Abstract

We develop an efficient method for accurately calculating the electric field of tightly focused laser beams in the presence of specific configurations of microscopic scatterers. This Huygens–Fresnel wave-based electric field superposition (HF-WEFS) method computes the amplitude and phase of the scattered electric field in excellent agreement with finite difference time-domain (FDTD) solutions of Maxwell’s equations. Our HF-WEFS implementation is 2–4 orders of magnitude faster than the FDTD method and enables systematic investigations of the effects of scatterer size and configuration on the focal field. We demonstrate the power of the new HF-WEFS approach by mapping several metrics of focal field distortion as a function of scatterer position. This analysis shows that the maximum focal field distortion occurs for single scatterers placed below the focal plane with an offset from the optical axis. The HF-WEFS method represents an important first step toward the development of a computational model of laser-scanning microscopy of thick cellular/tissue specimens.

© 2014 Optical Society of America

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2013 (1)

E. E. Hoover and J. A. Squier, “Advances in multiphoton microscopy technology,” Nat. Photonics 7, 93–101 (2013).
[CrossRef]

2012 (3)

P. Kanchanawong and C. M. Waterman, “Advances in light-based imaging of three-dimensional cellular ultrastructure,” Curr. Opin. Cell Biol. 24, 125–133 (2012).
[CrossRef]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a focused beam,” Phys. Rev. A 86, 033817 (2012).
[CrossRef]

Y. Jiang, Y. Shao, X. Qu, J. Ou, and H. Hua, “Scattering of a focused Laguerre–Gaussian beam by a spheroidal particle,” J. Opt. 14, 125709 (2012).
[CrossRef]

2011 (4)

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2011).
[CrossRef]

C. G. Koay, “A simple scheme for generating nearly uniform distribution of antipodally symmetric points on the unit sphere,” J. Comput. Sci. 2, 377–381 (2011).
[CrossRef]

C. K. Hayakawa, E. O. Potma, and V. Venugopalan, “Electric field Monte Carlo simulations of focal field distributions produced by tightly focused laser beams in tissues,” Biomed. Opt. Express 2, 278–290 (2011).
[CrossRef]

Z. Cui, Y. Han, and Q. Xu, “Numerical simulation of multiple scattering by random discrete particles illuminated by Gaussian beams,” J. Opt. Soc. Am. A 28, 2200–2208 (2011).
[CrossRef]

2010 (4)

S. A. Prahl, D. D. Duncan, and D. G. Fischer, “Stochastic Huygens and partial coherence propagation through thin tissues,” Proc. SPIE 7573, 75730D (2010).
[CrossRef]

B. J. Davis, P. S. Carney, and R. Bhargava, “Theory of midinfrared absorption microspectroscopy: I. Homogeneous samples,” Anal. Chem. 82, 3474–3486 (2010).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

2009 (3)

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Annu. Rev. Neurosci. 32, 435–506 (2009).
[CrossRef]

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103, 43903 (2009).
[CrossRef]

M. S. Starosta and A. K. Dunn, “Three-dimensional computation of focused beam propagation through multiple biological cells,” Opt. Express 17, 12455–12469 (2009).
[CrossRef]

2008 (4)

2007 (1)

2006 (2)

J. A. Lock, S. Y. Wrbanek, and K. E. Weiland, “Scattering of a tightly focused beam by an optically trapped particle,” Appl. Opt. 45, 3634–3645 (2006).
[CrossRef]

R. Heintzmann and G. Ficz, “Breaking the resolution limit in light microscopy,” Briefings Funct. Genomics Proteomics 5, 289–301 (2006).

2004 (1)

2003 (2)

X. Deng and M. Gu, “Penetration depth of single-, two-, and three-photon fluorescence microscopic imaging through human cortex structures: Monte Carlo simulation,” Appl. Opt. 42, 3321–3329 (2003).
[CrossRef]

H.-X. Xu, “A new method by extending Mie theory to calculate local field in outside/inside of aggregates of arbitrary spheres,” Phys. Lett. A 312, 411–419 (2003).
[CrossRef]

2000 (1)

1999 (1)

1997 (1)

1996 (2)

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

A. Dunn and R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

1991 (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

1988 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Aubry, A.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2011).
[CrossRef]

Bachelier, G.

Backman, V.

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 127, 363–379 (1996).
[CrossRef]

Berns, M. W.

Bhargava, R.

B. J. Davis, P. S. Carney, and R. Bhargava, “Theory of midinfrared absorption microspectroscopy: I. Homogeneous samples,” Anal. Chem. 82, 3474–3486 (2010).
[CrossRef]

Bigio, I. J.

Billaud, P.

Boccara, A. C.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2011).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bonnet, C.

Borghese, F.

F. Borghese, P. Denti, and R. Saija, Scattering from Model Nonspherical Particles, 2nd ed. (Springer, 2007).

Boyer, J.

Brock, R. S.

Broyer, M.

Burns, L. D.

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Annu. Rev. Neurosci. 32, 435–506 (2009).
[CrossRef]

Çapoglu, I. R.

Carney, P. S.

B. J. Davis, P. S. Carney, and R. Bhargava, “Theory of midinfrared absorption microspectroscopy: I. Homogeneous samples,” Anal. Chem. 82, 3474–3486 (2010).
[CrossRef]

Chen, X.

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a focused beam,” Phys. Rev. A 86, 033817 (2012).
[CrossRef]

Cižmár, T.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

Coleno, M.

Cottancin, E.

Cui, Z.

Davis, B. J.

B. J. Davis, P. S. Carney, and R. Bhargava, “Theory of midinfrared absorption microspectroscopy: I. Homogeneous samples,” Anal. Chem. 82, 3474–3486 (2010).
[CrossRef]

Deng, X.

Denti, P.

F. Borghese, P. Denti, and R. Saija, Scattering from Model Nonspherical Particles, 2nd ed. (Springer, 2007).

Dholakia, K.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

Dong, K.

Duncan, D. D.

S. A. Prahl, D. D. Duncan, and D. G. Fischer, “Stochastic Huygens and partial coherence propagation through thin tissues,” Proc. SPIE 7573, 75730D (2010).
[CrossRef]

Dunn, A.

A. Dunn and R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

Dunn, A. K.

Feld, M. S.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[CrossRef]

Ficz, G.

R. Heintzmann and G. Ficz, “Breaking the resolution limit in light microscopy,” Briefings Funct. Genomics Proteomics 5, 289–301 (2006).

Fink, M.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2011).
[CrossRef]

Fischer, D. G.

S. A. Prahl, D. D. Duncan, and D. G. Fischer, “Stochastic Huygens and partial coherence propagation through thin tissues,” Proc. SPIE 7573, 75730D (2010).
[CrossRef]

Fuselier, T.

Ghosh, K. K.

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Annu. Rev. Neurosci. 32, 435–506 (2009).
[CrossRef]

Gigan, S.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2011).
[CrossRef]

Gu, M.

Han, Y.

Hayakawa, C. K.

C. K. Hayakawa, E. O. Potma, and V. Venugopalan, “Electric field Monte Carlo simulations of focal field distributions produced by tightly focused laser beams in tissues,” Biomed. Opt. Express 2, 278–290 (2011).
[CrossRef]

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103, 43903 (2009).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Heintzmann, R.

R. Heintzmann and G. Ficz, “Breaking the resolution limit in light microscopy,” Briefings Funct. Genomics Proteomics 5, 289–301 (2006).

Ho, E. T. W.

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Annu. Rev. Neurosci. 32, 435–506 (2009).
[CrossRef]

Hoang, T. X.

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a focused beam,” Phys. Rev. A 86, 033817 (2012).
[CrossRef]

Hoekstra, A. G.

Hoover, E. E.

E. E. Hoover and J. A. Squier, “Advances in multiphoton microscopy technology,” Nat. Photonics 7, 93–101 (2013).
[CrossRef]

Hu, X. H.

Hua, H.

Y. Jiang, Y. Shao, X. Qu, J. Ou, and H. Hua, “Scattering of a focused Laguerre–Gaussian beam by a spheroidal particle,” J. Opt. 14, 125709 (2012).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Jiang, Y.

Y. Jiang, Y. Shao, X. Qu, J. Ou, and H. Hua, “Scattering of a focused Laguerre–Gaussian beam by a spheroidal particle,” J. Opt. 14, 125709 (2012).
[CrossRef]

Johnson, T. M.

Johnston, R. G.

E. K. Yen and R. G. Johnston, “The ineffectiveness of the correlation coefficient for image comparisons,” research paper (Los Alamos National Laboratory, Los Alamos, New Mexico, 1996).

Kanchanawong, P.

P. Kanchanawong and C. M. Waterman, “Advances in light-based imaging of three-dimensional cellular ultrastructure,” Curr. Opin. Cell Biol. 24, 125–133 (2012).
[CrossRef]

Koay, C. G.

C. G. Koay, “A simple scheme for generating nearly uniform distribution of antipodally symmetric points on the unit sphere,” J. Comput. Sci. 2, 377–381 (2011).
[CrossRef]

Krishnamachari, V. V.

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103, 43903 (2009).
[CrossRef]

Lagendijk, A.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

Lermé, J.

Lerosey, G.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2011).
[CrossRef]

Lock, J. A.

Lu, J. Q.

Mackowski, D. W.

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991).
[CrossRef]

Marhaba, S.

Mazilu, M.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

Mosk, A. P.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
[CrossRef]

Mourant, J. R.

Mukamel, E. A.

B. A. Wilt, L. D. Burns, E. T. W. Ho, K. K. Ghosh, E. A. Mukamel, and M. J. Schnitzer, “Advances in light microscopy for neuroscience,” Annu. Rev. Neurosci. 32, 435–506 (2009).
[CrossRef]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Ou, J.

Y. Jiang, Y. Shao, X. Qu, J. Ou, and H. Hua, “Scattering of a focused Laguerre–Gaussian beam by a spheroidal particle,” J. Opt. 14, 125709 (2012).
[CrossRef]

Pellarin, M.

Popoff, S. M.

S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Exploiting the time-reversal operator for adaptive optics, selective focusing, and scattering pattern analysis,” Phys. Rev. Lett. 107, 263901 (2011).
[CrossRef]

Potma, E. O.

C. K. Hayakawa, E. O. Potma, and V. Venugopalan, “Electric field Monte Carlo simulations of focal field distributions produced by tightly focused laser beams in tissues,” Biomed. Opt. Express 2, 278–290 (2011).
[CrossRef]

C. K. Hayakawa, V. Venugopalan, V. V. Krishnamachari, and E. O. Potma, “Amplitude and phase of tightly focused laser beams in turbid media,” Phys. Rev. Lett. 103, 43903 (2009).
[CrossRef]

Prahl, S. A.

S. A. Prahl, D. D. Duncan, and D. G. Fischer, “Stochastic Huygens and partial coherence propagation through thin tissues,” Proc. SPIE 7573, 75730D (2010).
[CrossRef]

Psaltis, D.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[CrossRef]

Qu, X.

Y. Jiang, Y. Shao, X. Qu, J. Ou, and H. Hua, “Scattering of a focused Laguerre–Gaussian beam by a spheroidal particle,” J. Opt. 14, 125709 (2012).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Richards-Kortum, R.

A. Dunn and R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

Saija, R.

F. Borghese, P. Denti, and R. Saija, Scattering from Model Nonspherical Particles, 2nd ed. (Springer, 2007).

Schaub, S. A.

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Figures (11)

Fig. 1.
Fig. 1.

HF plane wavelets in (a) nonscattering and (b) scattering media. The spherical surface produces a series of forward-propagating HF spherical waves shown as dashed semicircles. In nonscattering media, each HF plane wavelet provides a partial electric field at point D. In media containing scatterers, a HF plane wavelet incident upon a particle produces a complete 3D scattered field. The contribution from each scattered field provides a partial scattered electric field at D. Dashed lines show the boundaries of the scattered field (blue and green). Superposition of two scattered fields gives the total scattered field. Only two HF plane wavelets are shown in (b) for clarity.

Fig. 2.
Fig. 2.

(a) Geometrical representation of the focused x-polarized flat beam by an aplanatic lens. The x-polarized incident wave (Einc) that is defined by unit vectors i, j, and k is refracted by the reference spherical surface of radius f. E and E are parallel and perpendicular electric field components after refraction, respectively. (b) The relationship between global and local orthonormal coordinate systems. The initial local coordinate system (m0,n0,u0) aligned with the unit vectors of the global Cartesian coordinate system. The local coordinate system for the HF plane wavelet (m1,n1,u1) (red) can be calculated by rotating the initial local coordinate system by ϕ and θ, respectively. m is the projection of m1 on the xy plane.

Fig. 3.
Fig. 3.

Schematic of the simulation setup (not to scale). Single scatterers are placed (a) 6 μm below the focal plane for case A, (b) 6 μm below the focal plane and offset by 2.4 μm left of the optical axis for case B, and (c) 15 μm below the focal plane and offset by 2.4 μm right of the optical axis for case C. (d) Case D is a combination of cases B and C. An xy plane detector is placed at z=0. (e) A single scatterer is placed at different locations in the xz plane grid (y=0) and its effect on the focal field and the focal volume are considered. The scatterer is placed at the least distance r (radius of the scatterer) below the focal plane and the grid spacing is set to 500 nm. NA and focal length of the lens are 0.667 and 500 μm, respectively.

Fig. 4.
Fig. 4.

Unscattered electric field EUnscat(r) amplitude and phase at the focal plane predicted using the (a) HF-WEFS method and the (b) Richards and Wolf’s analytical solution [32]. (c) The error of the HF-WEFS predictions as a percentage of maximum amplitude of EUnscat(r) is shown on the left, and the absolute phase difference is shown on the right. The size of the xy plane detector is 10μm×10μm with a 20 nm resolution. Amplitude results in (a) and (b) are shown in Log10 scale. NA of the lens is 0.667.

Fig. 5.
Fig. 5.

Scattered electric field EScat(r) at the focal plane predicted using the HF-WEFS method [(a) Amplitude, (d) phase] and the FDTD [(b) Amplitude, (e) phase] for cases A–D. (c) amplitude difference between the methods expressed as the percentage of maximum amplitude of EScat(r). (f) Absolute phase difference. Scatterer diameter is 5 μm. The size of the xy plane detector is 10μm×10μm with nodes placed on a square grid with 20 nm spacing. Amplitude results in (a) and (b) are shown in Log10 scale. NA of the lens is 0.667.

Fig. 6.
Fig. 6.

x component of the scattered electric field amplitude ExScat(r) on the xz plane (y=0) predicted by (a) HF-WEFS and (b) FDTD for cases A–D. Diameters of the scatterers are 1, 2.5, and 5 μm. The horizontal dashed line shows the focal plane. The size of the xz plane detector is 10μm×10μm with nodes placed on a rectangular grid with 20 nm spacing. Amplitude results are shown in Log10 scale. NA of the lens is 0.667.

Fig. 7.
Fig. 7.

2D amplitude correlation coefficient as a function of the xz particle location of the total x component of the electrical field ExTot(r), as measured on a circular detector placed at the focal plane with respect to the nonscattering case. Scatterer diameters are (a) 1 μm, (b) 2.5 μm, and (c) 5 μm. Contour spacing is 0.01. Right: electric field amplitude distribution profiles in the 4 μm circular detector with nodes placed at 20 nm increments on a square grid for cases with no scatterers, a scatterer positioned at location L1(1.6,0,5)μm, and a scatterer positioned at location L2(0,0,8)μm in (c).

Fig. 8.
Fig. 8.

2D phase correlation coefficient (ExTot) of a circular detector at the focal plane with respect to the nonscattering case is mapped on the xz plane placement grid. Scatterer diameters are (a) 1 μm, (b) 2.5 μm, and (c) 5 μm. Contour spacing is 0.05. Right: phase distribution profiles in the 4 μm circular detector with nodes placed at 20 nm increments on a square grid for no scatterers, and a scatterer at location L1(0,0,2.5)μm, L2(0,0,8.5)μm, L3(2.5,0,8.5)μm, and L4(0,0,16)μm in (c).

Fig. 9.
Fig. 9.

Total electric field distribution [ETot(r)] in the focal vicinity (a) for the nonscattering case and (b) by a single scatterer (case B: 5 μm diameter). The rendered focal volumes have the same amplitude at the surfaces. Amplitude results are shown in Log10 scale.

Fig. 10.
Fig. 10.

Percentage change in the maximum electric field amplitude within the focal volume region due to the introduction of a single scatterer with a diameter of (a) 1 μm, (b) 2.5 μm, and (c) 5 μm as mapped on the xz placement grid. Contour spacing is 1.5% of the amplitude change. (d) Percentage of amplitude change for a scatterer moving along the optical axis (white dashed line).

Fig. 11.
Fig. 11.

(a)–(c) Lateral and (d)–(f) axial displacement of the largest amplitude point of the focal volume relative to the nominal focal point due to a single scatterer is mapped on the xz placement grid. Lateral displacement (DL) maps with contour spacing of 0.02 μm for a single scatterer with a diameter of (a) 1 μm, (b) 2.5 μm, and (c) 5 μm. Axial displacement (DA) maps with contour spacing of 0.1 μm for a single scatterer with a diameter of (a) 1 μm, (b) 2.5 μm, and (c) 5 μm. 1D plots show the displacement profile along the white dashed line.

Equations (8)

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|E(θ,ϕ)|=|Einc(θ,ϕ)|nincn(cosθ)12,
E(θ,ϕ)=Enθ+Enϕ,
E(r)=(ExEyEz)=(mxnxmynymznz)(EE),
(m1n1u1)=(cosθcosϕcosθsinϕsinθsinϕcosϕ0sinθcosϕsinθsinϕcosθ)(m0n0u0),
(E,1E,1)=Einc(θ,ϕ)(cosϕsinϕsinϕcosϕ)(1+i00+i0)×nincn(cosθ)12.
(EUnscatEUnscat)=(E,1E,1)exp(ikd),
(EScatEScat)=1krs(S2(rs,θs)00S1(rs,θs))×(cosϕssinϕssinϕscosϕs)(E,iE,i),
ETot(r)=EScat(r)+EUnscat(r).

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