Localizing a fluorescent particle by scanning a focused laser beam in its vicinity and analyzing the detected photon stream provides real-time information for a modern class of feedback control systems for particle tracking and trapping. We show for the full range of standard merit functions based on the Fisher information matrix (1) that the optimal path coincides with the positions of maximum slope of the square root of the beam intensity rather than with the intensity itself, (2) that this condition matches that derived from the theory describing the optimal design of experiments and (3) that in one dimension it is equivalent to maximizing the signal to noise ratio. The optimal path for a Gaussian beam scanned in two or three dimensions is presented along with the Cramér-Rao bound, which gives the ultimate localization accuracy that can be achieved by analyzing the detected photon stream. In two dimensions the optimum path is independent of the chosen merit function but this is not the case in three dimensions. Also, we show that whereas the optimum path for a Gaussian beam in two dimensions can be chosen to be continuous, it cannot be continuous in three dimensions.
© 2012 OSA
We address the general problem of determining the position of a fluorescent particle by scanning a focused laser beam in its vicinity (see Fig. 1) and detecting, on a single-pixel (i.e. non-imaging) detector such as a single-photon counter, the fluorescent photons generated by the particle. This method provides a modern, real-time alternative to image-based single-particle localization in optical microscopy [1–3] - which has been widely applied for studying colloidal and biophysical systems [4–7] and achieving nanometer-scale “super-resolution” single-molecule images  - and is critical to a new class of measurement technique [9–11] wherein a single fluorescent particle is located in real time while a feedback controller tracks or traps the particle to maintain its position in the observation volume of a microscope. Each experimental implementation uses a different combination of laser scanning or multiple detector placement, sample scanning, and localization algorithm to accomplish a real-time position measurement in two [12–18] or three [19–25] dimensions. While the referenced experimental implementations have each been used successfully, optimal strategies have been identified only for restricted cases of the laser scan pattern or estimation algorithm: under a Gaussian noise approximation the optimal circular scan path for two-dimensional localization was found in [26, 27]; under a Gaussian noise approximation and for a particular estimator - the fluoroBancroft algorithm -an optimal three-dimensional laser scan path was found in ; a complex two-dimensional scan geometry was studied in  for the case of a particle that cannot be localized within the “linear” estimator range of the focused beam. To date, however, a unified, global optimality strategy has not been identified nor have the fundamental limits to localization accuracy been established. We solve this problem here for scanned non-imaging systems like those described in [10, 13, 20, 22]. We show that for every standard scalar merit function that can be defined using the Fisher information matrix, the optimum trajectory is consistent with maintaining the particle at the positions of maximum slope of the square root of the intensity, rather than at the maximum slope of the intensity itself, but that the optimum time spent at these positions does depend on the particular choice of merit function. We also discuss how this is related to maximizing the signal to noise. For Gaussian beams we show that these results exactly match those that can be derived from the theory of the optimal design of experiments. Based on these results we introduce a simple test that can be applied to determine the degree of optimality of any experimental design with mathematical rigor. This test allows a user to ascertain quickly whether a particular laser scan path (in two or three dimensions) provides maximum information about a particle’s position for a particular focused-beam geometry. These results hold under the assumption that the particle is effectively stationary during the scan period which does not necessarily suffice for the situation studied in , where the beam is scanned over a relatively large range to accommodate a very fast moving particle. A more general and difficult problem, not addressed here, is to devise a strategy for locating a particle that explores a significant fraction of the excitation geometry during the scan time. Under this quasi-stationary assumption we solve the optimal scan path problem completely.
Specifically we find that for a Gaussian beam in two dimensions (2D) the fundamental bounds on the localization accuracy are30] and Nph the (average) number of photons collected during the scan time. Note that Eq. (1) is identical to the standard image-based result when a Gaussian point-spread function is assumed [1, 2], demonstrating the equivalent information content of a diffraction-limited image and the photon stream from an appropriately scanned diffraction-limited excitation beam.
For a Gaussian beam in three dimensions (3D) we find that for one particular choice of optimization function
As stated above, these limits apply to scanned non-imaging type systems such as those described in [10,13,20,22]. The one- and two-dimensional Cramer-Rao bounds for imaging systems where the point spread function, assumed to be Gaussian, generated by a fluorescing particle or molecule is projected onto a detector array were derived many years ago by Winnick . There is no result equivalent to Eq. (2) for image-based three-dimensional localization systems, such as those described in [32–35]. For imaging systems there is of course no trajectory to optimize but the approach used here to determine optimality and derive the Cramer-Rao bound, i.e., the Fisher information, can (and should) be evaluated for these system and the result compared to the bound given above.
The remainder of the paper is devoted to a derivation and discussion of these results. We specify the problem in terms of the Fisher information matrix, give global optimality conditions, and identify scan paths for the familiar Gaussian beam profiles in two- and three-dimensions. In Appendix A we justify the global optimality condition and in Appendix B we derive the linearized maximum likelihood estimator based on the detected photon stream and foreknowledge of the beam shape (gradients and curvatures of the intensity profile), which can be applied in experimental hardware for real-time localization.
2. Theory of optimal design of experments
Consider a laser beam with a position dependent intensity distribution I(r) where r = (r1, r2, r3) = (x, y, z). When the beam intensity is shifted to position rL and a fluorescent particle, is located at position r, the intensity at the particle position is I(r − rL) and so the (average) rate at which fluorescent photons are generated is given by ξI(r − rL) where ξ is the fluorescence cross-section (area) of the particle. Letting Γ = ξI with I in units of (incident photons)/(area×time) it follows that Γ has units of (fluorescent photons)/time. (Note: Although it is convenient to think of rL as the position of peak intensity or as the centroid of the beam, this is not necessary. The solution to the optimization problem will automatically take whatever position definition is used into account.) Now suppose the beam is scanned over a time-dependent path rL(t) for a time period τ. Our task is to determine which scan path encodes the most information about the fluorescent particles position r in the detected fluorescent photon arrival times; that is, which path enables the best unbiased estimate of r?
Fluorescently generated photons obey a Poisson distribution, i.e., over a time interval Δt for which rL (t) is approximately constant, the probability for detecting n photons when the particle is at position r is given by pn (t) Δt = (Γ(r − rL(t))Δt)n/n!exp[−Γ(r − rL(t))Δt]. The probability of counting n photons is proportional to Δtn and hence in the limit as Δt becomes extremely small, the probablity of counting more than one photon in time interval t − Δt/2 to t + Δt/2 effectively vanishes. The probability of counting no photons in the same interval is exp[−Γ(r − rL(t))Δt]. It follows from this that the probability of counting 1 photon in the time intervals tk − Δt/2 to tk + Δt/2 for k = 1, 2, ⋯K and t1 < t2 < ⋯ < tK and no photons in any time interval between t = 0 and τ = NΔt is given by p1 (t1) Δt p1 (t2) Δt ⋯ p1 (tK) Δt exp . Hence over a finite time τ = NΔt the statistical description of the measurement process is given by the the probability of observing K photon arrival times tk36], F which for p(t1, ...,tk|r) given above has j,k elements given by 36]. Thus, we seek the scan path rL(t) that maximizes F and correspondingly minimizes the covariance V. For one-dimensional estimation, this is a straightforward scalar maximization task, but in higher dimensions we must choose a scalar quantity that characterizes the “size” ϕ [F] of the matrix F. (Below we show that for the 1D case, maximizing F maximizes the signal to noise ratio. In higher dimensions there is more than one signal and choosing ϕ [F] is equivalent to choosing what function of these signals is to be maximized relative to the noise ) Two common choices for quantifying the size of F are the determinant ϕ0 [F] = det [F]1/d and the trace of its inverse ϕ−1 [F] = dTr [F−1]−1, which in turn bound the determinant and trace of the covariance matrix V. The trace of the covariance matrix, and hence ϕ−1 [F], is particularly important for our case since it is proportional to the localization accuracy (e.g. for D = 3, ). The functions ϕ0 and ϕ−1 are only two examples of the more general matrix information function ϕp used in , and defined for all p ≠ 0 by 37]. It is important to note that, except in special cases of high symmetry, the optimal scan path maximizing ϕp [F] depends on the choice of p as we will show explicitly below for a Gaussian beam in three dimensions.
For any pair of fluorophore coordinates xj and xk, a straightforward computation of F using the above definition yields
The theory of optimal experimental design provides a very general and extensive analysis of the solution to this type of problem . For the ϕp optimality criteria defined above an experimental design with associated Fisher matrix F* is ϕp-optimal if and only ifFig 2. We provide a justification for this form of the optimality condition in Appendix A.)
Substituting for g, we can also write that a laser scan path is ϕp-optimal for any finite p ≤ 1 if and only ifEq. (10)
3. Solution via the calculus of variations
F is a functional of the laser scan trajectory rL (t) and so the optimum trajectory with respect to ϕp [F] defined in Eq. (6) is the one for which the change in ϕp [F] vanishes to first order in δrL (t) when rL (t) → rL (t) + δrL (t). Of course this condition only gives an extremum of ϕp [F] and we must separately determine that a given solution maximizes ϕp [F]. Carrying out the variation and setting the result to zero yields
In 1 dimension (1D) F is a non-negative scalar, i.e., F = F and assuming it does not vanish Eq. (11) reduces to30], and we have ∂x2 a (x − xL) = 0 for with both solutions having the same value of |∂xa|. ϕp [F] is therefore a maximum for .where is the mean number of photons collected during the scan. Obviously |∂xa (x)| needs to maintain a large value over the range of uncertainty in the particle position Δx, i.e., |∂xa (x − εΔx − xL)| should be approximately constant, and large, for −1 ≲ ε ≲ +1.
Next consider the 1D case from the point of view of the intensity rather than the amplitude, i.e., from the point of view of the first line of Eq. (7). In this form F is maximized at positions where the slope of the intensity, ∂xI, is a maximum with the intensity I itself being a minimum. At first requiring I to be a minimum seems counterintuitive. But the signal amplitude is on the order of ∂xI × Δx while the absolute noise level is proportional to and so the signal to noise ratio for a given Δx is maximized at positions where is a maximum. Note that maximizing is equivalent to maximizing its square, , since . Hence maximizing F in 1D is equivalent to maximizing the signal to noise ratio.
This 1D solution generalizes directly to 2D and 3D. Equation (11) can be satisfied by choosing positions that have ∂i∂ja (r) = 0 for all i and j. Again the beam can be held stationary at a sufficient number of these positions during the scan time although a continuous trajectory which maintains these conditions may be easier to implement mechanically and/or optically. The condition ∂i∂ja (r) = 0 for i = j is exactly the same as the condition ∂2a(x) = 0 in 1D. For i ≠ j this condition effectively amounts to having the gradients of a(r) at the chosen positions be mutually orthogonal But, as opposed to 1D where the entire measurement time τ can be spent with the beam locked at one position, in 2D and 3D it is not immediately clear how to divide up the time among the different positions. To be specific let the positions which maximize ϕp [F] be , with s ranging from 1 to D where D is the number of dimensions and assume that ∇a at each point separately aligns with one of the coordinate axes so that points purely in the rs direction, i.e., points purely in the x direction, and so on, then
We now apply this solution technique in 3D to a Gaussian beam which has a field amplitude given by Eq. (12) yields dwell times at and which are both equal to τ/2.. For localization in all three directions the difference between the slope in the z direction and those in the x and y directions causes the dwell times to depend on p. For p = −1 Eq. (12) we get
In both 2D and 3D the ± signs lead to minor ambiguity in the particle position since nominally one cannot tell which side of the beam the particle is on. In many cases only the movement of the particle relative to it’s starting position is required and so the absolute position is not required. But the ambiguity can be lifted in any case simply by dithering the beam position slightly in each direction and determining the sign of the change in signal level.
In the next section we show that the 2D solution for arbitrary p and the 3D solution for p = −1 derived here are exactly the same as those given by Eq. (10)
3.1. Gaussian beam in two dimensions via optimal design
In the two-dimensional case, the experimental design problem is straightforward. The beam geometry assuming the particle is maintained at the beam waist (z = 0) is withe with I0 the peak intensity in terms of the number of photons/(area × time) . When this beam is offset by a position shift of r = (x, y) from the origin the regression vector g is given by
Note that the same Fisher matrix is achieved for any pair of orthogonal vectors or for a constant-speed circular scan about the origin so long as the scan points lie on the circle with radius .
For any other laser focal position r′ and for any finite p ≤ 1, the inequality that must be satisfied, Eq. (10), becomes
The Cramér-Rao bound on the two-dimensional localization accuracy corresponding to p = −1 becomes , so thatEq. (1). Note that due to the symmetry of the Gaussian in 2D this result is independent of p.
3.2. Gaussian beam in three dimensions via optimal design
The situation is more complex in three dimensions. Consider the beam profile given by the usual expression 
According to the optimality criteria, the scan path is optimal if and only if f (r) ≤ 1 for all r. Writing r = (x,y,z) in convenient dimensionless units where , r̄ = r/w0 and z̄ = z/zR we findFig. 2, where it is clearly seen not to exceed the value 1 (this can also be shown analytically). Thus, the proposed scan path is ϕ−1-optimal and therefore the three-dimensional localization is always limited by the Cramér-Rao bound, taking here the form Fig. 2, are separated by a valley (f < 1) there is no smoothly varying continuous path in 3D that is optimal for a Gaussian beam. This is in contrast to the 2D case with a Gaussian beam where all the points on the circle have f = 1 and hence a continuous path can be used if desired.  The fact that no smoothly varying continuous path is possible in 3D has obvious implications with respect to the practical implementation of these results. True optimality requires the beam to hop instantaneously between z = 0 with and r = 0 with . In a strict sense this is not possible physically but obviously any system with a hopping time which is tiny fraction of τ will for all practical purposes be optimal.
Computing the number of photons collected along the scan path during a single period from , we can rewrite the best possible three-dimensional localization accuracy, achieved for the optimal scan path given by the weighted solution as30], we recover Eq. (2), proving the initial claim.
We have used the calculus of variations to derive the optimum scan path for tracking and localizing a fluorescent particle and have shown that for Gaussian beams in two and three dimensions the calculus of variations result matches the solution derived from the optimal design of experiments. In one dimension this condition corresponds simply to maximizing the signal to noise ratio. In higher dimensions there are multiple signals, essentially one for each direction, and depending on how these are combined into a single signal there are different optimization criteria which is equivalent to having to choose the value of p in the merit function ϕp [F]. These results provide a simple, testable optimality criterion to determine whether a candidate laser scan path encodes maximal information about a fluorescent particles position in the detected photon stream. We presented optimal scan paths for two- and three-dimensional Gaussian beams and used these to derive the best possible localization accuracies, quoted in the introduction. We have shown that the optimal path for 2D localization using a Gaussian beam can be continuous if desired, but the optimal path in 3D for a Gaussian cannot be continuous. These results can be applied to other experimental geometries, including those where multiple detectors - rather than multiple beam positions - are used for real-time localization. Future work should focus on relaxing the assumption that the particle remains effectively stationary during each scan cycle so as to extend optimality results to cases where the particle is moving under a particular dynamic model (for example, free diffusion or diffusion plus flow) or where feedback control may not be sufficiently tight that the particle is well-localized relative to the beam size. Also, it would be worthwhile to determine if there are physically realizable intensity distributions which do allow for the optimal path to be continuous as this might aid the practical implementation of these results.
Appendix A: Justification of the global optimality criterion
The rigorous proof of the optimality condition Eq. (9) is rather complex and will not be presented here. Instead we will present a justification for it. By definition F* is ϕp optimal relative to all other F if and only ifEq. (6) into the above condition, raising both sides to the power p and cancelling factors of 1/d gives
Appendix B: Maximum likelihood position estimation for an arbitrary scan path
In earlier sections, we derived design criteria for determining which laser scan path contains the most information about a particle’s position, when the particle is near the origin of coordinates. In general, however, we also require an estimation procedure that can extract the position from the detected photon stream. This position estimator must, of course, include some information about the beam size, shape, and the scan path. One candidate is a maximum likelihood estimator, whose performance will be uncertain when the photon number is very small but will tend towards optimality for large photon numbers (how large cannot be determined a priori and is a common criticism of maximum likelihood). In this appendix, we derive a simple linear form for the maximum likelihood estimator of a fluorescent particle’s position for an arbitrary (2D or 3D) scan path, under the assumption that the particle position r is close to the origin. To do this, we can expand the time-detection rate function to second order in r asEq. (4) and setting the gradient with respect to particle position r to zero, we find the following linear equation for the maximum likelihood estimate rMLE of the particle position r:
As a simple example, consider the two-dimensional Gaussian beam example of Sect. with the circular scan path . By direct computation, we find26].
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