of the fast axis scan). We perform a 9-point first-order numerical derivative Display Formula
$\u2207\phi i=[Zi]\u2297[1280,\u22124105,15,\u221245,0,45,\u221215,4105,\u22121280]2\u2009\u2009h,$(15)
where $h$ is the distance between two points on the image plane and brackets indicate the Zernike mode $i$ value across the horizontal axis in matrix form. Now, we can write an equation to estimate the wavefront error caused by stepping or pixelation of correction due to having less frequency of correction than signal collection, in the form Display Formula$\sigma s=\u2211i(\u27e8\u2207\phi i\u27e9aNfsfc)2,$(16)
where $a$ is the pixel size, $N$ is the number of pixels horizontally, $fs$ is the scanning frequency, $fc$ is the frequency of the correction element, and the brackets indicate an average over the field of view. Where there is a significant difference between the frequencies of scanning and correction, the discrepancy between the sample aberrations modeled by Zernike modes 5 to 37 and the corrective elements on the scanner lead to an increase in the wavefront error. This error results in only a portion of the best-case theoretical correction (i.e., $\sigma res=1.25$, $Strehl ratio=20%$) being achieved. When quantified by $\sigma s$ as a function of correction frequency at a fixed laser scan rate [Fig. 10(a)], the additional absolute error approaches zero. Using the $\sigma s$ values, we calculate the Strehl percentage^{56} using $Sscan=exp(\u2212\sigma s2)$, assuming an 8-kHz resonant scanner and a 256 pixel width of the field of view [Fig. 10(a)], where 100% is full correction of Zernike modes 5 to 37. $Sscan$ therefore characterizes the reduction in Strehl ratio expected with inadequate scanning frequency.