Abstract

The interaction of a Gaussian laser beam with a particle that is located off axis is a fundamental problem encountered across many scientific fields, including biological physics, chemistry, and medicine. For spherical geometries, generalized Lorenz–Mie theory affords a solution of Maxwell’s equations for the scattering from such a particle. The solution can be obtained by expanding the laser fields in terms of vector spherical harmonics (VSHs). However, the computation of the VSH expansion coefficients for off-axis beams has proven challenging. In the present study, we provide a very viable, theoretical framework to efficiently compute the sought-after expansion coefficients with high numerical accuracy. We use the existing theory for the expansion of an on-axis laser beam and employ Cruzan’s translation theorems [Q. Appl. Math. 20, 33 (1962)QAMAAY0033-569X] for the VSHs to obtain a description for more general off-axis beams. The expansion coefficients for the off-axis laser beam are presented in an analytical form in terms of an infinite series over the underlying translation coefficients. A direct comparison of the electromagnetic fields of such a beam expansion with the original laser fields and with results obtained using numerical quadratures shows excellent agreement (relative errors are on the order of 103). In practice, the analytical approach presented in this study has numerous applications, reaching from multiparticle scattering problems in atmospheric physics and climatology to optical trapping, sorting, and sizing techniques.

© 2011 Optical Society of America

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References

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  1. A. Jonáš and P. Zemánek, “Light at work: the use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813–4851 (2008).
    [CrossRef]
  2. J. Guck, R. Ananthakrishnan, T. Moon, C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5454 (2000).
    [CrossRef] [PubMed]
  3. L. Boyde, K. Chalut, and J. Guck, “Interaction of Gaussian beam with near-spherical particle: an analytic-numerical approach for assessing scattering and stresses,” J. Opt. Soc. Am. A 26, 1814–1826 (2009).
    [CrossRef]
  4. L. Boyde, M. Kreysing, K. Chalut, and J. Guck, “Physical insight into light scattering by photoreceptor cell nuclei,” Opt. Lett. 35, 2639–2641 (2010).
    [CrossRef] [PubMed]
  5. L. Boyde, K. Chalut, and J. Guck, “Near- and far-field scattering from arbitrary 3D aggregates of coated spheres using parallel computing,” Phys. Rev. E 83, 026701 (2011).
    [CrossRef]
  6. L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
    [CrossRef]
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    [CrossRef]
  10. 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]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. O. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  14. Ø. Farsund and B. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,” Physica A 227, 108–130 (1996).
    [CrossRef]
  15. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  16. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  17. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles, 1st ed. (Wiley, 1983).
  18. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran 90: the Art of Parallel Scientific Computing, 2nd ed. (Cambridge University, 1996).
  19. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
    [CrossRef] [PubMed]
  20. Y. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
    [CrossRef]
  21. M. Melamed, T. Lindmo, and M. Mendelsohn, Flow Cytometry and Sorting, 2nd ed. (Wiley, 1990).

2011 (1)

L. Boyde, K. Chalut, and J. Guck, “Near- and far-field scattering from arbitrary 3D aggregates of coated spheres using parallel computing,” Phys. Rev. E 83, 026701 (2011).
[CrossRef]

2010 (1)

2009 (1)

2008 (1)

A. Jonáš and P. Zemánek, “Light at work: the use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813–4851 (2008).
[CrossRef]

2003 (1)

2000 (1)

J. Guck, R. Ananthakrishnan, T. Moon, C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5454 (2000).
[CrossRef] [PubMed]

1998 (2)

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Y. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

1997 (1)

1996 (1)

Ø. Farsund and B. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,” Physica A 227, 108–130 (1996).
[CrossRef]

1995 (1)

1994 (2)

1989 (1)

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

1988 (2)

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]

G. Gouesbet, G. Gréhan, and B. Maheu, “Computations of the gn-coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

1962 (1)

O. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Alexander, D. R.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

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]

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, T. Moon, C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5454 (2000).
[CrossRef] [PubMed]

Backman, V.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Barton, J. P.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

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]

Bohren, C.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles, 1st ed. (Wiley, 1983).

Boyde, L.

Chalut, K.

Crawford, J.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Cruzan, O.

O. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Cunningham, C.

J. Guck, R. Ananthakrishnan, T. Moon, C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5454 (2000).
[CrossRef] [PubMed]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Doicu, A.

Farsund, Ø.

Ø. Farsund and B. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,” Physica A 227, 108–130 (1996).
[CrossRef]

Feld, M.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Felderhof, B.

Ø. Farsund and B. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,” Physica A 227, 108–130 (1996).
[CrossRef]

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran 90: the Art of Parallel Scientific Computing, 2nd ed. (Cambridge University, 1996).

Gouesbet, G.

Gréhan, G.

Guck, J.

L. Boyde, K. Chalut, and J. Guck, “Near- and far-field scattering from arbitrary 3D aggregates of coated spheres using parallel computing,” Phys. Rev. E 83, 026701 (2011).
[CrossRef]

L. Boyde, M. Kreysing, K. Chalut, and J. Guck, “Physical insight into light scattering by photoreceptor cell nuclei,” Opt. Lett. 35, 2639–2641 (2010).
[CrossRef] [PubMed]

L. Boyde, K. Chalut, and J. Guck, “Interaction of Gaussian beam with near-spherical particle: an analytic-numerical approach for assessing scattering and stresses,” J. Opt. Soc. Am. A 26, 1814–1826 (2009).
[CrossRef]

J. Guck, R. Ananthakrishnan, T. Moon, C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5454 (2000).
[CrossRef] [PubMed]

Hamano, T.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Han, Y.

Huffman, D.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles, 1st ed. (Wiley, 1983).

Itzkan, I.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Jonáš, A.

A. Jonáš and P. Zemánek, “Light at work: the use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813–4851 (2008).
[CrossRef]

Käs, J.

J. Guck, R. Ananthakrishnan, T. Moon, C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5454 (2000).
[CrossRef] [PubMed]

Kreysing, M.

Lima, C.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Lindmo, T.

M. Melamed, T. Lindmo, and M. Mendelsohn, Flow Cytometry and Sorting, 2nd ed. (Wiley, 1990).

Lock, J.

Maheu, B.

Manoharan, R.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Melamed, M.

M. Melamed, T. Lindmo, and M. Mendelsohn, Flow Cytometry and Sorting, 2nd ed. (Wiley, 1990).

Mendelsohn, M.

M. Melamed, T. Lindmo, and M. Mendelsohn, Flow Cytometry and Sorting, 2nd ed. (Wiley, 1990).

Moon, T.

J. Guck, R. Ananthakrishnan, T. Moon, C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5454 (2000).
[CrossRef] [PubMed]

Nusrat, A.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Perelman, L.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran 90: the Art of Parallel Scientific Computing, 2nd ed. (Cambridge University, 1996).

Schaub, S. A.

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]

Seiler, M.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Shields, S.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran 90: the Art of Parallel Scientific Computing, 2nd ed. (Cambridge University, 1996).

Van Dam, J.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran 90: the Art of Parallel Scientific Computing, 2nd ed. (Cambridge University, 1996).

Wallace, M.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Wriedt, T.

Xu, Y.

Y. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
[CrossRef] [PubMed]

Zemánek, P.

A. Jonáš and P. Zemánek, “Light at work: the use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813–4851 (2008).
[CrossRef]

Zonios, G.

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

Appl. Opt. (4)

Electrophoresis (1)

A. Jonáš and P. Zemánek, “Light at work: the use of optical forces for particle manipulation, sorting, and analysis,” Electrophoresis 29, 4813–4851 (2008).
[CrossRef]

J. Appl. Phys. (2)

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

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]

J. Comput. Phys. (1)

Y. Xu, “Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories,” J. Comput. Phys. 139, 137–165 (1998).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Phys. Rev. A (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Phys. Rev. E (1)

L. Boyde, K. Chalut, and J. Guck, “Near- and far-field scattering from arbitrary 3D aggregates of coated spheres using parallel computing,” Phys. Rev. E 83, 026701 (2011).
[CrossRef]

Phys. Rev. Lett. (2)

L. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. Crawford, and M. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–630 (1998).
[CrossRef]

J. Guck, R. Ananthakrishnan, T. Moon, C. Cunningham, and J. Käs, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451–5454 (2000).
[CrossRef] [PubMed]

Physica A (1)

Ø. Farsund and B. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,” Physica A 227, 108–130 (1996).
[CrossRef]

Q. Appl. Math. (1)

O. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Other (3)

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles, 1st ed. (Wiley, 1983).

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in Fortran 90: the Art of Parallel Scientific Computing, 2nd ed. (Cambridge University, 1996).

M. Melamed, T. Lindmo, and M. Mendelsohn, Flow Cytometry and Sorting, 2nd ed. (Wiley, 1990).

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Figures (3)

Fig. 1
Fig. 1

Geometrical setup. A Gaussian laser beam with electromagnetic fields E and H, beam waist w 0 , and focal point F ( x 0 , y 0 , z 0 ) propagates parallel to the + z direction of the unprimed Cartesian coordinate system, S. The fields can also be expressed in the primed Cartesian coordinate system, S , which is translated by the vector R . In S , the position of the laser focus is shifted ( r 0 = R + r 0 ).

Fig. 2
Fig. 2

Comparison of the dimensionless intensity, S e , for the exact (left) and the expanded fields (middle) of a Gaussian laser beam ( λ = 550 nm , w 0 = 4 λ ) with an off-axis focus ( k x 0 = k y 0 = k z 0 = 5 ). The observation plane is perpendicular to the optical axis and includes the focus. The absolute value of the difference between the exact and expanded fields is plotted on the right.

Fig. 3
Fig. 3

Comparison of the dimensionless intensity, S e , for the exact (left) and the expanded fields (middle) of a Gaussian laser beam ( λ = 550 nm , w 0 = 2 λ ) with an off-axis focus ( k x 0 = k y 0 = 40 and k z 0 = 10 ). The observation plane is parallel to the x z plane and shifted from the origin by a distance k y = 40 , such that it includes the focus. The absolute value of the difference between the exact and expanded fields is plotted on the right.

Tables (4)

Tables Icon

Table 1 Comparison of the Exact ( E θ ) and Expanded Polar Laser Field Components ( E θ ) at Various Points and for Different Beam Waists w 0 a

Tables Icon

Table 2 Comparison of the Exact ( E θ ) and Expanded Laser Field Components for Zero ( E θ ) and Nonzero ( E θ , ( off ) ) Offset of the Laser Focus a

Tables Icon

Table 3 Comparison of Different Orders of Expansion Coefficients Computed Using Either Numerical Quadrature ( p m n quad , q m n quad ) or Analytical Expressions ( p m n , q m n ) a

Tables Icon

Table 4 Comparison of the Analytical Expansion Coefficients of a Plane Wave ( p m n wave ) with the Field Expansion Coefficients of a Gaussian Laser Beam ( p m n ) in the Plane Wave Limit ( w 0 , z R ) a

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

E x = E 0 Q exp [ Q ( x x 0 ) 2 + ( y y 0 ) 2 w 0 2 + i k ( z z 0 ) ] ,
E z = i Q x x 0 z R E x ,
H y = H 0 Q exp [ Q ( x x 0 ) 2 + ( y y 0 ) 2 w 0 2 + i k ( z z 0 ) ] ,
H z = i Q y y 0 z R H y .
Q ( z ) = ( 1 + i z z 0 z R ) 1 .
M m n ( j ) ( r , θ , ϕ ) = i m sin θ z n ( j ) ( k r ) P n m ( cos θ ) e i m ϕ e θ z n ( j ) ( k r ) P n m ( cos θ ) θ e i m ϕ e ϕ ,
N m n ( j ) ( r , θ , ϕ ) = n ( n + 1 ) k r z n ( j ) ( k r ) P n m ( cos θ ) e i m ϕ e r + 1 k r r [ r z n ( j ) ( k r ) ] P n m ( cos θ ) θ e i m ϕ e θ + i m k r sin θ r [ r z n ( j ) ( k r ) ] P n m ( cos θ ) e i m ϕ e ϕ .
P n m ( x ) = { ( 1 ) m 2 n n ! ( 1 x 2 ) m 2 d n + m d x n + m ( x 2 1 ) n m > 0 , ( 1 ) | m | ( n | m | ) ! ( n + | m | ) ! P n | m | ( x ) m < 0.
E = n = 0 m = n + n [ p m n N m n ( 1 ) + q m n M m n ( 1 ) ] ,
H = 1 i Z n = 0 m = n + n [ p m n M m n ( 1 ) + q m n N m n ( 1 ) ] ,
E r = E x sin θ cos ϕ + E y sin θ sin ϕ + E z cos θ
E r = E 0 Q ( 1 i Q r cos θ z R ) exp [ Q r 2 sin 2 θ w 0 2 ] × exp [ i k ( r cos θ z 0 ) ] sin θ cos ϕ ,
H r = H 0 Q ( 1 i Q r cos θ z R ) exp [ Q r 2 sin 2 θ w 0 2 ] × exp [ i k ( r cos θ z 0 ) ] sin θ sin ϕ .
E 0 Q ˜ exp [ Q r 2 sin 2 θ w 0 2 ] exp [ i k ( r cos θ z 0 ) ] sin θ cos ϕ = + n = 0 m = n + n p m n n ( n + 1 ) j n ( k r ) k r P n m ( cos θ ) e i m ϕ ,
E 0 Q ˜ exp [ Q r 2 sin 2 θ w 0 2 ] exp [ i k ( r cos θ z 0 ) ] sin θ sin ϕ = n = 0 m = n + n i q m n n ( n + 1 ) j n ( k r ) k r P n m ( cos θ ) e i m ϕ .
E 0 Q ˜ exp [ Q r 2 sin 2 θ w 0 2 ] exp [ i k ( r cos θ z 0 ) ] sin θ = n = 1 o n 2 n ( n + 1 ) j n ( k r ) k r P n 1 ( cos θ ) ,
o n p + 1 n = q + 1 n .
E 0 Q ˜ * exp [ Q * r 2 sin 2 θ w 0 2 ] exp [ i k ( r cos θ z 0 ) ] sin θ = n = 1 ( 1 ) n i n 1 ( 2 n + 1 ) g n j n ( k r ) k r P n 1 ( cos θ ) .
o n = i n + 1 2 n + 1 2 n ( n + 1 ) g n * .
g 2 ν + 1 * = E 0 e i k z 0 μ = 0 ν Γ ( ν + μ + 3 2 ) Γ ( ν + 3 2 ) × ν ! μ ! ( ν μ ) ! Q 0 μ + 1 ( 2 i s ) 2 μ ,
g 2 ν + 2 * = E 0 e i k z 0 μ = 0 ν Γ ( ν + μ + 5 2 ) Γ ( ν + 5 2 ) [ 1 ( 2 + μ ) Q 0 k z R ] × ν ! μ ! ( ν μ ) ! Q 0 μ + 1 ( 2 i s ) 2 μ ,
p 1 n = ( n + 1 ) ! ( n 1 ) ! p + 1 n ,
q 1 n = + ( n + 1 ) ! ( n 1 ) ! q + 1 n .
M m n ( j ) ( r , θ , ϕ ) = ν = 0 μ = ν + ν [ A μ ν m n M μ ν ( κ ) ( r , θ , ϕ ) + B μ ν m n N μ ν ( κ ) ( r , θ , ϕ ) ] ,
N m n ( j ) ( r , θ , ϕ ) = ν = 0 μ = ν + ν [ A μ ν m n N μ ν ( κ ) ( r , θ , ϕ ) + B μ ν m n M μ ν ( κ ) ( r , θ , ϕ ) ] ,
κ = 1 for     r | R | , κ = j for     r > | R | ,
E = n = 0 m = n + n [ p m n N m n ( 1 ) + q m n M m n ( 1 ) ] ,
H = 1 i Z n = 0 m = n + n [ p m n M m n ( 1 ) + q m n N m n ( 1 ) ] ,
p m n = ν = 1 μ = ν + ν [ p μ ν A m n μ ν ( R ) + q μ ν B m n μ ν ( R ) ] ,
q m n = ν = 1 μ = ν + ν [ q μ ν A m n μ ν ( R ) + p μ ν B m n μ ν ( R ) ] .
p m n = ν = 1 o ν [ A m n + 1 ν + B m n + 1 ν ν ( ν + 1 ) ( A m n 1 ν B m n 1 ν ) ] ,
q m n = ν = 1 o ν [ A m n + 1 ν + B m n + 1 ν + ν ( ν + 1 ) ( A m n 1 ν B m n 1 ν ) ] .
E r ( r , θ , ϕ ) = n = 0 m = n + n p m n N m n , r ( 1 ) ( r , θ , ϕ ) ,
H r ( r , θ , ϕ ) = 1 i Z n = 0 m = n + n q m n N m n , r ( 1 ) ( r , θ , ϕ ) .
O ^ = θ = 0 π d θ sin θ P n m ( cos θ ) ϕ = 0 2 π d ϕ e i m ϕ × ,
p m n q m n = lim r r 0 1 4 π 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! k r j n ( k r ) × ϕ = 0 2 π θ = 0 π d θ d ϕ sin θ P n m ( cos θ ) e i m ϕ E r ( r , θ , ϕ ) i Z H r ( r , θ , ϕ ) .
E r = [ sin θ cos ϕ i ( r sin θ cos ϕ x 0 ) / z R 1 + i ( r cos θ z 0 ) / z R cos θ ] E x ,
H r = [ sin θ sin ϕ i ( r sin θ sin ϕ y 0 ) / z R 1 + i ( r cos θ z 0 ) / z R cos θ ] E x Z ,
E x = E 0 exp [ i k ( z z 0 ) ] ,
H y = H 0 exp [ i k ( z z 0 ) ] .
p m n = E 0 i n 1 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! e i k z 0 × [ τ m n ( cos α ) cos β i π m n ( cos α ) sin β ] ,
q m n = E 0 i n 1 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! e i k z 0 × [ π m n ( cos α ) cos β i τ m n ( cos α ) sin β ] .
π m n ( 1 ) = { 1 2 m = 1 1 2 n ( n + 1 ) m = + 1 0 otherwise ,
τ m n ( 1 ) = { + 1 2 m = 1 1 2 n ( n + 1 ) m = + 1 0 otherwise .
p + 1 n = + q + 1 n = + E 0 i n + 1 2 n + 1 2 n ( n + 1 ) e i k z 0 ,
p 1 n = q 1 n = E 0 i n + 1 1 2 ( 2 n + 1 ) e i k z 0 .
S e = 1 2 Re { ϵ [ E r E r * + E θ E θ * + E ϕ E ϕ * ] } .

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