## Recovering fluorophore location and orientation from lifetimes |

Optics Express, Vol. 21, Issue 1, pp. 421-430 (2013)

http://dx.doi.org/10.1364/OE.21.000421

Acrobat PDF (746 KB)

### Abstract

In this paper, we study the possibility of using lifetime data to estimate the position and orientation of a fluorescent dipole source within a disordered medium. The vector Foldy-Lax equations are employed to calculate the interaction between the fluorescent source and the scatterers that are modeled as point-scatterers. The numerical experiments demonstrate that if good prior knowledge about the positions of the scatterers is available, the position and orientation of the dipole source can be retrieved from its lifetime data with precision. If there is uncertainty about the positions of the scatterers, the dipole source position can be estimated within the same level of uncertainty.

© 2013 OSA

## 1. Introduction

1. See K. Suhling, P.W. French, and D. Philipps, “Time-resolved fluorescence microscopy,” Photochem. Photobiol. Sci. **4**, 13–22 (2005) and references therein. [CrossRef]

2. K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. **1**, 693–701 (1970). [CrossRef]

3. R.R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. **37**, 1–65 (1978). [CrossRef]

4. J.P. Hoogenboom, G. Sanchez-Mosteiro, G. Colas des Francs, D. Heinis, G. Legay, A. Dereux, and N.F. van Hulst, “The single molecule probe: nanoscale vectorial mapping of photonic mode density in a metal nanocavity,” Nano Lett. **9**, 1189–1195 (2009). [CrossRef] [PubMed]

5. M. Frimmer, Y. Chen, and A.F. Koenderink, “Scanning emitter lifetime imaging microscopy for spontaneous emission control,” Phys. Rev. Lett. **107**, 123602 (2011). [CrossRef] [PubMed]

6. E.A. Donley and T. Plakhotnik, “Luminescence lifetimes of single molecules in disordered media,” J. Chem. Phys. **114**, 9993–9997 (2001). [CrossRef]

11. R. Sapienza, P. Bondareff, R. Pierrat, B. Habert, R. Carminati, and N. F. van Hulst, “Long-Tail statistics of the Purcell factor in disordered media driven by near-field interactions,” Phys. Rev. Lett. **106**, 163902 (2011). [CrossRef] [PubMed]

11. R. Sapienza, P. Bondareff, R. Pierrat, B. Habert, R. Carminati, and N. F. van Hulst, “Long-Tail statistics of the Purcell factor in disordered media driven by near-field interactions,” Phys. Rev. Lett. **106**, 163902 (2011). [CrossRef] [PubMed]

12. A. Cazé, R. Pierrat, and R. Carminati, “Near-field interactions and nonuniversality in speckle patterns produced by a point source in a disordered medium,” Phys. Rev. A **82**, 043823 (2010). [CrossRef]

13. L.S. Froufe-Pérez and R. Carminati, “Lifetime fluctuations of a single emitter in a disordered nanoscopic system: The influence of the transition dipole orientation,” Phys. Stat. Sol. (a) **205**, 1258–1265 (2008). [CrossRef]

*prior*information is available (in particular the geometry of the environment). The purpose of this work is to study this question on the basis of numerical experiments.

16. N. Irishina, M. Moscoso, and R. Carminati, “Source location from fluorescence lifetime in disordered media,” Optics Letters , **37**, 951–953 (2012). [CrossRef] [PubMed]

## 2. Numerical model

**r**

_{0}due to the interaction with the surrounding medium is given by [17

17. J.M. Wylie and J.E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A **30**, 1185–1193 (1984). [CrossRef]

_{0}is the decay rate in free space,

*ω*is the emission frequency,

**p**

*is the transition dipole between the excited and ground states,*

_{eg}**G⃡**

*is the scattered part of the dyadic Green function, and*

_{s}**u**=

**p**

*/|*

_{eg}**p**

*| is the normalized dipole direction. From this expression, it is apparent that decay rate data encode the flourophore position through the imaginary part of the scattered field*

_{eg}**E**

*(*

_{s}**r**

_{0},

*ω*) =

*μ*

_{0}

*ω*

^{2}

**G⃡**

*(*

_{s}**r**

_{0},

**r**

_{0},

*ω*)

**p**

*at its position.*

_{eg}*M*scatterers with polarizability

*α*(

*ω*) and that are small compared to the wavelength. The scatterers are distributed randomly in the free space an have positions

**r**

*,*

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

*M*. Wave propagation and multiple scattering in such a medium can be formulated in terms of the so-called Foldy-Lax equations [18

18. L.L. Foldy, “The multiple scattering of waves,” Phys. Rev. **67**, 107–119 (1945). [CrossRef]

19. M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. **23**, 287–310 (1951). [CrossRef]

**G⃡**

_{0}is the dyadic free space Green function,

*k*=

*ω*/

*c*is the wavenumber, and

*c*is the speed of light in free space. The summations are extended over the discrete set of scatterers. Eq. (2) gives the scattered field

**E**

*(*

_{s}**r**;

**r**

_{0},

**u**,

*ω*) at point

**r**and frequency

*ω*in terms of the exciting fields

**E**

*(*

_{exc}**r**

*;*

_{j}**r**

_{0},

**u**,

*ω*) at each scatterer position. Both depend on the position

**r**

_{0}, dipole direction

**u**, and emision frequency

*ω*of the fluorophore. The self-consistent system of

*M*equations (3) gives the exciting fields at each scatterer position

**r**

*. The exciting field*

_{j}**E**

*(*

_{exc}**r**

*;*

_{j}**r**

_{0},

**u**,

*ω*) is equal to the sum of the incident field

**E**

*(*

_{inc}**r**

*;*

_{j}**r**

_{0},

**u**,

*ω*) at

**r**

*(radiated by the fluorophore with dipole direction*

_{j}**u**placed at

**r**

_{0}) and the scattered fields at

**r**

*due to all scatterers at*

_{j}**r**

*≠*

_{m}**r**

*. This system can be written in matrix form and solved numerically by standard techniques if the number of scatterers*

_{j}*M*is not too large. In our experiments,

*M*is small (

*M*= 20) and we solve the linear system of equations (3) with a standard LU factorization technique. For

*M*large, Eqs. (3) can be solved efficiently using, for example, the Fast Multipole Method [20

20. S. Koc and W. C. Chew, “Calculation of acoustical scattering from a cluster of scatterers,” J. Acoust. Soc. Am. **103**, 721–734 (1998). [CrossRef]

*D*configuration. The scatterers lie in the

*xy*plane, and the dipole moment of the fluorophore is parallel to this plane (p polarization). In this case, the free-space dyadic Green function is given by Here,

*R*= ‖

**r**−

**r’**‖,

**r̂**⊗

**r̂**is the tensor product of

**r̂**with itself,

**I⃡**is the unit tensor, and

*i*= 0, 1. The scatterers are described by the resonant polarizability where

*ω*

_{0}is the resonance frequency and

*γ*is the linewidth. This form of the polarizability describes a lossless scatterer, and includes self interaction. This is the reason why only the exciting field on each scatterer is involved in Eq. (3). Including self-interaction (radiation reaction) is also a requirement for energy conservation, or equivalently the optical theorem [21

21. P. de Vries, D.V. van Coevordden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. **70**, 447–466 (1998). [CrossRef]

*δ*=

*ω*−

*ω*

_{0}, with

*δ*≪

*ω*

_{0}. As a measure of the scattering strength, we use (

*kl*)

_{s}^{−1}, where

*ℓ*= 1/[

_{s}*ρσ*(

_{s}*ω*)] is the scattering mean free path. In this last expression,

*ρ*denotes the density of scatterers and

*σ*(

_{s}*ω*) = (

*k*

^{3}/8)|

*α*(

*ω*)|

^{2}denotes the scattering cross section. In our experiments,

*ω*

_{0}= 2 · 10

^{6}GHz (

*λ*

_{0}= 0.94

*μ*m) and

*γ*= 5 GHz. In order to study scattering regimes with different strengths, we select appropriate ranges of frequencies. For frequencies

*ω*in the range (

*δ*

_{1},

*δ*

_{2}) = (1GHz, 2GHz), (

*kl*)

_{s}^{−1}varies from 0.23 to 0.18. These are frequencies close to the resonance frequency

*ω*

_{0}. For frequencies in the range (

*δ*

_{1},

*δ*

_{2}) = (3GHz, 5GHz), (

*kl*)

_{s}^{−1}decreases and varies from 0.11 to 0.09. For frequencies in the range (

*δ*

_{1},

*δ*

_{2}) = (10GHz, 12GHz), (

*kl*)

_{s}^{−1}decreases even more and varies from 0.02 to 0.01.

## 3. Numerical experiments

6. E.A. Donley and T. Plakhotnik, “Luminescence lifetimes of single molecules in disordered media,” J. Chem. Phys. **114**, 9993–9997 (2001). [CrossRef]

8. L.S. Froufe-Pérez, R. Carminati, and J.J. Sáenz, “Fluorescence decay rate statistics of a single molecule in a disordered cluster of nanoparticles,” Phys. Rev. A **76**, 013835 (2007). [CrossRef]

13. L.S. Froufe-Pérez and R. Carminati, “Lifetime fluctuations of a single emitter in a disordered nanoscopic system: The influence of the transition dipole orientation,” Phys. Stat. Sol. (a) **205**, 1258–1265 (2008). [CrossRef]

*θ*that defines the angle between the dipole direction

**u**and the

*x*-axis, that is, cos(

*θ*) =

**u**·

*x*̂. For a fluorophore placed at

**r**

_{0}= (

*x*,

*y*) with orientation parameter

*θ*, we have solved the

*M*×

*M*linear system (3) for the exciting fields, and we have computed the scattered field at

**r**

_{0}using Eq. (2). M=20 in all the numerical experiments shown in the paper. The fluorescent decay rates

*d*(

*ω*) = Γ(

**r**

_{0},

*θ*,

*ω*) at different frequencies

*ω*are then computed using Eq. (1). The results are shown in the bottom row of Fig. 1. The decay rate strongly depends on the frequency

*ω*because the interaction between the fluorophore and the environment is very sensitive to the frequency. This figure motivates the idea of recovering the fluorophore’s position

**r**

_{0}and its orientation

*θ*from the knowledge of the decay rate at different frequencies. In the next numerical experiments we show that this can, in fact, be accomplished easily.

**r**

_{0}and the dipole direction

*θ*that satisfy the nonlinear equations We consider a 2D configuration with

**r**

_{0}= (

*x*,

*y*) and, thus, we only seek for the values of three unknowns: (

*x*,

*y*) and

*θ*. Hence, we only need, in principle, data corresponding to three different frequencies to recover these three unknowns by solving Eq. (6). If desired, we can use data from more frequencies and compute the nonlinear least-squares solution to Eq. (6) for estimates of those unknowns. This will be the case when the data is contaminated with noise and/or there is uncertainty in the position of the scatterers.

*μ*m

^{2}around the true source position to restrict the search area. This can be done in practice using, for example, a coarse-grain for locating the fluorophore using a standard microscope fluorescence intensity image. Once the DOI is defined, the position and orientation of the dipole source are found by minimizing the residual using a standard iterative gradient method. We use the Matlab Optimization Toolbox that contains an efficient gradient method that is easy to use. We note, though, that this is a local method that finds only the nearest local minimum to the initial guess. Other local methods for solving nonlinear equations can be used as well [22]. A good alternative for finding the global minimum, at the cost of a much higher computational cost, is the use of global methods such as, for example, interval methods, simulating annealing or Monte-Carlo methods [23, 24]. These methods can only be used for small problems, though.

### 3.1. Noiseless data

*M*= 20 scatterers, as well as their polarizability. We do not know neither the position nor the dipole orientation of the fluorophore.

*kl*)

_{s}^{−1}decreases from the left column to the right column. For each frequency range (or each scattering regime), we use synthetic data corresponding to only

*N*= 3 frequencies within that range of frequency. We keep the same 3.7 × 3.7

_{ω}*μ*m

^{2}disordered medium in all the 6 experiments shown in the figure. The positions of the scatterers are shown by dots and the real fluorophore position by a star. The estimated position of the fluorophore found by solving Eq. (6) is shown by a circle. The orientations of both, the real and reconstructed dipole source, are represented by arrows. As reported in [16

16. N. Irishina, M. Moscoso, and R. Carminati, “Source location from fluorescence lifetime in disordered media,” Optics Letters , **37**, 951–953 (2012). [CrossRef] [PubMed]

*kl*)

_{s}^{−1}= 0.18 − 0.23 (left column) the estimates are very precised, for (

*kl*)

_{s}^{−1}= 0.01–0.02 the estimates are not as good. This is so because lifetime data is more sensitive to small changes in the position or orientation of the dipole source when its interaction with the environment is stronger and, hence, the global minimum of Eq. (7) is better defined. When the interaction between the dipole source an the environment is weak, many positions and orientations of the dipole source give rise to nearly the same lifetime data.

### 3.2. Noisy data

*δ*

_{1},

*δ*

_{2}) = (1GHz, 2GHz), so the scattering regime is (

*kl*)

_{s}^{−1}= 0.18 − 0.23.

### 3.3. Uncertainty with respect to the scatterers positions

*λ*/8 in the scatterer’s positions. We still use data from 3 frequencies corrupted with 1% of gaussian noise. We now observe that in almost all the cases the estimates of the positions and orientations of the dipole sources are not satisfactory.

*δ*

_{1},

*δ*

_{2}) = (1GHz, 2GHz), instead of only 3 frequencies. When data from more frequencies are used, the positions and orientations are better estimated. We note, however, that beyond a certain number of frequencies the values of the estimates do not improve anymore. In general, when enough frequencies are used the distances between the true and estimated positions are of the same order as the uncertainty on the scatterer’s positions.

*x*and

*y*cross-sections of the residual

*λ*/4 in the scatterers positions. We use lifetime data from 9 frequencies corrupted with 1% of gaussian noise. We have applied now a global optimization method since multiply local minima in the cost functional exist within the DOI and, therefore, a gradient method fails. The orientation of the dipoles are well estimated in all the cases, and their positions are recovered within a precision of ±

*λ*/4.

## 4. Conclusion

*x*,

*y*,

*z*) and two directional variables (

*θ*,

*ϕ*)) and, therefore, the resolution of the nonlinear optimization problem is more complicated. The numerical experiments show that the position and orientation of a dipole source inside a predetermined domain of interest can be retrieved with high accuracy from multi-frequency fluorescence lifetime data (except for the ambiguity under dipole inversion). The proposed strategy requires

*prior*knowledge about the scatterers position. Our results indicate, however, that the precision of the recovered source position deteriorates only at the same rate as the uncertainty on the scatterer’s position increases.

## References and links

1. | See K. Suhling, P.W. French, and D. Philipps, “Time-resolved fluorescence microscopy,” Photochem. Photobiol. Sci. |

2. | K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. |

3. | R.R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. |

4. | J.P. Hoogenboom, G. Sanchez-Mosteiro, G. Colas des Francs, D. Heinis, G. Legay, A. Dereux, and N.F. van Hulst, “The single molecule probe: nanoscale vectorial mapping of photonic mode density in a metal nanocavity,” Nano Lett. |

5. | M. Frimmer, Y. Chen, and A.F. Koenderink, “Scanning emitter lifetime imaging microscopy for spontaneous emission control,” Phys. Rev. Lett. |

6. | E.A. Donley and T. Plakhotnik, “Luminescence lifetimes of single molecules in disordered media,” J. Chem. Phys. |

7. | R.A.L. Vallée, N. Tomczak, L. Kuipers, G.J. Vancso, and N.F. van Hulst, “Single molecule lifetime fluctuations reveal segmental dynamics in polymers,”Phys. Rev. Lett. |

8. | L.S. Froufe-Pérez, R. Carminati, and J.J. Sáenz, “Fluorescence decay rate statistics of a single molecule in a disordered cluster of nanoparticles,” Phys. Rev. A |

9. | M. D. Birowosuto, S. E. Skipetrov, W. L. Vos, and A. P. Mosk, “Observation of spatial fluctuations of the local density of states in random media,” Phys. Rev. Lett. |

10. | V. Krachmalnicoff, E. Castanié, Y. De Wilde, and R. Carminati, “Fluctuations of the local density of states probe localized surface plasmons on disordered metal films,” Phys. Rev. Lett. |

11. | R. Sapienza, P. Bondareff, R. Pierrat, B. Habert, R. Carminati, and N. F. van Hulst, “Long-Tail statistics of the Purcell factor in disordered media driven by near-field interactions,” Phys. Rev. Lett. |

12. | A. Cazé, R. Pierrat, and R. Carminati, “Near-field interactions and nonuniversality in speckle patterns produced by a point source in a disordered medium,” Phys. Rev. A |

13. | L.S. Froufe-Pérez and R. Carminati, “Lifetime fluctuations of a single emitter in a disordered nanoscopic system: The influence of the transition dipole orientation,” Phys. Stat. Sol. (a) |

14. | G. Derveaux, G. Papanicolaou, and C. Tsogka, “Resolution and denoising in near-field imaging,” Inverse Problems |

15. | A. Chai, M. Moscoso, and G. Papanicolaou, “Array imaging using intensity-only measurements,” Inverse Problems |

16. | N. Irishina, M. Moscoso, and R. Carminati, “Source location from fluorescence lifetime in disordered media,” Optics Letters , |

17. | J.M. Wylie and J.E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A |

18. | L.L. Foldy, “The multiple scattering of waves,” Phys. Rev. |

19. | M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. |

20. | S. Koc and W. C. Chew, “Calculation of acoustical scattering from a cluster of scatterers,” J. Acoust. Soc. Am. |

21. | P. de Vries, D.V. van Coevordden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. |

22. | A. Ruszczynski, |

23. | E. R. Hansen, |

24. | R. Horst, P. M. Pardalos, and N. V. Thoai, eds., |

**OCIS Codes**

(260.2510) Physical optics : Fluorescence

(290.3200) Scattering : Inverse scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: September 18, 2012

Revised Manuscript: November 15, 2012

Manuscript Accepted: November 16, 2012

Published: January 4, 2013

**Virtual Issues**

Vol. 8, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

N. Irishina, M. Moscoso, and R. Carminati, "Recovering fluorophore location and orientation from lifetimes," Opt. Express **21**, 421-430 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-421

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### References

- See K. Suhling, P.W. French, and D. Philipps, “Time-resolved fluorescence microscopy,” Photochem. Photobiol. Sci.4, 13–22 (2005) and references therein. [CrossRef]
- K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin.1, 693–701 (1970). [CrossRef]
- R.R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys.37, 1–65 (1978). [CrossRef]
- J.P. Hoogenboom, G. Sanchez-Mosteiro, G. Colas des Francs, D. Heinis, G. Legay, A. Dereux, and N.F. van Hulst, “The single molecule probe: nanoscale vectorial mapping of photonic mode density in a metal nanocavity,” Nano Lett.9, 1189–1195 (2009). [CrossRef] [PubMed]
- M. Frimmer, Y. Chen, and A.F. Koenderink, “Scanning emitter lifetime imaging microscopy for spontaneous emission control,” Phys. Rev. Lett.107, 123602 (2011). [CrossRef] [PubMed]
- E.A. Donley and T. Plakhotnik, “Luminescence lifetimes of single molecules in disordered media,” J. Chem. Phys.114, 9993–9997 (2001). [CrossRef]
- R.A.L. Vallée, N. Tomczak, L. Kuipers, G.J. Vancso, and N.F. van Hulst, “Single molecule lifetime fluctuations reveal segmental dynamics in polymers,”Phys. Rev. Lett.91, 038301 (2003). [CrossRef] [PubMed]
- L.S. Froufe-Pérez, R. Carminati, and J.J. Sáenz, “Fluorescence decay rate statistics of a single molecule in a disordered cluster of nanoparticles,” Phys. Rev. A76, 013835 (2007). [CrossRef]
- M. D. Birowosuto, S. E. Skipetrov, W. L. Vos, and A. P. Mosk, “Observation of spatial fluctuations of the local density of states in random media,” Phys. Rev. Lett.105, 013904 (2010). [CrossRef] [PubMed]
- V. Krachmalnicoff, E. Castanié, Y. De Wilde, and R. Carminati, “Fluctuations of the local density of states probe localized surface plasmons on disordered metal films,” Phys. Rev. Lett.105, 183901 (2010). [CrossRef]
- R. Sapienza, P. Bondareff, R. Pierrat, B. Habert, R. Carminati, and N. F. van Hulst, “Long-Tail statistics of the Purcell factor in disordered media driven by near-field interactions,” Phys. Rev. Lett.106, 163902 (2011). [CrossRef] [PubMed]
- A. Cazé, R. Pierrat, and R. Carminati, “Near-field interactions and nonuniversality in speckle patterns produced by a point source in a disordered medium,” Phys. Rev. A82, 043823 (2010). [CrossRef]
- L.S. Froufe-Pérez and R. Carminati, “Lifetime fluctuations of a single emitter in a disordered nanoscopic system: The influence of the transition dipole orientation,” Phys. Stat. Sol. (a)205, 1258–1265 (2008). [CrossRef]
- G. Derveaux, G. Papanicolaou, and C. Tsogka, “Resolution and denoising in near-field imaging,” Inverse Problems22, 1437–1456 (2006). [CrossRef]
- A. Chai, M. Moscoso, and G. Papanicolaou, “Array imaging using intensity-only measurements,” Inverse Problems27, 015005 (2011). [CrossRef]
- N. Irishina, M. Moscoso, and R. Carminati, “Source location from fluorescence lifetime in disordered media,” Optics Letters, 37, 951–953 (2012). [CrossRef] [PubMed]
- J.M. Wylie and J.E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A30, 1185–1193 (1984). [CrossRef]
- L.L. Foldy, “The multiple scattering of waves,” Phys. Rev.67, 107–119 (1945). [CrossRef]
- M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys.23, 287–310 (1951). [CrossRef]
- S. Koc and W. C. Chew, “Calculation of acoustical scattering from a cluster of scatterers,” J. Acoust. Soc. Am.103, 721–734 (1998). [CrossRef]
- P. de Vries, D.V. van Coevordden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys.70, 447–466 (1998). [CrossRef]
- A. Ruszczynski, Nonlinear Optimization (Princeton University Press, Princeton, 2006).
- E. R. Hansen, Global Optimization using Interval Analysis (Marcel Dekker, New York, 1992).
- R. Horst, P. M. Pardalos, and N. V. Thoai, eds., Introduction to Global Optimization (Kluwer Academic, 2nd ed., 2000).

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