1. Introduction

Optical microscopy in thick biological samples, such as tissues, is limited by wavefront distortions as light propagates into these heterogeneous media, causing a decrease of the signal-to-noise ratio, contrast and spatial resolution. To correct these effects, adaptive optics (AO) approaches have been proposed that consist in compensating the phase distortion by a wavefront modulator such as a deformable mirror or a spatial light modulator. Although, in some cases, wavefront sensors are included to measure sample-induced aberrations [1–3], most AO systems in microscopy rely on sensorless optimization-based strategies [4–9] since biological samples do not generally provide point emitters that could be used as guide-stars. In sensorless AO, different aberrations are sequentially applied by a wavefront shaping device and a quality metric, which maximum corresponds to optimal correction, is measured. The choice of this metric and of the optimization procedure is important to increase correction speed, accuracy and limit the cost of AO correction in terms of photobleaching and phototoxicity, which are detrimental to the study of living samples.

Microscopy methods that aim at providing, not images, but quantitative information are particularly affected by optical aberrations, since their results can be biased and rendered useless. This is the case of fluorescence fluctuations methods, such as fluorescence correlation spectroscopy (FCS), which are important to study the dynamics of cellular processes [10]. These methods allow to measure the concentration and mobility of fluorescent molecules by analyzing the temporal fluctuations that appear when they diffuse in and out of an observation volume, corresponding to the point spread function (PSF) of the confocal or two-photon microscope. Therefore, their accuracy critically depends on the size and shape of the PSF. We have shown that even small amounts of optical aberrations (i.e. rms < 100 nm), that can hardly be perceived in images, cause sufficient distortion of the PSF to result in erroneous estimation of the concentration and diffusion constant. A sensorless adaptive optics approach has proved successful to stabilize the FCS observation volume in solution, through cells and within tissues [11–13].

To improve the accuracy and speed of aberration correction and ensure its reliability in presence of e.g. large or complex wavefront distortions, the optimization process should be adapted to the application and in particular an adequate metric should be chosen. We have previously introduced a metric extracted from fluctuation analysis, the molecular brightness, which dependence on aberration amplitude is well-described by the Strehl ratio squared, similarly to the signal from a point-emitter [11]. Its great sensitivity is advantageous when correcting small aberrations for FCS measurements. Another quantity, the fluorescence photon count, can also be readily measured from a solution of mobile fluorophores and used as an optimization metric. The use of these metrics may be interest not only for fluctuation techniques but for biological microscopy at large. Indeed, many implementations of sensorless AO rely on maximizing an image-based metric [7–9,14,15]. However, the sensitivity of such metrics depends on the imaged region (small and bright structures affording the highest sensitivity) and varies across the sample. Using mobile molecules for optimization is an interesting alternative since most living samples either contain endogeneous fluorescent molecules or can be easily doped with freely-diffusing fluorophores. Additional advantages are the reduced impact of photobleaching when measuring mobile as opposed to immobilized fluorophores, and the possibility of a truly local measurement, while image-based metrics entails spatial averaging.

In this work, we compare two metrics that can be used for sensorless adaptive optics in a solution of fluorescent molecules: fluorescence count rate and molecular brightness. We first investigate their measurement noise and sensitivity to aberrations. In the small aberrations range, the AO correction accuracy is experimentally measured and theoretically predicted, for each metric, from its measurement noise and sensitivity to aberrations. We show that the metric that optimizes correction accuracy differs depending on the concentration and brightness of the fluorophores. Finally we discuss the properties of these metrics in the case of larger aberrations. Our results provide guidelines to optimize the efficiency and minimize the cost of sensorless AO correction in solution, for fluorescence fluctuation measurements but also image acquisition in confocal or two-photon microscopes.

2. Experimental setup and methods

For these experiments we used a custom built confocal microscope with an adaptive optics system, which consists of a Deformable Mirror (DM) (Alpao) and a Shack-Hartmann Wavefront Sensor (Alpao) for DM calibration [11–13,16]. The setup is controlled via Matlab (Mathworks) integrating Alpao Core Engine Software (ACE, Alpao) for the AO system control. The confocal microscope consists of a 561 nm-diode pumped solid state laser (Cobolt) which output is directed to, successively, a deformable mirror (Hi-speed 97 DM, Alpao) and galvanometric mirrors used for scanning (6215H, Cambridge technology) before reaching the microscope base (Olympus X71) where it is focused by the objective (water-immersion, ×63, NA=1,2). Both the deformable mirror and the scan mirrors middle position are conjugated with the objective back pupil plane, in position and magnification. The collected fluorescence light travels back through the deformable mirror, which is used to correct both the excitation and the fluorescence paths. A dichroic mirror (Chroma ZT561rdc, cutoff wavelength 574 nm) separates the fluorescence emission from excitation beam. The fluorescence light is detected a avalanche photodiode (SPCM-AQRH-13, Perkin Elmer) through a multimode fiber that acts as the pinhole with diameter 0.7 times that of the diffraction-limited spot. The pulses delivered by the APD are sent to a data acquisition board (PCIe-6321, National Instrument) for counting.

The Zernike modal base was chosen for open loop modal control of the system for convenience due to its ubiquitous use and their orthogonality relative to the Strehl ratio (hence the brightness metric). Alpao DMs are well suited for creating Zernike modes. With our Alpao DM 97-15, with 97 actuators, we are able to accurately create Zernike aberrations up to the 8th radial order with an amplitude up to 0.5 μm.

The fluorescent solutions used in this work were obtained from Sulforhodamine B sodium salt (Sigma-Aldrich, St. Louis, USA), used without further purification, diluted in pure water at concentrations from a few to 200 nM.

The modal adaptive optics scheme used in this work is similar to the one described for image-based modal aberration correction [7,15]. The DM shape is optimized by sequentially testing up to eleven Zernike modes (excepting tip, tilt and defocus). For each Zernike mode, the metric (count rate or brightness) is measured while the DM is used to introduce different bias amplitudes of the mode. In the following, the number of measurements was set at N_mes=3, corresponding to amplitudes: zero, positive and negative bias and the bias was chosen to be 50 nm. The optimal amplitude is then determined from a least-square fit, assuming a quadratic variation of the metric around the maximum and this new optimal shape is updated on the DM before correcting the next mode. The whole correction cycle was repeated 3 times.

3. Measuring signals from fluorophores in solution

3.1. Definition of fluorescence count rate and brightness

When performing fluorescence fluctuation measurements, the fluorescence light emitted by molecules in the observation volume is detected as a series of photon counts values k, each integrated over a sampling time δt (which should be short enough to capture the fluctuations arising from the phenomenon of interest, e.g. diffusion). In order to evaluate fluctuations, a large number N of values is acquired corresponding to a total acquisition time of T, the measurement duration.

As an example, Fig. 1 depicts the fluorescence time trace and the corresponding histogram of photon counts detected with 300 μs sampling time for two solutions. These solutions have different concentrations, but the excitation power has been adjusted so that the average fluorescence signal is similar in both cases.

 

Fig. 1 Fluorescence time trace (left) and photon count histogram (right) measured on two solutions of Sulforhodamine B in a water/glycerol mixture: the 20 nM solution (blue) exhibits dilute and bright molecules, while the 200 nM solution (red) is characterized by concentrated but dimmer molecules. FCS fits (not shown) yield average numbers of molecules of 8 and 68, and brightness values of 18 kHz/molecule and 2.5 kHz/molecules respectively for the 20 nM and 200 nM solutions, while the diffusion time is 11 ms for both solutions. The brightness is varied by adjusting the excitation power.

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We extract two parameters after each period T that can be used as metrics for AO: the fluorescence count rate and the molecular brightness. The count rate (CR) is simply the average number of photons detected per second and is obtained from the mean photon count 〈k〉 calculated over T: CR = 〈k〉/δt. The molecular brightness (or count rate per molecule) ∊ is the count rate divided by the mean number of molecules N_mol in the observation volume. It is estimated from both the mean and the variance of photon counts values.

Figure 1 illustrates how the molecular brightness is related to the second moment of the photon count distribution: a sample presenting the same average level of fluorescence signal can consist of concentrated but relatively dim molecules, corresponding to a smaller variance of photon counts as in the case of the 200 nM-solution, or dilute but brighter molecules, corresponding to a larger variance (or broader distribution) of photon counts as shown for the 20 nM-solution.

The molecular brightness can be obtained by evaluating the mean and variance of the photon distribution: ∊ = (1/δt)(〈Δk²〉 − 〈k〉)/〈k〉. The term 〈k〉 is subtracted from the numerator to remove the contribution of shot noise from the variance, so that only the contribution of the molecular occupation number is considered. Another way to remove the shot noise contribution is using the autocorrelation at the shorter lag time (i.e. δt) to estimate the brightness: ∊ = 〈k〉(G(δt) − 1), where G(δt) = 〈k_ik_i+1〉/〈k〉². In practice, this expression yields a more stable result and will be used throughout this work.

3.2. Measurement noise

The measurement noise of the chosen metric is an important ingredient in the estimation of the AO correction accuracy. We investigate the signal-to-noise ratio (S/N) of count rate and molecular brightness measurements experimentally and compare with theoretical predictions.

3.2.1. Count rate

The fluorescence count rate is given by the total photon counts, K, during the integration time T. To predict the measurement noise of K, we will calculate its variance using the derivation presented in [17]. The probability distribution function of K is related to the integrated light intensity, W, incident on the detector by Mandel’s formula, which states that the photon counts distribution results from a superposition of Poisson distributions obtained for each possible value of W. A consequence is that the cumulant value of W is equal to the factorial cumulant of K of the same order. Therefore their variance (second-order cumulant) are related by: 〈ΔW²〉 = 〈ΔK²〉 − 〈K〉, so that the variance of K can be obtained from the variance of W.

If the experiment is assumed to be stationary, the integrated light intensity (for an integration time T) is:

If the integration time T is short compared to the fluctuation time, B(T) scales as T² and the variance is a function of (∊T)². Since ∊T is the number of photons detected per molecule during T, this expression is the one used in Number and Brightness analysis [18]. If the integration time is long compared to the fluctuations time, B(T) scales as T, which can be understood as the fact that the variance is governed by the number of molecules diffusing through the observation volume during T (molecular ’occupation’ noise).

The signal-to-noise ratio (S/N) is given by $〈K〉/\sqrt{〈\mathrm{\Delta }{K}^2〉}$. Using the expression of B(T) obtained in [17] when assuming free diffusion and a 3D Gaussian PSF, we found a good agreement between the calculated and experimental S/N, as shown on Fig. 2(A) for two solutions of Sulforhodamine B of different concentrations. When comparing the two graphs, it is clear that the S/N of the count rate is not limited by the number of detected photons but by the stochastic process of molecules diffusing through the observation volume, since the weakly concentrated solution exhibits lower S/N at equivalent number of photons.

 

Fig. 2 Signal-to-noise ratio (S/N) of measured count rate and molecular brightness in different experimental conditions. (A) S/N measured (symbols) and calculated (solid lines) as a function of integration time T for both metrics estimated on two solutions of Sulforhodamine B of different concentrations. (B) Calculated S/N for count rate measurements and (C) calculated S/N for brightness measurements, as a function of the average number of molecules and brightness (integration time set at T = 0.1 s). The two solutions shown in (A) are depicted as crosses in the 2D maps.

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It is important to realize that, contrary to the case of immobilized fluorophores, the count rate measured in a solution of diffusing fluorophores is not subject solely to shot noise, even for a relatively long integration time. The noise also depends on molecular concentration, brightness and diffusion time. The S/N map in Fig. 2(B) shows that the accuracy of such measurement is mainly limited by the number of molecules as soon as the brightness is above a few kHz/molecule. At the limit of infinite concentration, the S/N becomes shot noise limited.

3.2.2. Molecular brightness

As mentioned previously, the brightness is estimated by: ∊ = 〈k_ik_i+1〉/〈k〉. Noting S = 〈k_ik_i+1〉, the variance of the brightness can be written:

The variance of 〈k〉 can be directly derived from the previous paragraph since 〈k〉 = K(δt/T). The variance of S has been studied by several authors [19–21]. Koppel provided the first analytical expression of the noise in FCS measurement [19]. He considered an exponentially decaying correlation function and assumed Gaussian statistics for the fluorescence signal which is only valid for large numbers of molecules in the observation volume. Later, Qian [20] has extended this analysis to a hyperbolic correlation function which is more realistic for diffusion and Poissonian statistics. We have previously shown that Qian’s expression is in good agreement with our experimental data [16]. Hence, in the following, the variance 〈ΔS²〉 is obtained from equation 6 of [20].

Equation (3) yields the following expression:

Figure 2(A) shows that this calculated S/N is in good agreement with experimental results for two different concentrations. When comparing the two solutions, we can see that, although the count rate S/N is always above the brightness S/N, the difference is reduced in the dilute solution.

The S/N map in Fig. 2(C) reveals that brightness measurements accuracy strongly depends on the molecular brightness itself, while the count rate measurement noise is mainly governed by the number of molecules (Fig. 2(B)). These maps confirm that, in standard conditions, the measurement of the count rate (which is based on the estimation of the mean of the photon count) always displays a higher S/N than that of the brightness (which is based on the estimation of the variance of the photon-count).

3.3. Sensitivity to aberrations: count rate vs molecular brightness

The second important consideration when choosing a metric is how the metric value varies with aberration amplitude. As will be detailed in section 4, the sensitivity of the metric is directly related to the accuracy of aberration corrections. Here we investigate experimentally the sensitivity of the two proposed metrics, count rate and molecular brightness. These values will be required to predict and compare their performance in terms of AO correction accuracy.

These sensitivity measurements are performed by applying various amounts of aberrations with the deformable mirror and measuring the metric value. For each rms amplitude value, 20 random combinations of 11 Zernike modes (up to the 4th radial order excepting tip-tilt and defocus) are generated and the corresponding metric values are averaged. Although the sensitivity may depend on the shape of the wavefront (hence the mode), we would consider a ’mean’ sensitivity value for simplicity. The choice of 11 modes is justified in microscopy AO applications where a small number of low-order aberrations was generally sufficient to account for most of the aberration [7,9,22].

Figure 3(A) shows the decay of both count rate and brightness as the aberration magnitude is increased. Comparing the two metrics, we note that the count rate is not as strongly affected by aberrations as the brightness, which sensitivity is well approximated by the Strehl ratio squared as previously shown [11,16]. This can be easily understood: as wavefront aberrations increase, the PSF is enlarged leading to, on one hand, a lower excitation intensity and thus a lower brightness for individual molecules, but on the other hand, a larger observation volume with more molecules that contribute to the signal. These two opposing effects cause the total count rate to decrease relatively slowly as a function of aberration magnitude.

 

Fig. 3 Sensitivity to aberrations (A) Measured value of both metrics (count rate and brightness) as a function of aberration magnitude for random combinations of 11 Zernike modes introduced by the deformable mirror. The sample is a aqueous 10 nM solution of Sulforhodamine B (N_mol=1.6). The Strehl ratio squared is plotted as a solid line. Quadratic fits in the small aberration range for both metrics give sensitivity values of 0.37 rad⁻² for CR and 1.5 rad ⁻² for brightness (B) Variation of other parameters extracted from fluctuation analysis, the number of molecules (top) and the diffusion time (bottom) as a function of aberration amplitude. For each rms, the parameters are measured with 20 random aberrations. The graphs depict the mean and standard deviation of these measurements.

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For small aberrations, we approximate the metric variation by a quadratic function:

In Fig. 3(B), we present the variation of the number of molecules and the diffusion time measured by FCS as a function of aberration amplitude. The number of molecules increases rapidly in presence of aberrations, denoting the enlargement of the observation volume. The diffusion time, τ_D, also increases with aberration amplitude, though more moderately. For small aberrations, τ_D can be approximated by a quadratic function (with sensitivity β_τ=−0.90 rad⁻²). This approximation will be used in the next two sections to calculate the measurement noise in presence of aberrations.

4. Correction accuracy in case of small aberrations

In this section, we present how to measure and calculate the residual aberrations after AO correction, using either the count rate or the brightness as metric, as a function of the sample properties, in the case of small aberrations.

The experiments are carried with four solutions of Sulforhodamine B of different concentrations corresponding to N_mol= 30, 8, 2 and 0.5 molecules in the observation volume (without aberration). The molecular brightness is varied between 3 and 80 kHz/molecules by adjusting the laser power. The correction accuracy is evaluated using the following procedure: first the DM is set to induce random (but known) aberrations distributed over 10 Zernike modes, then the optimization process is launched to blindly correct these aberrations. The residual error is obtained by confronting the initial and corrected aberrations. The amplitude of each mode is estimated from 3 measurements at −b, 0 and +b (b=0.54 rad). The modes are corrected sequentially and we loop three times through all of them. Initial aberration amplitude is chosen small enough so that a quadratic approximation can be used: the total rms is set to be 50 nm (0.54 rad). For each condition (molecular brightness and average number of molecules), the residual error is estimated from 50 optimizations with different sets of initial aberrations.

Figure 4(A) shows examples of the evolution of count rate or brightness as the AO optimization proceeds: in the first sample that contains dilute but bright molecules, the signal is not improved when the count rate is used as metric, whereas AO correction is successful with the brightness. For the second sample however, which exhibits a higher concentration of dimmer molecules, the count rate is a more efficient metric than the brightness.

 

Fig. 4 Comparison of AO correction using either the count rate or the molecular brightness as optimization metric. (A) Examples of optimization in either a dilute sample with relatively bright molecules (top) or a more concentrated solution with dim molecules (bottom). The left graphs depict the count rate measured during AO optimization using the count rate as metric. The right graphs depict the measured brightness during AO optimization using the brightness as metric. (B) Residual aberrations after correcting with the count rate (red circles) or the brightness (blue triangles) as metric, in four samples with different concentrations (corresponding to N_mol=30, 8, 2 and 0.5 molecules in the observation volume) at various laser power. The error bars are the standard deviation of 50 optimizations per point for each metric. The continuous line are residuals calculated with the method presented in the text. The correction is performed on 10 modes using 3 measurements per mode for bias −b, 0 and +b (b=0.54 rad) and an initial aberration of 0.54 rad.

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The correction accuracy achieved by both metrics is experimentally assessed in different conditions as presented in Fig. 4(B). The residual aberration after AO is depicted as a function of brightness (without aberration) for the four solutions. While, at N_mol=30 the count rate is a better metric at all laser powers, the brightness allows more accurate corrections as the number of molecules N_mol decreases. We also note that, like the signal-to-noise ratio, the residual aberration using the count rate mainly depends on the number of molecules as it is rather flat on the graphs (plotted as a function of the brightness). In contrast, the residual obtained with the brightness varies chiefly with the brightness itself as it is almost unchanged for the four concentrations.

To theoretically predict the residual aberration, we first determine the uncertainty of the mode amplitude estimation at each correction step. Since we are concerned with small aberrations (<0.5 rad), the quadratic approximation (Eq. (5)) is considered valid. We also assume that Zernike modes, which we used here, are orthogonal for both metrics. Although this is not strictly true, our experimental tests show that mode coupling are weak and could be overlooked when several correction loops are performed.

Let us consider a single correction step: a mode of initial amplitude a_m is tested. A number N_mes (3 in our case) of preset bias amplitudes x_i (i = 1...N_mes) are sequentially introduced by the DM and the metric is measured for each DM shape. Then a_m is determined by a least-square fit of the metric values y_i with a function of the form (at the bias amplitudes x_i): y_i == p − q(x_i − a_m)² (where, following the notation of Eq. (5), $p=M_0\left(1-\beta \left({\sigma }^2-a_m^2\right)\right)$ and q = M₀β). We note f the fit function: y_i = f(x_i, p, q, a_m).

The accuracy of the estimated value of a_m depends on the noise of the measurements y_i which have been discussed in section 3.2. To determine the variance of a_m, we first write the Jacobian matrix that relates small variations of the measured values y_i to that of the function parameters (p, q and a_m).

By inverting J, the variation of the parameters can be expressed as a function of the measurement noise. We note L the following matrix (J^T is the transpose of J):

If the errors on the measured y_i values are assumed to be uncorrelated (the corresponding variance-covariance matrix is then diagonal), the variance of the estimated amplitude a_m is given by

Therefore, the error on the estimation of a_m is proportional to the measurement noise with prefactors that depend on the derivatives of the fit function. In particular, the factors L_3i depend on the inverse of q. Thus, the error Δa_m scales approximately as (S/N × β)⁻¹ with S/N the signal-to-noise ratio of the metric and β the sensitivity. That is why, due to its higher sensitivity, the brightness can allow more accurate corrections in some cases, even though its measurement is always more noisy than that of the count rate.

We simulated the whole correction process. At the beginning, the signal level (count rate or brightness) and signal-to-noise ratio of the metric is set by the amount of initial rms aberration. Then each mode is sequentially optimized. At the end of the first step, the mode that has been corrected has a amplitude statistically distributed around zero with a width given by the error Δa_m. The process is repeated for the next mode until N_modes (N_modes=10 in this experiment) are corrected. In the calculation, we assumed that the measurement noise does not change as successive modes are corrected within one loop. Therefore, the rms amplitude of the residual aberration is: $\mathit{rms}=\sqrt{N_{\mathit{modes}}}$ Δa_m at the end of the loop. This new rms is used to compute the S/N of the metric for the next loop, which results in a updated error per mode Δa_m. The process is iterated 3 times since 3 loops are used in the experiments.

The assumption that the measurement noise remains constant during one loop does not change substantially the results here since aberrations are uniformly distributed on all corrected modes, but it would not be valid if some modes are predominant (such as spherical aberration), which is often the case in real experiments. In the latter case, correcting first the predominant modes would improve substantially the total rms and hence the S/N of subsequent measurements. Another remark could be made: attempting to correct modes that are initially absent in noisy conditions leads to an increase of residual aberration. Therefore, predicting a priori the nature of aberrations encountered in specific samples would help improve the correction efficiency. Some work has been performed in this direction [23].

As shown on Fig. 4(B), the residual aberration calculated in this way is in good agreement with experimental results. A fixed error E₀ = 2 nm has been added to account for errors in the estimation of system aberrations, which was performed before starting the experiment. The 2D maps depicted in Fig. 5(A) are obtained by computing the residual aberration in a range of brightness values and sample concentrations. These maps bear close resemblance to the S/N maps of Fig. 2(B): the correction accuracy obtained with the count rate mainly depends on the number of molecules while the one obtained with the brightness as metric depends on the brightness. To better compare the performance of the two metrics, Fig. 5(B) show the ratio of the residual aberration with the brightness to that obtained with the count rate. If this ratio is smaller than 1, the brightness allows more accurate AO corrections, while if it is larger than 1, it is more advantageous to use the count rate as metric. The brightness metric leads to higher correction accuracy only in the bottom right corner of the map corresponding to rather bright molecules in a dilute solution. These conditions are relevant for typical FCS experiments in solution where nM-solutions are used and the brightness can reach tens of Hz/molecule. For more concentrated or less bright molecules, which is often the case of proteins in live cells, the count rate should allow more accurate corrections.

 

Fig. 5 Predicted AO correction accuracy. (A) Calculated 2D maps of residual aberrations in rad as a function of the molecular brightness and the number of molecules, when using the count rate (left) or the brightness (right) as metric. The black dashed lines show the concentrations of the solutions used in the experiments. The correction accuracy needed for a Strehl ration of 0.9 is shown by a dashed white line. (B) Ratio of the calculated residual with the brightness metric to the one obtained with the count rate metric. When this ratio is smaller than 1, brightness leads to more accurate correction. When it is larger than 1, count rate yields more accurate correction.

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5. Correction of large aberrations

A metric with high sensitivity, such as the molecular brightness, is advantageous in the small aberration range, since sensitivity can compensate for lower S/N and improve correction accuracy. However, this high sensitivity causes the brightness to vanish when the rms amplitude is above 1 rad, as shown on Fig. 3(A), suggesting that it would not be appropriate for correcting large aberrations. Here, we discuss the possibility to correct larger aberrations and the most adequate metric in this case.

First, we investigated how the signal-to-noise ratios of the two metrics depend on aberration magnitude. The expected S/N for both count rate and brightness were calculated as described in section 3.2 using measured values of the count rate, number of molecules and diffusion time at various aberration amplitudes (for these measurements, the DM was used to produce 20 sets of random aberrations at each rms amplitude). As shown on Fig. 6(A), while the expected S/N of the brightness decreases with aberration magnitude, the S/N of the count rate increases as aberration becomes larger. This counter-intuitive trend can be explained by considering that, as shown previously, the noise of the count rate is dominated by molecular ’occupation’ noise, especially in such a dilute sample (N_mol=1.6). As a consequence, the S/N mainly depends on the number of molecules in the observation volume. When aberration magnitude increases, the observation volume is enlarged and contains a larger number of molecules, resulting in a reduced ’occupation’ noise. Hence, count rate measurements tend to be more accurate for larger aberrations, while the S/N of brightness measurements rapidly collapses with the brightness value.

 

Fig. 6 Performance of the two metrics in case of large aberrations. (A) Expected signal-to-noise ratio of Count Rate (red circles) and Brightness (blue triangles) measurements as a function of total aberration amplitude. These S/N are calculated from measured values of CR, brightness and τ_D (Fig. 3) obtained for 20 random aberrations at each total rms amplitude. (B) Expected error on each mode estimation as a function of the total aberration amplitude, calculated with 3 measurements per mode and 0.5 rad bias, using either the Count Rate (red circles) or the Brightness (blue triangles) as metric. If the aberration is uniformly distributed over N_modes modes, an improvement can only be obtained if the uncertainty for one mode is below $\mathit{rms}=\sqrt{N_{\mathit{modes}}}$. The corresponding area is shown for N_modes = 10 (left) and N_modes = 50 (right). For all graphs in this figure, the number of molecules is N_mol=1.6 and the brightness is 37 kHz/molecule without aberration.

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When starting the AO optimization, the amount of initial aberration defines how accurately the metric can be measured and thus the error on the amplitude of the first corrected mode. If this error is larger than the initial amplitude of this mode, the total amount of aberration is not reduced. In this case, the whole correction would be inefficient, if all tested modes have similar initial amplitude (i.e. the total rms aberration is uniformly distributed). Hence, the result of this first step can be used as a criterion to determine whether AO can improve optical quality.

In Fig. 6(B), the calculated error of the first optimization step is shown as a function of the initial rms aberration. This error is obtained with the formalism described in the previous section. (However, we did not use a quadratic approximation, no longer valid for large aberration. Instead, a Gaussian function was considered in the least square fits.) The AO correction will lead to an improvement if this error is smaller than the initial amplitude of one mode. If the aberration is uniformly distributed on all N_modes modes, each mode has an amplitude of $\mathit{rms}=\sqrt{N_{\mathit{modes}}}$, where rms is the rms amplitude of the total aberration. This is shown as a line in Fig. 6(B): there will be no progression if the error is in the area depicted in gray.

When N_modes=10, both the count rate and the brightness can reduce aberrations, although in different rms ranges. The brightness leads to better accuracy (this case corresponds to few but bright molecules) and can be used in the small aberration range where the lower correction accuracy provided by count rate would not improve optical quality. However, as the aberration magnitude increases, the brightness correction error rises as its S/N decays. The range over which the count rate is efficient extends to larger aberrations. Ultimately, for even larger amplitude, the estimation error grows as the measurements are too far away from the optimum.

The range over which aberration correction is efficient also depends on the complexity of the initial wavefront. In the case where aberrations are spread over more numerous modes (right side of Fig. 6(B) corresponding to N_modes=50), the gray area covers almost all the curves, showing that correction is mostly inefficient. Indeed, since each mode only bears a small fraction of the total aberration, extremely accurate corrections are needed to reduce aberrations mode by mode. Due to the higher accuracy of brightness-based corrections in the present case, it can still be used in a small region.

Finally, the count rate is usually a more advantageous metric for large aberrations since its S/N is not reduced when aberration magnitude increases, contrary to the brightness. The range of amplitude that can be corrected for given conditions (sample concentration, brightness) also depends on the mode decomposition of the aberration of interest: easier situations are when a few modes bear most of the aberration, while aberrations spread over a high number of modes require very accurate measurements if they are to be corrected.

6. Conclusion

We presented a general theoretical framework to predict the correction accuracy in a standard sensorless AO scheme where the amplitude of each mode is sequentially estimated by fitting metric measurements at given bias amplitudes. We show that the error on the estimation of a mode scales inversely with the signal-to-noise ratio of the measured metric times its sensitivity.

We investigated the case of AO correction in a fluorescent solution which is particularly relevant to fluorescence fluctuation microscopy. In this case, two quantities can be readily used as optimization metrics, the count rate and the molecular brightness. We compared these quantities in terms of measurement noise and sensitivity in the small aberration range. Although the brightness measurement is always more noisy than that of the count rate, a higher sensitivity allows the brightness to reach better correction accuracies in samples containing dilute but relatively bright molecules.

In relatively dilute solutions, we found that the count rate S/N is not degraded (it may even improve) as aberration amplitude increases, while the brightness S/N decreases as the brightness itself. Consequently, the optimization progresses differently between the two metrics. When using the count rate as metric, one loop through all the modes is sufficient to reach the best correction (provided mode coupling can be neglected), whereas, when the brightness is used as metric, one should consider an iterative process: each loop reduces the aberration, hence improves the measurement S/N, which in turn allow residual aberrations to be further reduced in the next loop. Several loops may be required to reach the best correction. Besides, due to the better persistence of its S/N, the count rate is a more reliable metric in the case of large aberrations. Both these remarks suggest that using the count rate as metric in the first loop to quickly reduce the amount of aberration is an interesting strategy. Depending on the sample and experiment conditions, additional optimization loops can be performed with the brightness to reach a better correction accuracy.

It is important to keep in mind that a mode can only be corrected with a limited accuracy, so that attempting to correct a mode that has a small initial amplitude may degrade the total rms. Therefore, it is desirable to reduce the number of corrected modes to those with substantial amplitude. Computational approaches that use prior knowledge on the sample geometry to predict those modes are promising as a complement to sensorless AO correction.