Abstract

Numerical computations in the framework of the generalized Lorenz–Mie theory require the evaluation of a new double set of coefficients gn,TMm and gn,TEm (n = 1, …, ∞; m = − n, … +n). A localized interpretation of these coefficients is designed to permit fast and accurate computations, even on microcomputers. When the scatter center is located on the axis of the beam, a previously published localized approximation for a simpler set of coefficients gn is recovered as a special case. The subscript n in coefficients gn and gnm is associated with ray localization and discretization of space in directions perpendicular to the beam axis, while superscript m in coefficients gnm is associated with azimuthal wave modes.

© 1990 Optical Society of America

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References

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  1. G. Gouesbet, G. Gréhan, eds., Proceedings of the International Symposium on Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988).
    [CrossRef]
  2. G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
    [CrossRef]
  3. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [CrossRef]
  4. G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 99–194 (1977).
    [CrossRef]
  5. G. Gréhan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
    [CrossRef] [PubMed]
  6. B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: numerical results and application to optical sizing,” Part. Part. Syst. Character. 4, 141–147 (1987).
    [CrossRef]
  7. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
    [CrossRef]
  8. G. Gouesbet, B. Maheu, G. Gréhan, “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]
  9. B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
    [CrossRef]
  10. G. Gouesbet, “Advanced theory of light scattering. Application to optical sizing,” in 6th Symposium of Japan Association of Aerosol Science and Technology (JAAST, Osaka, 1988), pp. 106–108.
  11. G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz-Mie theory and application to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, to be published).
  12. F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computation on a microcomputer,” Part. Part. Syst. Character. 5, 103–108 (1988).
    [CrossRef]
  13. G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4833 (1988).
    [CrossRef] [PubMed]
  14. G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef] [PubMed]
  15. G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
    [CrossRef]
  16. G. Gouesbet, G. Gréhan, B. Maheu, “The localized interpretation to compute the coefficients gn, gn1, and gn−1 in the framework of the generalized Lorenz-Mie theory,” in Optical Methods in Flows and Particle Diagnostics (International Congress on Applications of Lasers and Electro-Optics, Santa Clara, Calif., 1988), Vol. 67, pp. 263–273.
  17. G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz-Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
    [CrossRef]
  18. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  19. G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 2, 83–93 (1985).
  20. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

1989

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz-Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

1988

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “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]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computation on a microcomputer,” Part. Part. Syst. Character. 5, 103–108 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4833 (1988).
[CrossRef] [PubMed]

1987

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: numerical results and application to optical sizing,” Part. Part. Syst. Character. 4, 141–147 (1987).
[CrossRef]

1986

1985

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 2, 83–93 (1985).

1982

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

1980

1979

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

1977

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 99–194 (1977).
[CrossRef]

1970

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

1908

G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Ashkin, A.

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

Corbin, F.

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computation on a microcomputer,” Part. Part. Syst. Character. 5, 103–108 (1988).
[CrossRef]

Davis, L. W.

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

Gouesbet, G.

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz-Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computation on a microcomputer,” Part. Part. Syst. Character. 5, 103–108 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4833 (1988).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “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]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: numerical results and application to optical sizing,” Part. Part. Syst. Character. 4, 141–147 (1987).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 2, 83–93 (1985).

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gréhan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
[CrossRef] [PubMed]

G. Gouesbet, “Advanced theory of light scattering. Application to optical sizing,” in 6th Symposium of Japan Association of Aerosol Science and Technology (JAAST, Osaka, 1988), pp. 106–108.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz-Mie theory and application to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, to be published).

G. Gouesbet, G. Gréhan, B. Maheu, “The localized interpretation to compute the coefficients gn, gn1, and gn−1 in the framework of the generalized Lorenz-Mie theory,” in Optical Methods in Flows and Particle Diagnostics (International Congress on Applications of Lasers and Electro-Optics, Santa Clara, Calif., 1988), Vol. 67, pp. 263–273.

Gréhan, G.

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz-Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4833 (1988).
[CrossRef] [PubMed]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computation on a microcomputer,” Part. Part. Syst. Character. 5, 103–108 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “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]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: numerical results and application to optical sizing,” Part. Part. Syst. Character. 4, 141–147 (1987).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 2, 83–93 (1985).

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gréhan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz-Mie theory and application to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, to be published).

G. Gouesbet, G. Gréhan, B. Maheu, “The localized interpretation to compute the coefficients gn, gn1, and gn−1 in the framework of the generalized Lorenz-Mie theory,” in Optical Methods in Flows and Particle Diagnostics (International Congress on Applications of Lasers and Electro-Optics, Santa Clara, Calif., 1988), Vol. 67, pp. 263–273.

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz-Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4833 (1988).
[CrossRef] [PubMed]

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computation on a microcomputer,” Part. Part. Syst. Character. 5, 103–108 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “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]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: numerical results and application to optical sizing,” Part. Part. Syst. Character. 4, 141–147 (1987).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 2, 83–93 (1985).

G. Gouesbet, G. Gréhan, B. Maheu, “The localized interpretation to compute the coefficients gn, gn1, and gn−1 in the framework of the generalized Lorenz-Mie theory,” in Optical Methods in Flows and Particle Diagnostics (International Congress on Applications of Lasers and Electro-Optics, Santa Clara, Calif., 1988), Vol. 67, pp. 263–273.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz-Mie theory and application to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, to be published).

Mie, G.

G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Roosen, G.

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 99–194 (1977).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Ann. Phys.

G. Mie, “Beiträge zur Optik Trüber Medien, Speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Appl. Opt.

J. Opt. (Paris)

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz-Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 2, 83–93 (1985).

J. Opt. Soc. Am. A

Opt. Commun.

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 99–194 (1977).
[CrossRef]

Part. Part. Syst. Character.

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: numerical results and application to optical sizing,” Part. Part. Syst. Character. 4, 141–147 (1987).
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computation on a microcomputer,” Part. Part. Syst. Character. 5, 103–108 (1988).
[CrossRef]

Phys. Rev. A

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

Phys. Rev. Lett.

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

Other

G. Gouesbet, G. Gréhan, eds., Proceedings of the International Symposium on Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988).
[CrossRef]

G. Gouesbet, “Advanced theory of light scattering. Application to optical sizing,” in 6th Symposium of Japan Association of Aerosol Science and Technology (JAAST, Osaka, 1988), pp. 106–108.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz-Mie theory and application to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, to be published).

G. Gouesbet, G. Gréhan, B. Maheu, “The localized interpretation to compute the coefficients gn, gn1, and gn−1 in the framework of the generalized Lorenz-Mie theory,” in Optical Methods in Flows and Particle Diagnostics (International Congress on Applications of Lasers and Electro-Optics, Santa Clara, Calif., 1988), Vol. 67, pp. 263–273.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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Figures (10)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

R n 0 ¯ ± n versus n in which R ¯ is the average of R n 0 obtained for 54 dimensionless values and σ is the corresponding standard deviation.

Fig. 3
Fig. 3

R n m ¯ ± σ versus n, for m = 0,2,3,4, and 5. Curves ( R ¯ + σ ) and ( R ¯ σ ) cannot be distinguished. The value for m = 1 has been added as a constant line.

Fig. 4
Fig. 4

Real part of ϕ n 0 ¯ , ϕ n 0 ¯ ± σ versus n. The value Re ( ϕ n 0 ) = 0 from Eq. (45) is also indicated.

Fig. 5
Fig. 5

Imaginary part of ϕ n 0 ¯ ± σ versus n. The value Im ( ϕ n 0 ) = 1 from Eq. (45) is also indicated.

Fig. 6
Fig. 6

Percentage of error for R n 0.

Fig. 7
Fig. 7

Imaginary part of g n , TM 0 versus real part of g n , TM 0. The comparisons are between the finite series (+) and the localized approximation (□).

Fig. 8
Fig. 8

Imaginary part of g n , TM m versus real part of g n , TM m for m = 2 and −2. The comparisons are between the finite series (+) and the localized approximation (□).

Fig. 9
Fig. 9

Imaginary part of g n , TM m versus real part of g n , TM m for m = 3 and −3. The comparisons are between the finite series (+) and the localized approximation (□).

Fig. 10
Fig. 10

Imaginary part of g n , TM m versus real part of g n , TM m, for m = 4 and −4. The comparisons are between the finite series (+) and the localized approximation (□).

Tables (3)

Tables Icon

Table 1 Example of Evaluation of the Normalization, Real Factors R n 0

Tables Icon

Table 2 Computation Times in Seconds for the Computation of All the g n m’sa

Tables Icon

Table 3 Computation Times in Secondsa

Equations (51)

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

s = w 0 / l = 1 / ( k w 0 ) ,
O P O G = ( x 0 , y 0 , z 0 ) .
( E r H r ) = { ( E 0 cos φ H 0 sin φ ) sin θ exp ( i k z ) } × [ 1 L 2 Q l z ] i Q exp [ i Q r 2 sin 2 θ w 0 2 ] exp ( i k z 0 ) ,
Q = 1 i + 2 z z 0 l
g n = f ( A ) ,
g n = i Q ¯ exp [ i Q ¯ ( ρ n w 0 ) 2 ] exp ( i k z 0 ) ,
Q ¯ = Q ( z = 0 ) = 1 i 2 z 0 l ,
ρ n = ( n + 1 / 2 ) 2 π λ .
g n = exp [ ( ρ n w 0 ) 2 ] ,
g n , TM 1 = g n , TM 1 = g n / 2 = g n , TM / 2 ,
g n , TE 1 = g n , TE 1 = i g n / 2 = i g n , TE / 2 .
E r = E r + 1 + E r 1 = m = 1 , 1 E r m ,
E r + 1 = E 0 2 e i φ sin θ exp ( i k z ) A ,
E r 1 = E 0 2 e i φ sin θ exp ( i k z ) A ,
g n , TM ± 1 = 1 2 f ( A ) .
H r = H r + 1 + H r 1 = m = 1 , 1 H r m ,
H r + 1 = H 0 2 i e i φ sin θ exp ( i k z ) A ,
H r 1 = H 0 2 i ( e i φ ) sin θ exp ( i k z ) A ,
g n , TE 1 = 1 2 i f ( A ) ,
g n , TE 1 = 1 2 i f ( A ) .
E r = E 0 F 2 [ Σ j p Ψ j p exp ( i j + φ ) + Σ j p Ψ j p exp ( i j φ ) ] + E 0 x 0 G L Σ j p Ψ j p exp ( i j 0 φ ) ,
H r = H 0 F 2 i [ Σ j p Ψ j p exp ( i j + φ ) Σ j p Ψ j p exp ( i j φ ) ] + H 0 y 0 G L Σ j p Ψ j p exp ( i j 0 φ ) ,
F = Ψ 0 0 sin θ ( 1 2 Q l L z ) exp ( i k z ) exp ( i k z 0 ) ,
G = Ψ 0 0 2 Q l cos θ exp ( i k z ) exp ( i k z 0 ) ,
Ψ 0 0 = i Q exp ( i Q r 2 sin 2 θ w 0 2 ) exp ( i Q x 0 2 + y 0 2 w 0 2 ) ,
Ψ j p = ( i Q r sin θ w 0 2 ) j ( x 0 i y 0 ) j p ( x 0 + i y 0 ) p ( i p ) ! p ! ,
Σ j p = j = 0 p = 0 j ,
j + = j + 1 2 p = j 0 + 1 ,
j = j 1 2 p = j 0 1 .
( E r H r ) = m = + ( E r m H r m ) .
E r m = { E 0 exp ( i k z ) exp ( i m φ ) sin θ } exp ( i k z 0 ) i Q × exp ( i Q r 2 sin 2 θ w 0 2 ) exp ( i Q x 0 2 + y 0 2 w 0 2 ) [ 1 2 ( 1 2 Q l L z ) × ( j + = m j p Ψ j p + j = m j p Ψ j p ) + x 0 L 2 Q l z r sin θ j 0 = m j p Ψ j p ] ,
H r m = { H 0 exp ( i k z ) exp ( i m φ ) sin θ } exp ( i k z 0 ) i Q × exp ( i Q r 2 sin 2 θ w 0 2 ) exp ( i Q x 0 2 + y 0 2 w 0 2 ) [ 1 2 i ( 1 2 Q l L z ) × ( j + = m j p Ψ j p j = m j p Ψ j p ) + y 0 L 2 Q l z r sin θ j 0 = m j p Ψ j p ] ,
g n , TM m , old = exp ( i k z 0 ) i Q ¯ exp [ i Q ¯ ( ρ n w 0 ) 2 ] × exp ( i Q ¯ x 0 2 + y 0 2 w 0 2 ) 1 2 ( j + = m j p Ψ ¯ j p + j = m j p Ψ ¯ j p ) ,
g n , TE m , old = exp ( i k z 0 ) i Q ¯ exp [ i Q ¯ ( ρ n w 0 ) 2 ] exp ( i Q ¯ x 0 2 + y 0 2 w 0 2 ) × 1 2 i ( j + = m j p Ψ ¯ j p j = m j p Ψ ¯ j p ) ,
g n , TM m , old = g n , TE m , old = 0 , | m | 1 , g n , TM 1 , old = g n , TE 1 , old = 1 2 g n , g n , TE 1 , old = g n , TE 1 , old = i 2 g n ,
( g n , TM m g n , TE m ) = ( Z n , TM m Z n , TE m ) ( g n , TM m , old g n , TE m , old ) .
Z n m = R n m ϕ n m ,
ϕ n m = exp ( i ϕ n m ) .
R n 0 = | Z n 0 | = | g n 0 | | g n 0 , old | .
R 1 0 = 1 R n 0 = R n 1 0 + n } ,
R n 0 = p = 1 n p = n ( n + 1 ) 2 .
R n 0 = 2 n ( n + 1 ) ( 2 n + 1 ) .
R n , TM 0 = R n , TE 0 = 2 n ( n + 1 ) ( 2 n + 1 ) ,
R n , TM m = R n , TE m = ( 2 2 n + 1 ) | m | 1 | m | 1 ,
ϕ n , TM m = ϕ n , TE m = i ( i ) | m | m .
Z n , TM 1 = Z n , TM 1 = Z n , TE 1 = Z n , TE 1 = 1
g n , TM m , old = exp ( i z 0 a d / s / s ) i Q ¯ exp { i Q ¯ [ ( n + 1 / 2 ) s ] 2 } × exp [ i Q ¯ ( x 0 ad 2 + y 0 ad 2 ) ] 1 2 ( j + = m j p Ψ ¯ j p + j = m j p Ψ ¯ j p ) ,
x 0 a d = x 0 / w 0 , y 0 ad = y 0 / w 0 , z 0 ad = z 0 / l ,
Q ¯ = 1 ( i 2 z 0 ad ) ,
Ψ ¯ j p = [ i Q ¯ ( n + 1 / 2 ) s ] j ( x 0 ad i y 0 ad ) j p j p ! ( x 0 ad + i y 0 ad ) p p ! .
= 100 × R n 0 ¯ R n 0 R n 0 ¯ .

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