## Optimal laser scan path for localizing a fluorescent particle in two or three dimensions |

Optics Express, Vol. 20, Issue 15, pp. 16381-16393 (2012)

http://dx.doi.org/10.1364/OE.20.016381

Acrobat PDF (850 KB)

### Abstract

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

## 1. Introduction

*σ*is the standard deviation in position measured along

_{k}*k*∈ {

*x*,

*y*,

*z*},

*w*

_{0}is the focused Gaussian beam waist [30] and

*N*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

_{ph}1. R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. **82**, 2775–2783 (2002). [CrossRef] [PubMed]

2. R. Ober, S. Ram, and E. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. **86**, 1185–1200 (2004). [CrossRef] [PubMed]

*λ*is is the wavelength. Making the following Gaussian approximation to the focused Airy diffraction pattern,

*w*

_{0}≈ 0.4

*λ*/NA, where NA = sin[

*θ*

_{max}] with

*θ*

_{max}being the maximum angle the light illuminating the sample makes with respect to the

*z*axis, the three-dimensional localization accuracy becomes

10. H. Cang, C. Shan Xu, and H. Yang, “Progress in single-molecule tracking spectroscopy,” Chem. Phys. Lett. **457**, 285–291 (2008). [CrossRef]

13. A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence correlation spectroscopy of individual particles,” Opt. Express **13**, 8069–8082 (2005). [CrossRef] [PubMed]

20. V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. **88**, 2919–2928 (2005). [CrossRef] [PubMed]

22. K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. **7**, 3535–3539 (2007). [CrossRef] [PubMed]

31. K. A. Winnick, “Cramer-Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A **3**, 1809–1815 (1986). [CrossRef]

32. H. Kao and A. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. **67**, 1291–1300 (1994). [CrossRef] [PubMed]

35. M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, “3D Particle trajectories observed by orthogonal tracking microscopy,” ACS Nano **3**, 609–614 (2009). [CrossRef] [PubMed]

## 2. Theory of optimal design of experments

*I*(

**r**) where

**r**= (

*r*

_{1},

*r*

_{2},

*r*

_{3}) = (

*x*,

*y*,

*z*). When the beam intensity is shifted to position

**r**

_{L}and a fluorescent particle, is located at position

**r**, the intensity at the particle position is

*I*(

**r**−

**r**

_{L}) and so the (average) rate at which fluorescent photons are generated is given by

*ξI*(

**r**−

**r**

_{L}) 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

**r**

_{L}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

**r**

_{L}(

*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**?

*t*for which

**r**

_{L}(

*t*) is approximately constant, the probability for detecting

*n*photons when the particle is at position

**r**is given by

*p*(

_{n}*t*) Δ

*t*= (Γ(

**r**−

**r**

_{L}(

*t*))Δ

*t*)

*/n!exp[−Γ(*

^{n}**r**−

**r**

_{L}(

*t*))Δ

*t*]. The probability of counting

*n*photons is proportional to Δ

*t*and hence in the limit as Δ

^{n}*t*becomes extremely small, the probablity of counting more than one photon in time interval

*t*− Δ

*t*/2 to

*t*+ Δ

*t*/2 effectively vanishes. The probability of counting no photons in the same interval is exp[−Γ(

**r**−

**r**

_{L}(

*t*))Δ

*t*]. It follows from this that the probability of counting 1 photon in the time intervals

*t*− Δ

_{k}*t*/2 to

*t*+ Δ

_{k}*t*/2 for

*k*= 1, 2, ⋯

*K*and

*t*

_{1}<

*t*

_{2}< ⋯ <

*t*and no photons in any time interval between

_{K}*t*= 0 and

*τ*=

*N*Δ

*t*is given by

*p*

_{1}(

*t*

_{1}) Δ

*t p*

_{1}(

*t*

_{2}) Δ

*t*⋯

*p*

_{1}(

*t*) Δ

_{K}*t*exp

*τ*=

*N*Δ

*t*the statistical description of the measurement process is given by the the probability of observing

*K*photon arrival times

*t*Here the arrival times

_{k}*t*are taken to be unordered by which we mean they are not required to obey

_{k}*t*

_{1}<

*t*

_{2}< ⋯ <

*t*The factor 1/

_{K}*K*! accounts for the switch from ordered to unordered. The product over

*k*is understood to be unity for

*K*= 0 and we have replaced

**r**of a particle in

*D*dimensions contained in a scan of the laser position

**r**

_{L}(

*t*) is quantified by the associated

*D*×

*D*Fisher information matrix [36],

**F**which for

*p*(

*t*

_{1,}...,

*t*|

_{k}**r**) given above has

*j,k*elements given by where

*∂*≡

_{j}*∂*/

*∂r*. The Cramér-Rao bound is the statement that the best unbiased estimator of

_{j}**r**has a covariance matrix given by

**V**=

**F**

^{−1}[36]. Thus, we seek the scan path

**r**

_{L}(

*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

*ϕ*

_{0}[

**F**] = det [

**F**]

^{1/d}and the trace of its inverse

*ϕ*

_{−1}[

**F**] =

*d*Tr [

**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,

*ϕ*

_{0}and

*ϕ*

_{−1}are only two examples of the more general matrix information function

*ϕ*used in [37

_{p}37. F. Pukelsheim, *Optimal design of experiments* (Society for Industrial and Applied Mathematics, 2006). [CrossRef]

*p*≠ 0 by where “

*·*” indicates matrix multiplication. These provide a sensible measure of the information content for all

*p*≤ 1 [37

37. F. Pukelsheim, *Optimal design of experiments* (Society for Industrial and Applied Mathematics, 2006). [CrossRef]

*ϕ*[

_{p}**F**] depends on the choice of

*p*as we will show explicitly below for a Gaussian beam in three dimensions.

*x*and

_{j}*x*, a straightforward computation of

_{k}**F**using the above definition yields Using

*a*(

**r**) the modulus of the field amplitude we can define where ∇ is the gradient, i.e., ∇

*=*

_{i}*∂*. Treating

_{i}**g**as a column vector and indicating the transpose with a superscript T, If the beam is moved in

*N*discrete steps, dwelling at each position

*t*, we can define a vector

_{n}**g**

*for each laser position and write the Fisher information matrix simply a This is the same Fisher information matrix obtained in a classical linear regression model where the unknown particle position*

_{n}**r**is projected onto a set of regression vectors

**g**

*with corresponding weights*

_{n}*c*, with observations corrupted by zero-mean measurement errors of unit magnitude. In this representation, the

_{n}*length*of a regression vector determines the

*precision*of a measurement along that direction. The optimal experimental design problem is to choose a set of

*N*vectors

**g**

*and corresponding weights*

_{n}*c*- or equivalently, a set of

_{n}*N*laser positions

*t*- that maximizes the information matrix

_{n}**F**relative to the criterion

*ϕ*[

_{p}**F**].

37. F. Pukelsheim, *Optimal design of experiments* (Society for Industrial and Applied Mathematics, 2006). [CrossRef]

*ϕ*optimality criteria defined above an experimental design with associated Fisher matrix

_{p}**F**

*is*

_{*}*ϕ*-optimal if and only if for

_{p}*all*possible regression vectors

**g**; in this case all possible scan paths, with equality being achieved only for vectors

**g**that are part of an optimal scan path. (For a Gaussian beam in 3D this is shown explicitly below in Section 3.2, see in particular Fig 2. We provide a justification for this form of the optimality condition in Appendix A.)

**g**, we can also write that a laser scan path is

*ϕ*-optimal for any finite

_{p}*p*≤ 1 if and only if for all

**r**; again, equality is achieved only for

**r**values that lie on an optimal scan path. Any laser scan path can be tested for optimality by computing the information matrix

**F**and testing for optimality using this criterion. Unfortunately, although it is reasonably easy to apply, it is not at all obvious how this process yields an optimum scan path. So in order to gain insight into the solution we will first solve the optimization problem by the more conventional approach of using the calculus of variations. This will not only specify the optimum scan path in a obvious way, it will also provide significant insight into the solution including indicating how to alter the intensity distribution Γ to improve the tracking accuracy. We will then show that for the Gaussian beam this yields precisely same result that is found using Eq. (10)

## 3. Solution via the calculus of variations

**F**is a functional of the laser scan trajectory

**r**

_{L}(

*t*) and so the optimum trajectory with respect to

*ϕ*[

_{p}**F**] defined in Eq. (6) is the one for which the change in

*ϕ*[

_{p}**F**] vanishes to first order in

*δ*

**r**

_{L}(

*t*) when

**r**

*(*

_{L}*t*) →

**r**

_{L}(

*t*) +

*δ*

**r**

_{L}(

*t*). Of course this condition only gives an extremum of

*ϕ*[

_{p}**F**] and we must separately determine that a given solution maximizes

*ϕ*[

_{p}**F**]. Carrying out the variation and setting the result to zero yields with repeated indices,

*j*,

*k*,... summed over the appropriate range and we have used that fact that

**F**is symmetric..

**F**is a non-negative scalar, i.e.,

**F**=

*F*and assuming it does not vanish Eq. (11) reduces to for all

*p*which shows that

*ϕ*[

_{p}**F**] is maximized at positions

*x*(

_{L}*t*) where

*∂*

_{x}^{2}

*a*(

*x*−

*x*

_{L}(

*t*)) = 0 with

*|∂*(

_{x}a*x*−

*x*

_{L}(

*t*))

*|*≠ 0. But

*∂*

_{x}^{2}

*a*(

*x*−

*x*

_{L}(

*t*)) = 0 is simply the condition that

*|∂*(

_{x}a*x*−

*x*

_{L}(

*t*))| is, neglecting inflection points, a maximum. Interestingly this does not correspond to the maximum slope of the intensity itself. Thus we can improve the localization accuracy by maximizing the absolute slope beam amplitude and if there are multiple positions where

*∂*

_{x}^{2}

*a*(

*x*−

*x*

_{L}(

*t*)) = 0 then the global optimum is achieved by using the one with the largest value of |

*∂*(

_{x}a*x*−

*x*

_{L}(

*t*))|. Under the assumption that during the scan time the particle position

*x*is essentially constant it follows that

*x*(

_{L}*t*) can also be held constant. For a Gaussian beam in 1D [30],

*∂*

_{x}^{2}

*a*(

*x*−

*x*

_{L}) = 0 for

*∂*|.

_{x}a*ϕ*[

_{p}**F**] is therefore a maximum for

*∂*(

_{x}a*x*)| needs to maintain a large value over the range of uncertainty in the particle position Δ

*x,*i.e., |

*∂*(

_{x}a*x*−

*ε*Δ

*x*−

*x*

_{L})

*|*should be approximately constant, and large, for −1 ≲

*ε*≲ +1.

*F*is maximized at positions where the slope of the intensity,

*∂*, is a maximum with the intensity

_{x}I*I*itself being a minimum. At first requiring

*I*to be a minimum seems counterintuitive. But the signal amplitude is on the order of

*∂*Δ

_{x}I ×*x*while the absolute noise level is proportional to

*x*is maximized at positions where

*F*in 1D is equivalent to maximizing the signal to noise ratio.

*∂*(

_{i}∂_{j}a**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}∂_{j}a**r**) = 0 for

*i*=

*j*is exactly the same as the condition

*∂*

^{2}

*a*(

*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

*ϕ*[

_{p}**F**] be

*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

*r*direction, i.e.,

_{s}*x*direction, and so on, then with the constraint that

*t*at each position can be chosen by solving Note that if the values of

_{p}*t*are equal to

_{s}*τ*/

*D*independent of the value of

*p*.

*λ*is the wavelength and

**r**is approximately zero on the scale of the width of the Gaussian. Then the positions of center of the Gaussian beam that have the maximum slope in each of the

*x*,

*y*and

*z*directions at the origin are

*xy*plane Eq. (12) yields dwell times at

*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 whereas for

*p*= 0 we get Δ

*t*

_{1}= Δ

*t*

_{2}= Δ

*t*

_{3}=

*τ*/3.

*±*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.

*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

*z*= 0) is

*I*

_{0}the peak intensity in terms of the number of photons/(area × time) [30]. When this beam is offset by a position shift of

**r**= (

*x*,

*y*) from the origin the regression vector

**g**is given by Scanning at the two (

*x*,

*y*) positions

*t*

_{1}= Δ

*t*

_{2}=

*τ*/2, we find the Fisher information matrix where the (mean) number of photons collected during a single scan period is

26. A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B **83**, 127–133 (2006). [CrossRef]

**r**′ and for any finite

*p*≤ 1, the inequality that must be satisfied, Eq. (10), becomes This is satisfied for all

**r**′ = (

*x*′,

*y*′) and so both the two point trajectory,

*t*

_{1}= Δ

*t*

_{2}=

*τ*/2 and the circular trajectory with

*p*= −1 becomes

*p*.

### 3.2. Gaussian beam in three dimensions via optimal design

*ϕ*

_{0}and

*ϕ*

_{−1}optimality are

*not*achieved by the same scan path, so we focus on

*ϕ*

_{−1}, which bounds the localization accuracy

**F**

*, is diagonal and is given by*

_{*}*c.f.*Eqs. (9) and (10) with

*p*= −1]:

*f*(

**r**) ≤ 1 for all

**r**. Writing

**r**= (

*x*,

*y*,

*z*) in convenient dimensionless units where

*r*̄ =

*r*/

*w*

_{0}and

*z*̄ =

*z*/

*z*we find This function is plotted in Fig. 2, where it is clearly seen not to exceed the value 1 (this can also be shown analytically). Thus, the proposed scan path is

_{R}*ϕ*

_{−1}-optimal and therefore the three-dimensional localization is always limited by the Cramér-Rao bound, taking here the form Note that because the two optimal points, i.e., the peaks in the graph in 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

*f*= 1 and hence a continuous path can be used if desired. [26

26. A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B **83**, 127–133 (2006). [CrossRef]

*z*= 0 with

*r*= 0 with

*τ*will for all practical purposes be optimal.

## 4. Conclusions

*p*in the merit function

*ϕ*[

_{p}**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

**F**

*is*

_{*}*ϕ*optimal relative to all other

_{p}**F**if and only if Substituting the definition of

*ϕ*from Eq. (6) into the above condition, raising both sides to the power

_{p}*p*and cancelling factors of 1/

*d*gives Writing

*c*into the definition of the

_{n}**g**

*and rearranging gives All*

_{n}**F**are real and symmetric and so can be diagonalized by a similarity transformation. Let the similarity transformation which diagonalizes

**F**

*be*

_{*}**S**whose rows are the orthonormal eigenvectors

**e**

*of*

_{i}**F**

*with*

_{*}*i*= 1,...,

*D*in

*D*dimensions. Then

**S**·

**F**

*·*

_{*}**S**

^{T}=

**f**

*is diagonal and*

_{*}**S**

^{T}·

**S**is the identity matrix. The diagonal elements of

**f**

*given by f*

_{*}*are real and positive since for any real nonzero*

_{*i}**v**. Writing

**g**

_{*n}in terms of the eigenvectors

**e**

*(written as column vectors) gives As we have seen above, in*

_{i}*D*dimensions we only need

*D*independent measurements to determine the particles position, i.e.,

*N*=

*D*, and that in the representation where

**F**is diagonal that these optimum positions are orthogonal to one another which means Substituting this into

**S**·

**F**

_{*}·

**S**

^{T}=

**f**

*we find with no sum on*

_{*}*i*. or

*j*which gives

**g**

_{*}with an arbitrary

**g**and undo the similarity transformation we have by definition

## Appendix B: Maximum likelihood position estimation for an arbitrary scan path

*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**as where

**H**(Γ)|

_{−rL}(

*t*) is the Hessian matrix of partial derivatives of the laser intensity function evaluated at the point −

**r**

_{L}(

*t*). For any function

*f*(

**r**), the

*jk*entry of the Hessian matrix is

**r**to zero, we find the following linear equation for the

*maximum likelihood estimate*

**r**

_{MLE}of the particle position

**r**: where the

*D*×

*D*matrix

**A**and

*D*×1 vector

**b**depend on the laser scan path and the measurement result {

*t*

_{1},...,

*t*} through The sums over

_{k}*k*are understood to be 0 when

*K*= 0. When the functional form of the terms in

**A**and

**b**can be precomputed or approximated, a real-time position estimate can be formed by computing

**A**and

**b**in real time (through the sums over

*k*) and solving the 2- or 3-dimensional linear system.

*t*[26

_{k}26. A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B **83**, 127–133 (2006). [CrossRef]

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21. | H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. |

22. | K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. |

23. | G. Lessard, P. Goodwin, and J. Werner, “Three-dimensional tracking of individual quantum dots,” Appl. Phys. Lett. |

24. | H. Cang, D. Montiel, C. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys. |

25. | K. McHale and H. Mabuchi, “Precise characterization of the conformation fluctuations of freely diffusing DNA: beyond Rouse and Zimm,” J. Am. Chem. Soc. |

26. | A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B |

27. | A. J. Berglund, K. McHale, and H. Mabuchi, “Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit,” Opt. Lett. |

28. | Z. Shen and S. Andersson, “Optimal measurement constellation of the fluoroBancroft localization algorithm for position estimation in tracking confocal microscopy,” Mechatronics |

29. | Q. Wang and W. Moerner, “Optimal strategy for trapping single fluorescent molecules in solution using the ABEL trap,” Appl. Phys. B |

30. | A. E. Siegman, |

31. | K. A. Winnick, “Cramer-Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A |

32. | H. Kao and A. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. |

33. | S. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express |

34. | K. T. Seale, R. S. Reiserer, D. A. Markov, I. A. Ges, C. Wright, C. Janetopoulos, and J. P. Wikswo, “Mirrored pyramidal wells for simultaneous multiple vantage point microscopy,” J. Microsc. |

35. | M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, “3D Particle trajectories observed by orthogonal tracking microscopy,” ACS Nano |

36. | S. Zacks, |

37. | F. Pukelsheim, |

**OCIS Codes**

(180.2520) Microscopy : Fluorescence microscopy

(110.3055) Imaging systems : Information theoretical analysis

**ToC Category:**

Microscopy

**History**

Original Manuscript: May 17, 2012

Revised Manuscript: June 7, 2012

Manuscript Accepted: June 8, 2012

Published: July 3, 2012

**Virtual Issues**

Vol. 7, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Gregg M. Gallatin and Andrew J. Berglund, "Optimal laser scan path for localizing a fluorescent particle in two or three dimensions," Opt. Express **20**, 16381-16393 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16381

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