Abstract

The efficiency of trapping an on-axis spherical particle by use of laser tweezers for a particle size from the Rayleigh limit to the ray optics limit is calculated from generalized Lorenz-Mie light-scattering theory and the localized version of a Gaussian beam that has been truncated and focused by a high-numerical-aperture lens and that possesses spherical aberration as a result of its transmission through the wall of the sample cell. The results are compared with both the experimental trapping efficiency and the theoretical efficiency obtained from use of the localized version of a freely propagating focused Gaussian beam. The predicted trapping efficiency is found to decrease as a function of the depth of the spherical particle in the sample cell owing to an increasing amount of spherical aberration. The decrease in efficiency is also compared with experiment.

© 2004 Optical Society of America

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References

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  1. J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. 1. Localized model description of a tightly focused laser beam with spherical aberration,” Appl Opt. 43, 2532–2544 (2004).
    [CrossRef] [PubMed]
  2. W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
    [CrossRef]
  5. J. S. Kim, S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
    [CrossRef]
  6. G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  7. R. Pobre, C. Saloma, “Radiation force on a nonlinear microsphere by a tightly focused Gaussian beam,” Appl. Opt. 41, 7694–7701 (2002).
    [CrossRef]
  8. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 668, Eqs. (12.81) and (12.81a) and footnote 2.
  9. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 123.
  10. J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  11. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  12. Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
    [CrossRef]
  13. A. Rohrbach, E. H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18, 839–853 (2001).
    [CrossRef]
  14. Ref. 9, pp. 143–144.
  15. Ref. 9, p. 127.
  16. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
    [CrossRef] [PubMed]
  17. R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9, 1922–1930 (1992).
    [CrossRef]
  18. S. Nemoto, H. Togo, “Axial force acting on a dielectric sphere in a focused laser beam,” Appl. Opt. 37, 6386–6394 (1998).
    [CrossRef]
  19. W. J. Glantschnig, S.-H. Chen, “Light scattering from water droplets in the geometrical optics approximation,” Appl. Opt. 20, 2499–2509 (1981).
    [CrossRef] [PubMed]
  20. A. Ungut, G. Grehan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz-Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
    [CrossRef] [PubMed]
  21. Y. Takano, M. Tanaka, “Phase matrix and cross sections for single scattering by circular cylinders: a comparison of ray optics and wave theory,” Appl. Opt. 19, 2781–2793 (1980).
    [CrossRef] [PubMed]
  22. E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
    [CrossRef]
  23. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  24. A. C. Dogariu, R. Rajagopalan, “Optical traps as force transducers: the effects of focusing the trapping beam through a dielectric interface,” Langmuir 16, 2770–2778 (2000).
    [CrossRef]
  25. A. Rohrbach, E. H. K. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. 41, 2494–2507 (2002).
    [CrossRef] [PubMed]
  26. P. Torok, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  27. P. Torok, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation (errata),” J. Opt. Soc. Am. A 12, 1605 (1995).
    [CrossRef]
  28. 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]
  29. R. Richards, 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]
  30. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 634, Eq. (5.52.1).
  31. Ref. 30, p. 692, Eq. (6.574.2).
  32. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), pp. 369–370, Eqs. (9.4.1) and (9.4.3).
  33. Ref. 32, pp. 360 and 364, Eqs. (9.1.10), (9.2.5), (9.2.9), and (9.2.10).
  34. M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1998), p. 477, Fig. 9.3.
  35. P. Torok, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I.” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  36. A. M. MacRobert, “Star-test your telescope,” Sky Telescope 89(3), 42–47 (1995), unnumbered figure on p. 46.
  37. G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991).
    [CrossRef]
  38. G. Videen, “Light scattering from a sphere on or near a surface (errata),” J. Opt. Soc. Am. A 9, 844–845 (1992).
    [CrossRef]
  39. B. R. Johnson, “Calculation of light scattering from a spherical particle on a surface by the multipole expansion method,” J. Opt. Soc. Am. A 13, 326–337 (1996).
    [CrossRef]
  40. E. Fucile, P. Denti, F. Borghese, R. Saija, O. I. Sindoni, “Optical properties of a sphere in the vicinity of a plane surface,” J. Opt. Soc. Am. A 14, 1505–1514 (1997).
    [CrossRef]

2004 (1)

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. 1. Localized model description of a tightly focused laser beam with spherical aberration,” Appl Opt. 43, 2532–2544 (2004).
[CrossRef] [PubMed]

2002 (2)

2001 (1)

2000 (1)

A. C. Dogariu, R. Rajagopalan, “Optical traps as force transducers: the effects of focusing the trapping beam through a dielectric interface,” Langmuir 16, 2770–2778 (2000).
[CrossRef]

1998 (1)

1997 (1)

1996 (2)

B. R. Johnson, “Calculation of light scattering from a spherical particle on a surface by the multipole expansion method,” J. Opt. Soc. Am. A 13, 326–337 (1996).
[CrossRef]

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

1995 (5)

1994 (2)

1992 (4)

1991 (1)

1989 (2)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

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

1988 (1)

1986 (1)

1983 (1)

1981 (2)

1980 (1)

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]

R. Richards, 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, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

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

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 668, Eqs. (12.81) and (12.81a) and footnote 2.

Asakura, T.

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
[CrossRef] [PubMed]

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

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

Berns, M. W.

Bjorkholm, J. E.

Booker, G. R.

Borghese, F.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1998), p. 477, Fig. 9.3.

Brevik, I.

Chen, S.-H.

Chu, S.

Denti, P.

Dogariu, A. C.

A. C. Dogariu, R. Rajagopalan, “Optical traps as force transducers: the effects of focusing the trapping beam through a dielectric interface,” Langmuir 16, 2770–2778 (2000).
[CrossRef]

Dziedzic, J. M.

Felgner, H.

Fucile, E.

Glantschnig, W. J.

Gouesbet, G.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 634, Eq. (5.52.1).

Grehan, G.

Gussgard, R.

Harada, Y.

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

Hovenac, E. A.

Johnson, B. R.

Kim, J. S.

Laczik, Z.

Lee, S. S.

Lindmo, T.

Lock, J. A.

MacRobert, A. M.

A. M. MacRobert, “Star-test your telescope,” Sky Telescope 89(3), 42–47 (1995), unnumbered figure on p. 46.

Maheu, B.

Muller, O.

Nemoto, S.

Pobre, R.

Rajagopalan, R.

A. C. Dogariu, R. Rajagopalan, “Optical traps as force transducers: the effects of focusing the trapping beam through a dielectric interface,” Langmuir 16, 2770–2778 (2000).
[CrossRef]

Richards, R.

R. Richards, 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]

Rohrbach, A.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 634, Eq. (5.52.1).

Saija, R.

Saloma, C.

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Schliwa, M.

Sindoni, O. I.

Sonek, G. J.

Stelzer, E. H. K.

Takano, Y.

Tanaka, M.

Togo, H.

Torok, P.

Ungut, A.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 123.

Varga, P.

Videen, G.

Wolf, E.

R. Richards, 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]

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]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1998), p. 477, Fig. 9.3.

Wright, W. H.

Appl Opt. (1)

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. 1. Localized model description of a tightly focused laser beam with spherical aberration,” Appl Opt. 43, 2532–2544 (2004).
[CrossRef] [PubMed]

Appl. Opt. (8)

Biophys. J. (1)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

J. Appl. Phys. (2)

J. P. Barton, 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, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (11)

G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

A. Rohrbach, E. H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18, 839–853 (2001).
[CrossRef]

G. Videen, “Light scattering from a sphere on or near a surface (errata),” J. Opt. Soc. Am. A 9, 844–845 (1992).
[CrossRef]

B. R. Johnson, “Calculation of light scattering from a spherical particle on a surface by the multipole expansion method,” J. Opt. Soc. Am. A 13, 326–337 (1996).
[CrossRef]

P. Torok, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

P. Torok, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation (errata),” J. Opt. Soc. Am. A 12, 1605 (1995).
[CrossRef]

P. Torok, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I.” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

E. Fucile, P. Denti, F. Borghese, R. Saija, O. I. Sindoni, “Optical properties of a sphere in the vicinity of a plane surface,” J. Opt. Soc. Am. A 14, 1505–1514 (1997).
[CrossRef]

G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991).
[CrossRef]

E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
[CrossRef]

J. Opt. Soc. Am. B (1)

Langmuir (1)

A. C. Dogariu, R. Rajagopalan, “Optical traps as force transducers: the effects of focusing the trapping beam through a dielectric interface,” Langmuir 16, 2770–2778 (2000).
[CrossRef]

Opt. Commun. (1)

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

Opt. Lett. (1)

Proc. R. Soc. London Ser. A (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]

R. Richards, 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]

Sky Telescope (1)

A. M. MacRobert, “Star-test your telescope,” Sky Telescope 89(3), 42–47 (1995), unnumbered figure on p. 46.

Other (9)

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 668, Eqs. (12.81) and (12.81a) and footnote 2.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 123.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 634, Eq. (5.52.1).

Ref. 30, p. 692, Eq. (6.574.2).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), pp. 369–370, Eqs. (9.4.1) and (9.4.3).

Ref. 32, pp. 360 and 364, Eqs. (9.1.10), (9.2.5), (9.2.9), and (9.2.10).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1998), p. 477, Fig. 9.3.

Ref. 9, pp. 143–144.

Ref. 9, p. 127.

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Tables (4)

Tables Icon

Table 1 Contribution to Minimum Radiation Trapping Efficiency Qmin of External Reflection (ER), Transmission (T), and Transmission Followed by p - 1 Internal Reflections (IR p-1) for the Localized Version of a Focused Gaussian Beam with λ = 0.488 μm, n = 1.33, m = 1.2, a = 5.0 μm, w i = 0.172 μm, w a = 0.200 μm, and z0 max = -5.21 μm

Tables Icon

Table 2 Minimum Value of Radiation Trapping Efficiency Qmin As a Function of Particle Radius a for the Localized Version of a Focused Gaussian Beam with λ = 1.06 μm, n = 1.33, m = 1.18, and w a = 0.390 μm Incident upon the Particlea

Tables Icon

Table 3 Minimum Value of Radiation Trapping Efficiency Qmin As a Function of Particle Radius a for a Gaussian Beam Truncated and Focused by a Lens and Transmitted through a Flat Interface with λ = 1.06 μm, w a = 0.390 μm, W/A = 1.5, α = 60°, n1 = 1.5, n2= 1.33, and m = 1.18 Incident upon the Particlea

Tables Icon

Table 4 Minimum Value of Radiation Trapping Efficiency Qmin As a Function of Interface Position d for a Gaussian Beam Truncated and Focused by a Lens and Transmitted through a Flat Interface with λ = 1.06 μm, W/A = 1.5, α = 60°, n1 = 1.5, n2 = 1.33, m = 1.18, and a = 4.935 μma

Equations (45)

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

I=E*×B/μ0,
Fz=n/cnE02/μ0cπ/n2k2,
=l=1ll+2/l+1glgl+1*Ul+gl*gl+1Ul*+hlhl+1*Vl+hl*hl+1Vl*+2l+1/ll+1×glhl*Wl+gl*hlWl*,
Ul=al+al+1*-2alal+1*,
Vl=bl+bl+1*-2blbl+1*,
Wl=al+bl*-2albl*.
al=ΨlXΨlY-mΨlXΨlY/ζlXΨlY-mζlXΨlY,
bl=mΨlXΨlY-ΨlXΨlY/mζlXΨlY-ζlXΨlY,
Xnka,
YmX,
Ψlzzjlz,
ζlzzhl1z,
FznP/cQ,
gl=hl =exp-inkz0exp-si2l+2l-1/1-2isiz0/wi/1-2isiz0/wi,
si=1/nkwi.
sa=1/nkwa,
PnE02/μ0cπwa2/2.
Q=2sa2.
F=n/cnE02/μ0c2πa3m2-1/m2+2e*·eparticle+n/cnE02/μ0c×8πn4k4a6/3m2-1/m2+22e*×bparticle,
E=E0e,
B=nE0/cb.
e=uxD expinkz-z0exp-Dx2+y2/w2,
b=uyD expinkz-z0exp-Dx2+y2/w2,
D=1/1+2isz-z0/w.
Q=32X3s5m2-1/m2+2z0/w/1+4s2z02/w22+16/3X6s2m2-1/m2+22/1+4s2z02/w2,
a1=-2i/3m2-1/m2+2X3-2i/5×m2-1m2-2/m2+22X5+4/9×m2-1/m2+22X6+OiX7,
b1=-i/45m2-1X5+OiX7,
a2=-i/15m2-1/2m2+3X5+OiX7,
b2=OiX7.
=3/2g1g2*a1+g1*g2a1*+g1h1*a1+g1*h1a1*.
=X3m2-1/m2+22K sinΘ+4/3X3m2-1/m2+21+K cosΘ/1+4s2z02/w2,
Kexp-4s2/1+4s2z02/w2,
Θ8s3z0/w/1+4s2z02/w2.
=16s3X3m2-1/m2+2z0/w/1+4s2z02/w22+8/3X6m2-1/m2+22/1+4s2z02/w2.
gl=-in1kF 0αsinθ1dθ1cosθ11/2×expin2k cosθ2z-d-n1k cosθ1×z0-d1/2exp-A/W2 tan2θ1/tan2α×tTE+tTM cosθ2J0n1/n2×l+1/2sinθ1+tTE-tTM cosθ2×J2n1/n2l+1/2sinθ1,
hl=-in1kF 0αsinθ1dθ1cosθ11/2×expin2k cosθ2z-d-n1k cosθ1×z0-d1/2exp-A/W2 tan2θ1/tan2α×tTM+tTE cosθ2J0n1/n2l+1/2sinθ1+tTM-tTE cosθ2J2n1/n2×l+1/2sinθ1.
NA=n1 sinα;
n1 sinθ1=n2 sinθ2;
tTE=2 cosθ1/cosθ1+n2/n1cosθ2,
tTM=2 cosθ1/n2/n1cosθ1+cosθ2.
Pn1E02/μ0cπF2 sin2α.
Gα=1-A/W2 cos2α-2/3A/W2 sin2αcos2α+2/3A/W4 cos4α+.
T12=n2/2n10αsinθ1dθ1 cosθ2tTE2+tTM2×exp-2A/W2 tan2θ1/tan2α0αsinθ1dθ1 cosθ1×exp-2A/W2 tan2θ1/tan2α.
Q=/n1n2k2F2 sin2αGαT12.
F=A tanα.

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