Abstract

Direct measurement of optical forces based on recording the change of momentum between the in- and outgoing light does not have specific requirements on particle size or shape, or on beam shape. Thus this approach overcomes many of the limitations of force measurements based on position measurements, which require frequent calibration. In this work we validate the achievable accuracy for direct force measurements in the axial direction for a single beam optical tweezers setup, based on numerical simulations and experimental investigations of situations, where the true force is known. We find that for typical experimental situations a good accuracy with an error of less than 1 % of the maximum force can be achieved, independent of particle size or refractive index, provided that the total amount of light scattered in the backward direction is also taken into account, which is easy to accomplish experimentally. Due to the inherent particle shape independence of the direct force measurement method, these findings support that it provides accurate results for 3D force measurements for particles of arbitrary shape.

© 2015 Optical Society of America

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References

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [Crossref]
  2. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
    [Crossref]
  3. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594– 612 (2004).
    [Crossref]
  4. S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
    [Crossref]
  5. S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
    [Crossref] [PubMed]
  6. S. B. Smith, Y. Cui, and C. Bustamante, “Optical-trap force transducer that operates by direct measurement of light momentum,” in “Methods in Enzymology,”, vol. 361 of Biophotonics, Part B, Gerard Marriott and Ian Parker, eds. (Academic, 2003), pp. 134–162.
    [Crossref]
  7. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
    [Crossref]
  8. A. Farré and M. Montes-Usategui, “A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,” Opt. Express 18, 11955–11968 (2010).
    [Crossref] [PubMed]
  9. A. Farré, F. Marsà, and M. Montes-Usategui, “Optimized back-focal-plane interferometry directly measures forces of optically trapped particles,” Opt. Express 20, 12270–12291 (2012).
    [Crossref] [PubMed]
  10. Y. Jun, S. K. Tripathy, B. R. J. Narayanareddy, M. K. Mattson-Hoss, and S. P. Gross, “Calibration of optical tweezers for in vivo force measurements: How do different approaches compare?” Biophys. J. 107, 1474–1484 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  22. N. Schaeffer, “Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations,” Geochem. Geophys. Geosyst. 14, 751–758 (2013).
    [Crossref]

2014 (2)

Y. Jun, S. K. Tripathy, B. R. J. Narayanareddy, M. K. Mattson-Hoss, and S. P. Gross, “Calibration of optical tweezers for in vivo force measurements: How do different approaches compare?” Biophys. J. 107, 1474–1484 (2014).
[Crossref] [PubMed]

D. Ott, S. Nader, S. Reihani, and L. B. Oddershede, “Simultaneous three-dimensional tracking of individual signals from multi-trap optical tweezers using fast and accurate photodiode detection,” Opt. Express 22, 23661– 23672 (2014).
[Crossref] [PubMed]

2013 (2)

F. Marsà, A. Farré, E. Martín-Badosa, and M. Montes-Usategui, “Holographic optical tweezers combined with back-focal-plane displacement detection,” Opt. Express 21, 30282–30294 (2013).
[Crossref]

N. Schaeffer, “Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations,” Geochem. Geophys. Geosyst. 14, 751–758 (2013).
[Crossref]

2012 (1)

2011 (3)

2010 (1)

2007 (1)

2006 (1)

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

2004 (3)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594– 612 (2004).
[Crossref]

J. K. Dreyer, K. Berg-Sørensen, and L. Oddershede, “Improved axial position detection in optical tweezers measurements,” Appl. Opt. 43, 1991–1995 (2004).
[Crossref] [PubMed]

2002 (2)

P. Bartlett and S. Henderson, “Three-dimensional force calibration of a single-beam optical gradient trap,” J. Phys.: Condens. Matter 14, 7757–7768 (2002).

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[Crossref]

2001 (1)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[Crossref]

1996 (1)

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[Crossref] [PubMed]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

Ashkin, A.

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

Bartlett, P.

P. Bartlett and S. Henderson, “Three-dimensional force calibration of a single-beam optical gradient trap,” J. Phys.: Condens. Matter 14, 7757–7768 (2002).

Behrndt, M.

Berg-Sørensen, K.

J. K. Dreyer, K. Berg-Sørensen, and L. Oddershede, “Improved axial position detection in optical tweezers measurements,” Appl. Opt. 43, 1991–1995 (2004).
[Crossref] [PubMed]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594– 612 (2004).
[Crossref]

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

Bustamante, C.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[Crossref] [PubMed]

S. B. Smith, Y. Cui, and C. Bustamante, “Optical-trap force transducer that operates by direct measurement of light momentum,” in “Methods in Enzymology,”, vol. 361 of Biophotonics, Part B, Gerard Marriott and Ian Parker, eds. (Academic, 2003), pp. 134–162.
[Crossref]

Cui, Y.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[Crossref] [PubMed]

S. B. Smith, Y. Cui, and C. Bustamante, “Optical-trap force transducer that operates by direct measurement of light momentum,” in “Methods in Enzymology,”, vol. 361 of Biophotonics, Part B, Gerard Marriott and Ian Parker, eds. (Academic, 2003), pp. 134–162.
[Crossref]

Dreyer, J. K.

Farré, A.

Flyvbjerg, H.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594– 612 (2004).
[Crossref]

Grier, D. G.

Grill, S. W.

Gross, S. P.

Y. Jun, S. K. Tripathy, B. R. J. Narayanareddy, M. K. Mattson-Hoss, and S. P. Gross, “Calibration of optical tweezers for in vivo force measurements: How do different approaches compare?” Biophys. J. 107, 1474–1484 (2014).
[Crossref] [PubMed]

Heckenberg, N. R.

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58, 528–544 (2011).
[Crossref]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[Crossref]

Henderson, S.

P. Bartlett and S. Henderson, “Three-dimensional force calibration of a single-beam optical gradient trap,” J. Phys.: Condens. Matter 14, 7757–7768 (2002).

Howard, J.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975), 2nd ed.

Jahnel, M.

Jannasch, A.

Jülicher, F.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Jun, Y.

Y. Jun, S. K. Tripathy, B. R. J. Narayanareddy, M. K. Mattson-Hoss, and S. P. Gross, “Calibration of optical tweezers for in vivo force measurements: How do different approaches compare?” Biophys. J. 107, 1474–1484 (2014).
[Crossref] [PubMed]

Kim, S.-H.

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002), 3 electronic release ed.

Lee, S.-H.

Loke, V. L.

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58, 528–544 (2011).
[Crossref]

Marsà, F.

Martín-Badosa, E.

Mattson-Hoss, M. K.

Y. Jun, S. K. Tripathy, B. R. J. Narayanareddy, M. K. Mattson-Hoss, and S. P. Gross, “Calibration of optical tweezers for in vivo force measurements: How do different approaches compare?” Biophys. J. 107, 1474–1484 (2014).
[Crossref] [PubMed]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002), 3 electronic release ed.

Montes-Usategui, M.

Nader, S.

Narayanareddy, B. R. J.

Y. Jun, S. K. Tripathy, B. R. J. Narayanareddy, M. K. Mattson-Hoss, and S. P. Gross, “Calibration of optical tweezers for in vivo force measurements: How do different approaches compare?” Biophys. J. 107, 1474–1484 (2014).
[Crossref] [PubMed]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

Nieminen, T. A.

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58, 528–544 (2011).
[Crossref]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[Crossref]

Oddershede, L.

Oddershede, L. B.

Ott, D.

Pavone, F. S.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Reihani, S.

Reihani, S. N. S.

Rohrbach, A.

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[Crossref]

Roichman, Y.

Rubinsztein-Dunlop, H.

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58, 528–544 (2011).
[Crossref]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[Crossref]

Samadi, A.

Schaeffer, N.

N. Schaeffer, “Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations,” Geochem. Geophys. Geosyst. 14, 751–758 (2013).
[Crossref]

Schäffer, E.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Schffer, E.

Smith, S. B.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[Crossref] [PubMed]

S. B. Smith, Y. Cui, and C. Bustamante, “Optical-trap force transducer that operates by direct measurement of light momentum,” in “Methods in Enzymology,”, vol. 361 of Biophotonics, Part B, Gerard Marriott and Ian Parker, eds. (Academic, 2003), pp. 134–162.
[Crossref]

Stelzer, E. H. K.

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[Crossref]

Stilgoe, A. B.

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58, 528–544 (2011).
[Crossref]

Tolic-Nørrelykke, S. F.

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002), 3 electronic release ed.

Tripathy, S. K.

Y. Jun, S. K. Tripathy, B. R. J. Narayanareddy, M. K. Mattson-Hoss, and S. P. Gross, “Calibration of optical tweezers for in vivo force measurements: How do different approaches compare?” Biophys. J. 107, 1474–1484 (2014).
[Crossref] [PubMed]

van Blaaderen, A.

van Oostrum, P.

Yang, S.-M.

Yi, G.-R.

Appl. Opt. (1)

Biophys. J. (1)

Y. Jun, S. K. Tripathy, B. R. J. Narayanareddy, M. K. Mattson-Hoss, and S. P. Gross, “Calibration of optical tweezers for in vivo force measurements: How do different approaches compare?” Biophys. J. 107, 1474–1484 (2014).
[Crossref] [PubMed]

Geochem. Geophys. Geosyst. (1)

N. Schaeffer, “Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations,” Geochem. Geophys. Geosyst. 14, 751–758 (2013).
[Crossref]

J. Appl. Phys. (1)

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
[Crossref]

J. Mod. Opt. (1)

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58, 528–544 (2011).
[Crossref]

J. Phys.: Condens. Matter (1)

P. Bartlett and S. Henderson, “Three-dimensional force calibration of a single-beam optical gradient trap,” J. Phys.: Condens. Matter 14, 7757–7768 (2002).

J. Quant. Spectrosc. Radiat. Transfer (1)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

Rev. Sci. Instrum. (3)

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594– 612 (2004).
[Crossref]

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006).
[Crossref]

Science (1)

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching b-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[Crossref] [PubMed]

Other (3)

S. B. Smith, Y. Cui, and C. Bustamante, “Optical-trap force transducer that operates by direct measurement of light momentum,” in “Methods in Enzymology,”, vol. 361 of Biophotonics, Part B, Gerard Marriott and Ian Parker, eds. (Academic, 2003), pp. 134–162.
[Crossref]

J. D. Jackson, Classical Electrodynamics (Wiley, 1975), 2nd ed.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002), 3 electronic release ed.

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Figures (8)

Fig. 1
Fig. 1

Principle of the direct force measurement method. The optical force exerted by transfer of momentum from the light field to the particle can be directly deduced from the angular or the far-field intensity distribution of the in- and outgoing trapping light. Here ϑ and φ denote the usual polar and azimuthal angles of spherical coordinates for a ray, which is mapped to a position x, y (in cartesian coordinates) in the back focal plane of the condenser.

Fig. 2
Fig. 2

Experimental setup.

Fig. 3
Fig. 3

Calculated axial force profile for a 3 μm diameter polystyrene bead. (a) True axial force Qz (solid blue line) and estimate Qz,f from forward scattered light only (red dashed line). Also shown (black dotted line) is the axial BFPI signal (for a condenser with NA = 0.8), scaled and shifted such that value and slope coincides with the true force at the zero crossing. (b) Difference ΔQ of the estimated and the true force (red dashed line), and normalized total power Pr of the light that is scattered in the backward direction and that is collected by the objective lens with NA = 1.2.

Fig. 4
Fig. 4

Calculation of the accuracy for polystyrene beads from 0.5 μm to 4 μm diameter. (a) Measurement error ΔQz = Qz,fQz for the direct force measurement method versus true force, without correction (blue lines) and with additional correction (green lines) based on the total amount of backward scatterd light. The results for some specific bead sizes are highlighted. (b) Error for force measurements based on BFPI (for a condenser with NA = 0.8), assuming a linear relationship between force and axial BFPI signal. For each bead size the slope and value at equilibrium was matched to the true force, mimicking a force calibration.

Fig. 5
Fig. 5

Experimental comparison of the direct force measurement method with drag forces for 1.8 μm and 3 μm diameter polystyrene beads (left column), and corresponding simulations (right column).

Fig. 6
Fig. 6

Simulation of the accuracy of the direct force measurement method depending on the particles refractive index. Every single point gives the maximum error relative to the maximum force for a certain particle size in the range from 0.5 μm to 4 μm. (a) Error without correction, (b) with an offset compensation at the equlibrium position, (c) with correction based on the total power of the back scattered light (note the different scale).

Fig. 7
Fig. 7

Measurement error for a 2.1 μm diameter silica bead. Due to the low refractive index (n = 1.44) scattering of the trapping light is reduced, and the direct force measurement method yields accurate results even without correction.

Fig. 8
Fig. 8

Direct measurement of the total optical force acting on two simultaneously trapped beads (polystyrene, 3 μm diameter).

Equations (17)

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

p ˙ = 1 c I ( ϑ , φ ) ( sin ϑ cos φ sin ϑ sin φ cos ϑ )
F f = ( F f , x F f , y F f , z ) = 1 c m I ( x , y ) ( x / R y / R 1 ( x 2 + y 2 ) / R 2 ) d x d y F 0 .
F ^ z = F z , f + k P P r / c m
I ( x 1 , y 1 ) = I 0 exp ( x 1 2 + y 1 2 σ 2 ) ,
E x = N I exp ( i ψ )
ψ ( x , y ) = k ( x 0 x 1 + y 0 y 1 + z 0 1 x 1 2 y 1 2 ) = k ( x 0 sin ϑ cos φ + y 0 sin ϑ sin φ + z 0 cos ϑ ) .
( E ϑ E φ ) = cos ϑ ( cos φ sin φ sin φ cos φ ) ( E x E y ) .
E = l = 1 l max m = l l s lm S lm ( ϑ , φ ) + t lm T lm ( ϑ , φ )
Y lm = 2 l + 1 4 π ( l m ) ! ( l + m ) ! P l m ( cos ϑ ) exp ( i m φ ) ,
P l m ( x ) = ( 1 ) m ( 1 x 2 ) m / 2 d m d x m P l ( x ) .
s lm = 1 l ( l + 1 ) S lm * E d Ω t lm = 1 l ( l + 1 ) T lm * E d Ω
S lm * S l m d Ω = l ( l + 1 ) δ l l δ m m ,
s lm = { s lm , R + i s lm , I if m 0 s l | m | , R * + i s l | m | , I * if m < 0
s lm , R = 1 2 ( s lm + s l m ) , s lm , R = 1 2 ( s lm s l m ) , s lm , I = 1 2 ( s lm + s l m ) , s lm , I = 1 2 ( s lm s l m ) .
s lm out = ( 1 ) l ( 1 2 α l ) s lm , t lm out = ( 1 ) l + 1 ( 1 2 β l ) t lm ,
α l = n r ψ l ( n r ρ ) ψ l ( ρ ) ψ l ( ρ ) ψ l ( n r ρ ) n r ψ l ( n r ρ ) ξ l ( ρ ) ξ l ( ρ ) ψ l ( n r ρ ) , β l = ψ l ( n r ρ ) ψ l ( ρ ) n r ψ l ( ρ ) ψ l ( n r ρ ) ψ l ( n r ρ ) ξ l ( ρ ) n r ξ l ( ρ ) ψ l ( n r ρ ) ,
Q z = ( | E out | 2 | E in | 2 ) cos ϑ d Ω ,

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