Fluorescence phase-shifting interferometry for axial single particle tracking: a numerical simulation study E. Arbel,1 A. Praiz,1 and A. Bilenca1,2,* 1
2
Biomedical Engineering Department, Ben-Gurion University of the Negev, Be’er-Sheva 84105, Israel Ilse Katz Institute for Nanoscale Science and Technology, Ben-Gurion University of the Negev, Be’er-Sheva 84105, Israel * [email protected]
Abstract: Tracking of single fluorescent probes along the axial (depth) dimension is an important task in the biological and physical sciences. In this paper, we propose and analyze the use of fluorescence phase-shifting interferometry (FPSI) for axial single particle tracking (SPT) along 1 μmdepth (z) trajectories. FPSI is a photon-efficient, self-interference method that collects and coherently combines the 4π steradian emission wavefronts of a single fluorescent particle while introducing multiple phase shifts between the wavefronts to axially localize the particle with high precision over an extended depth-of-field. We employ vectorial imaging analysis and Monte-Carlo simulations of diffusive and directed motions to present a detailed comparative study of spatial and temporal FPSI for axial SPT based on simultaneous and time sequential collection of four phase-shifted interferograms using a single camera, respectively. The results of the numerical simulations show that for ≤0.105 μm2/s diffusion, spatial FPSI attains a maximal twofold improvement in the trajectory reconstruction precision at the expense of a fourfold reduced field-of-view compared to temporal FPSI. Furthermore, the analysis predicts that for sufficiently slow random linear motions, temporal FPSI is superior to spatial FPSI and achieves a smaller trajectory reconstruction error. © 2014 Optical Society of America OCIS codes: (260.2510) Fluorescence; (110.4980) Partial coherence in imaging.
(100.3175)
Interferometric
imaging;
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9.
A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science 300(5628), 2061–2065 (2003). L. S. Churchman, Z. Okten, R. S. Rock, J. F. Dawson, and J. A. Spudich, “Single molecule high-resolution colocalization of Cy3 and Cy5 attached to macromolecules measures intramolecular distances through time,” Proc. Natl. Acad. Sci. U.S.A. 102(5), 1419–1423 (2005). L. Holtzer and T. Schmidt, “The tracking of individual molecules in cells and tissues,” in Single Particle Tracking and Single Molecule Energy Transfer, C. Bräuchle, D. C. Lamb, and J. Michaelis, eds. (Wiley-VCH, 2010). 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(4), 2919–2928 (2005). T. Ragan, H. Huang, P. So, and E. Gratton, “3D particle tracking on a two-photon microscope,” J. Fluoresc. 16(3), 325–336 (2006). H. Cang, C. S. Xu, D. Montiel, and H. Yang, “Guiding a confocal microscope by single fluorescent nanoparticles,” Opt. Lett. 32(18), 2729–2731 (2007). W. Supatto, S. E. Fraser, and J. Vermot, “An all-optical approach for probing microscopic flows in living embryos,” Biophys. J. 95(4), L29–L31 (2008). M. F. Juette and J. Bewersdorf, “Three-dimensional tracking of single fluorescent particles with submillisecond temporal resolution,” Nano Lett. 10(11), 4657–4663 (2010). V. Levi and E. Gratton, “Three-dimensional particle tracking in a laser scanning fluorescence microscope,” in Single Particle Tracking and Single Molecule Energy Transfer, C. Bräuchle, D. C. Lamb, and J. Michaelis, eds. (Wiley-VCH, 2010).
#211457 - $15.00 USD Received 9 May 2014; revised 21 Jun 2014; accepted 27 Jun 2014; published 7 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019641 | OPTICS EXPRESS 19641
10. D. D. Li, J. Xiong, A. L. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live PC12 cells,” Biophys. J. 87(3), 1991–2001 (2004). 11. R. M. Dickson, D. J. Norris, Y. L. Tzeng, and W. E. Moerner, “Three-dimensional imaging of single molecules solvated in pores of poly(acrylamide) gels,” Science 274(5289), 966–969 (1996). 12. M. Speidel, A. Jonás, and E. L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. 28(2), 69–71 (2003). 13. E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7(7), 2043–2045 (2007). 14. S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95(12), 6025– 6043 (2008). 15. H. P. Kao and A. S. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67(3), 1291–1300 (1994). 16. L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. 90(5), 053902 (2007). 17. M. A. Thompson, J. M. Casolari, M. Badieirostami, P. O. Brown, and W. E. Moerner, “Three-dimensional tracking of single mRNA particles in Saccharomyces cerevisiae using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 107(42), 17864–17871 (2010). 18. J. H. Spille, T. Kaminski, H. P. Königshoven, and U. Kubitscheck, “Dynamic three-dimensional tracking of single fluorescent nanoparticles deep inside living tissue,” Opt. Express 20(18), 19697–19707 (2012). 19. B. Nitzsche, F. Ruhnow, and S. Diez, “Quantum-dot-assisted characterization of microtubule rotations during cargo transport,” Nat. Nanotechnol. 3(9), 552–556 (2008). 20. C. von Middendorff, A. Egner, C. Geisler, S. W. Hell, and A. Schönle, “Isotropic 3D Nanoscopy based on single emitter switching,” Opt. Express 16(25), 20774–20788 (2008). 21. G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent superresolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009). 22. D. Aquino, A. Schönle, C. Geisler, C. V. Middendorff, C. A. Wurm, Y. Okamura, T. Lang, S. W. Hell, and A. Egner, “Two-color nanoscopy of three-dimensional volumes by 4Pi detection of stochastically switched fluorophores,” Nat. Methods 8(4), 353–359 (2011). 23. P. R. T. Munro, “Application of numerical methods to high numerical aperture imaging,” PhD Thesis, University of London (2007).
1. Introduction One of the most powerful tools for investigating structure and function of nano- and microenvironments of physical and biological soft matter is single particle tracking (SPT) of fluorescent emitters. In particular, fluorescence SPT features high labeling specificity and single-trajectory reconstruction with high precision. While particle tracking with λ/2 axial range using solely Δφ i or Δφ o due to phase wrapping. To overcome this limitation, the wrapped pair of phases [Δφ i, Δφ o] should be employed. Effectively, [Δφ i, Δφ o] can localize the emitter along ~2λ axial range
#211457 - $15.00 USD Received 9 May 2014; revised 21 Jun 2014; accepted 27 Jun 2014; published 7 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019641 | OPTICS EXPRESS 19645
unambiguously as depicted by the Δφ o-Δφ i lookup curves of a static emitter located at (0, 0, ζ) in Fig. 2(b).
Fig. 2. The behavior of the integrated inner and outer imaged wavefront phase differences Δφ . (a) Dependence of Δφ i/o on the linear wavefront phase difference Δφl = 4πζ /λ that corresponds for a low-angle imaging scenario. (b) Δφ o-Δφ i lookup curves of a static emitter located at (0, 0, ζ). The parameters used in these figures were the same as in Fig. 1(b).
i/o
3. Simulation results and discussion Two scenarios of axial SPT by spatial and temporal FPSI are examined in this section throughout vectorial imaging computations and Monte-Carlo simulations using Matlab/C + + software running on a computer cluster. These scenarios include diffusive and directed motions and serve as demonstrations for the potential extension of FPSI to particle tracking applications. The simulation study provides a fundamental comparison of temporal and spatial FPSI and sets an upper limit on their performance for axial tracking of a moving fluorescent particle having different emission rates. We point out that our FPSI simulations assumed an ideal 4π interferometeric system using a four-step PSI algorithm on a single camera. Also, the same magnification and number of camera pixels were used in both spatial and temporal FPSI simulations. It is worth noting that here temporal FPSI records the set of four phase-shifted interferograms of a single emitter time-sequentially but can use the entire camera sensor to image various interferogram sets of multiple optically resolved emitters, whereas spatial FPSI acquires the set of four phase-shifted interferograms of a single emitter on different quadrants of the camera simultaneously and consequently has a fourfold smaller field-of-view as illustrated in Figs. 3(a) and 3(b), respectively. 3.1 FPSI-based axial SPT of diffusive motion To study and compare the use of spatial and temporal FPSI for axial SPT of diffusive particles, we first generated a large number of one-dimensional axial (z) Brownian trajectories of a single dipole emitter oriented along the x-axis in the object plane. As a result, a set of positions {0, 0, ζn}, n = 0,1,…,N was produced. We point out that linearly polarized emission was assumed to reduce computational load; yet, simulations with various linear polarization states yielded similar trends to those reported below. Next, we evaluated the electric field distributions of the dipole emitter originating from {0, 0, ζn} and {0, 0, -ζn}, n = 0,1,…,N in the image plane of an ideal, telecentric and aplanatic high numerical aperture imaging system.
#211457 - $15.00 USD Received 9 May 2014; revised 21 Jun 2014; accepted 27 Jun 2014; published 7 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019641 | OPTICS EXPRESS 19646
Fig. 3. Temporal and spatial FPSI using a four-step PSI algorithm on a single camera. (a) Temporal FPSI (tFPSI) employs a time-sequential collection of the four phase-shifted interferograms of multiple optically resolved emitters, whereas (b) spatial FPSI (sFPSI) uses a simultaneous interferogram collection, but at the expense of a smaller field-of-view compared to tFPSI with an identical sensor array.
To simulate spatial FPSI, four relative phase shifts of δ k = 0, π/2, π, 3π /2, k = 1, 2, 3, 4 were introduced simultaneously between the self-interfering fluorescent wavefronts at each emitter position, yielding the phase-shifted interferograms {I1n, I2n, I3n, I4n }, n = 0,1,…,N of Eq. (2), each imaged on a different quadrant of the single camera simultaneously. Then, each set of M phase-shifted interferograms {IkMm, …, IkMm + M-1} with m = 0, 1, …, N/M-1 for each and every k = 1, 2, 3, 4 was averaged out to model the finite exposure time of the camera, resulting in four spatially phase-shifted interferograms {ISk = 1,2,3,4}m per the m-th acquisition interval. Finally, four noisy phase-shifted inetrferograms were formed by computing the statistical average intensity of I Sk, k = 1, 2, 3, 4 integrated over a square pixel, replacing the average with the numerical realization of a Poisson random variable of identical average intensity to simulate photon-counting noise, and adding a zero-mean, σ b2-Gaussian distributed random number to simulate background noise. These noisy spatially phase-shifted interferograms were the basis for computation of the integrated inner and outer imaged wavefront phase differences Δφ i/o using Eqs. (4)–(6). To finally provide the estimation for the axial position of the dipole emitter, we identified the extracted phase pair [Δφ i, Δφ o] in the static lockup curve (e.g., Fig. 2(b)) using least-squares optimization. Figure 4(a) shows a typical mapping of a single Brownian trajectory over ~1 μm extended depth-of-field by spatial FPSI (sFPSI). Abrupt peaks corresponding to mislocalizations are clearly observed. These mislocalizations are due to photon-counting and background noise that leads to the incorrect evaluation of the wrapped pair of phases [Δφ i, Δφ o] and consequently to an erroneous estimate of ζ. Temporal FPSI was simulated similarly to spatial FPSI with the same magnification and number of pixels that sample the interferograms of a single emitter, but with the exception of the manner in which the relative phase shifts δ k were introduced between the self-interfering fluorescent wavefronts of the emitter. Here, the relative phase shifts δ k = 0, π/2, π, 3π /2, k = 1, 2, 3, 4 were introduced serially to blocks of M successive emitter positions. The switching time between adjacent phase shifts was assumed to be much smaller than the camera exposure time and therefore its effect is negligible – an assumption that is fairly realistic when employing high-speed phase shifters with
2
Biomedical Engineering Department, Ben-Gurion University of the Negev, Be’er-Sheva 84105, Israel Ilse Katz Institute for Nanoscale Science and Technology, Ben-Gurion University of the Negev, Be’er-Sheva 84105, Israel * [email protected]
Abstract: Tracking of single fluorescent probes along the axial (depth) dimension is an important task in the biological and physical sciences. In this paper, we propose and analyze the use of fluorescence phase-shifting interferometry (FPSI) for axial single particle tracking (SPT) along 1 μmdepth (z) trajectories. FPSI is a photon-efficient, self-interference method that collects and coherently combines the 4π steradian emission wavefronts of a single fluorescent particle while introducing multiple phase shifts between the wavefronts to axially localize the particle with high precision over an extended depth-of-field. We employ vectorial imaging analysis and Monte-Carlo simulations of diffusive and directed motions to present a detailed comparative study of spatial and temporal FPSI for axial SPT based on simultaneous and time sequential collection of four phase-shifted interferograms using a single camera, respectively. The results of the numerical simulations show that for ≤0.105 μm2/s diffusion, spatial FPSI attains a maximal twofold improvement in the trajectory reconstruction precision at the expense of a fourfold reduced field-of-view compared to temporal FPSI. Furthermore, the analysis predicts that for sufficiently slow random linear motions, temporal FPSI is superior to spatial FPSI and achieves a smaller trajectory reconstruction error. © 2014 Optical Society of America OCIS codes: (260.2510) Fluorescence; (110.4980) Partial coherence in imaging.
(100.3175)
Interferometric
imaging;
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9.
A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-overhand: single fluorophore imaging with 1.5-nm localization,” Science 300(5628), 2061–2065 (2003). L. S. Churchman, Z. Okten, R. S. Rock, J. F. Dawson, and J. A. Spudich, “Single molecule high-resolution colocalization of Cy3 and Cy5 attached to macromolecules measures intramolecular distances through time,” Proc. Natl. Acad. Sci. U.S.A. 102(5), 1419–1423 (2005). L. Holtzer and T. Schmidt, “The tracking of individual molecules in cells and tissues,” in Single Particle Tracking and Single Molecule Energy Transfer, C. Bräuchle, D. C. Lamb, and J. Michaelis, eds. (Wiley-VCH, 2010). 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(4), 2919–2928 (2005). T. Ragan, H. Huang, P. So, and E. Gratton, “3D particle tracking on a two-photon microscope,” J. Fluoresc. 16(3), 325–336 (2006). H. Cang, C. S. Xu, D. Montiel, and H. Yang, “Guiding a confocal microscope by single fluorescent nanoparticles,” Opt. Lett. 32(18), 2729–2731 (2007). W. Supatto, S. E. Fraser, and J. Vermot, “An all-optical approach for probing microscopic flows in living embryos,” Biophys. J. 95(4), L29–L31 (2008). M. F. Juette and J. Bewersdorf, “Three-dimensional tracking of single fluorescent particles with submillisecond temporal resolution,” Nano Lett. 10(11), 4657–4663 (2010). V. Levi and E. Gratton, “Three-dimensional particle tracking in a laser scanning fluorescence microscope,” in Single Particle Tracking and Single Molecule Energy Transfer, C. Bräuchle, D. C. Lamb, and J. Michaelis, eds. (Wiley-VCH, 2010).
#211457 - $15.00 USD Received 9 May 2014; revised 21 Jun 2014; accepted 27 Jun 2014; published 7 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019641 | OPTICS EXPRESS 19641
10. D. D. Li, J. Xiong, A. L. Qu, and T. Xu, “Three-dimensional tracking of single secretory granules in live PC12 cells,” Biophys. J. 87(3), 1991–2001 (2004). 11. R. M. Dickson, D. J. Norris, Y. L. Tzeng, and W. E. Moerner, “Three-dimensional imaging of single molecules solvated in pores of poly(acrylamide) gels,” Science 274(5289), 966–969 (1996). 12. M. Speidel, A. Jonás, and E. L. Florin, “Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging,” Opt. Lett. 28(2), 69–71 (2003). 13. E. Toprak, H. Balci, B. H. Blehm, and P. R. Selvin, “Three-dimensional particle tracking via bifocal imaging,” Nano Lett. 7(7), 2043–2045 (2007). 14. S. Ram, P. Prabhat, J. Chao, E. S. Ward, and R. J. Ober, “High accuracy 3D quantum dot tracking with multifocal plane microscopy for the study of fast intracellular dynamics in live cells,” Biophys. J. 95(12), 6025– 6043 (2008). 15. H. P. Kao and A. S. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67(3), 1291–1300 (1994). 16. L. Holtzer, T. Meckel, and T. Schmidt, “Nanometric three-dimensional tracking of individual quantum dots in cells,” Appl. Phys. Lett. 90(5), 053902 (2007). 17. M. A. Thompson, J. M. Casolari, M. Badieirostami, P. O. Brown, and W. E. Moerner, “Three-dimensional tracking of single mRNA particles in Saccharomyces cerevisiae using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A. 107(42), 17864–17871 (2010). 18. J. H. Spille, T. Kaminski, H. P. Königshoven, and U. Kubitscheck, “Dynamic three-dimensional tracking of single fluorescent nanoparticles deep inside living tissue,” Opt. Express 20(18), 19697–19707 (2012). 19. B. Nitzsche, F. Ruhnow, and S. Diez, “Quantum-dot-assisted characterization of microtubule rotations during cargo transport,” Nat. Nanotechnol. 3(9), 552–556 (2008). 20. C. von Middendorff, A. Egner, C. Geisler, S. W. Hell, and A. Schönle, “Isotropic 3D Nanoscopy based on single emitter switching,” Opt. Express 16(25), 20774–20788 (2008). 21. G. Shtengel, J. A. Galbraith, C. G. Galbraith, J. Lippincott-Schwartz, J. M. Gillette, S. Manley, R. Sougrat, C. M. Waterman, P. Kanchanawong, M. W. Davidson, R. D. Fetter, and H. F. Hess, “Interferometric fluorescent superresolution microscopy resolves 3D cellular ultrastructure,” Proc. Natl. Acad. Sci. U.S.A. 106(9), 3125–3130 (2009). 22. D. Aquino, A. Schönle, C. Geisler, C. V. Middendorff, C. A. Wurm, Y. Okamura, T. Lang, S. W. Hell, and A. Egner, “Two-color nanoscopy of three-dimensional volumes by 4Pi detection of stochastically switched fluorophores,” Nat. Methods 8(4), 353–359 (2011). 23. P. R. T. Munro, “Application of numerical methods to high numerical aperture imaging,” PhD Thesis, University of London (2007).
1. Introduction One of the most powerful tools for investigating structure and function of nano- and microenvironments of physical and biological soft matter is single particle tracking (SPT) of fluorescent emitters. In particular, fluorescence SPT features high labeling specificity and single-trajectory reconstruction with high precision. While particle tracking with λ/2 axial range using solely Δφ i or Δφ o due to phase wrapping. To overcome this limitation, the wrapped pair of phases [Δφ i, Δφ o] should be employed. Effectively, [Δφ i, Δφ o] can localize the emitter along ~2λ axial range
#211457 - $15.00 USD Received 9 May 2014; revised 21 Jun 2014; accepted 27 Jun 2014; published 7 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019641 | OPTICS EXPRESS 19645
unambiguously as depicted by the Δφ o-Δφ i lookup curves of a static emitter located at (0, 0, ζ) in Fig. 2(b).
Fig. 2. The behavior of the integrated inner and outer imaged wavefront phase differences Δφ . (a) Dependence of Δφ i/o on the linear wavefront phase difference Δφl = 4πζ /λ that corresponds for a low-angle imaging scenario. (b) Δφ o-Δφ i lookup curves of a static emitter located at (0, 0, ζ). The parameters used in these figures were the same as in Fig. 1(b).
i/o
3. Simulation results and discussion Two scenarios of axial SPT by spatial and temporal FPSI are examined in this section throughout vectorial imaging computations and Monte-Carlo simulations using Matlab/C + + software running on a computer cluster. These scenarios include diffusive and directed motions and serve as demonstrations for the potential extension of FPSI to particle tracking applications. The simulation study provides a fundamental comparison of temporal and spatial FPSI and sets an upper limit on their performance for axial tracking of a moving fluorescent particle having different emission rates. We point out that our FPSI simulations assumed an ideal 4π interferometeric system using a four-step PSI algorithm on a single camera. Also, the same magnification and number of camera pixels were used in both spatial and temporal FPSI simulations. It is worth noting that here temporal FPSI records the set of four phase-shifted interferograms of a single emitter time-sequentially but can use the entire camera sensor to image various interferogram sets of multiple optically resolved emitters, whereas spatial FPSI acquires the set of four phase-shifted interferograms of a single emitter on different quadrants of the camera simultaneously and consequently has a fourfold smaller field-of-view as illustrated in Figs. 3(a) and 3(b), respectively. 3.1 FPSI-based axial SPT of diffusive motion To study and compare the use of spatial and temporal FPSI for axial SPT of diffusive particles, we first generated a large number of one-dimensional axial (z) Brownian trajectories of a single dipole emitter oriented along the x-axis in the object plane. As a result, a set of positions {0, 0, ζn}, n = 0,1,…,N was produced. We point out that linearly polarized emission was assumed to reduce computational load; yet, simulations with various linear polarization states yielded similar trends to those reported below. Next, we evaluated the electric field distributions of the dipole emitter originating from {0, 0, ζn} and {0, 0, -ζn}, n = 0,1,…,N in the image plane of an ideal, telecentric and aplanatic high numerical aperture imaging system.
#211457 - $15.00 USD Received 9 May 2014; revised 21 Jun 2014; accepted 27 Jun 2014; published 7 Aug 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019641 | OPTICS EXPRESS 19646
Fig. 3. Temporal and spatial FPSI using a four-step PSI algorithm on a single camera. (a) Temporal FPSI (tFPSI) employs a time-sequential collection of the four phase-shifted interferograms of multiple optically resolved emitters, whereas (b) spatial FPSI (sFPSI) uses a simultaneous interferogram collection, but at the expense of a smaller field-of-view compared to tFPSI with an identical sensor array.
To simulate spatial FPSI, four relative phase shifts of δ k = 0, π/2, π, 3π /2, k = 1, 2, 3, 4 were introduced simultaneously between the self-interfering fluorescent wavefronts at each emitter position, yielding the phase-shifted interferograms {I1n, I2n, I3n, I4n }, n = 0,1,…,N of Eq. (2), each imaged on a different quadrant of the single camera simultaneously. Then, each set of M phase-shifted interferograms {IkMm, …, IkMm + M-1} with m = 0, 1, …, N/M-1 for each and every k = 1, 2, 3, 4 was averaged out to model the finite exposure time of the camera, resulting in four spatially phase-shifted interferograms {ISk = 1,2,3,4}m per the m-th acquisition interval. Finally, four noisy phase-shifted inetrferograms were formed by computing the statistical average intensity of I Sk, k = 1, 2, 3, 4 integrated over a square pixel, replacing the average with the numerical realization of a Poisson random variable of identical average intensity to simulate photon-counting noise, and adding a zero-mean, σ b2-Gaussian distributed random number to simulate background noise. These noisy spatially phase-shifted interferograms were the basis for computation of the integrated inner and outer imaged wavefront phase differences Δφ i/o using Eqs. (4)–(6). To finally provide the estimation for the axial position of the dipole emitter, we identified the extracted phase pair [Δφ i, Δφ o] in the static lockup curve (e.g., Fig. 2(b)) using least-squares optimization. Figure 4(a) shows a typical mapping of a single Brownian trajectory over ~1 μm extended depth-of-field by spatial FPSI (sFPSI). Abrupt peaks corresponding to mislocalizations are clearly observed. These mislocalizations are due to photon-counting and background noise that leads to the incorrect evaluation of the wrapped pair of phases [Δφ i, Δφ o] and consequently to an erroneous estimate of ζ. Temporal FPSI was simulated similarly to spatial FPSI with the same magnification and number of pixels that sample the interferograms of a single emitter, but with the exception of the manner in which the relative phase shifts δ k were introduced between the self-interfering fluorescent wavefronts of the emitter. Here, the relative phase shifts δ k = 0, π/2, π, 3π /2, k = 1, 2, 3, 4 were introduced serially to blocks of M successive emitter positions. The switching time between adjacent phase shifts was assumed to be much smaller than the camera exposure time and therefore its effect is negligible – an assumption that is fairly realistic when employing high-speed phase shifters with
