Abstract

The generalized Lorenz–Mie theory describes the electromagnetic scattering of a Gaussian laser beam by a spherical particle. The most intensive computational aspect of the theory concerns the evaluation of the beam-shape coefficients in the general case of an off-axis location of the scatterer. These beam-shape coefficients can be computed starting from the set of beam-shape coefficients for an on-axis location by using the addition theorem for the spherical vector wave functions of the first kind under a translation of the coordinate origin.

© 1997 Optical Society of America

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References

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  1. G. Gouesbet, G. Grehan, “Sur la generalisation de la theorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
    [CrossRef]
  2. G. Gouesbet, G. Grehan, B. Maheau, “Scattering of a Gaussian beam by a Mie scatterer centre using a Bromwich formalism,” J. Opt. (Paris) 16, 89–93 (1985).
  3. G. Grehan, B. Maheau, G. Gouesbet, “Scattering of laser beams by Mie scatterer centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef]
  4. G. Gouesbet, B. Maheau, 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 (1989).
    [CrossRef]
  5. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  6. E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered an internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
    [CrossRef]
  7. 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]
  8. G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  9. B. Maheau, G. Gouesbet, G. Grehan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident beam,” J. Opt. (Paris) 19, 59–67 (1988).
    [CrossRef]
  10. G. Grehan, B. Maheau, G. Gouesbet, “Scattering of laser beams by Mie scatterer centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef]
  11. G. Gouesbet, G. Grehan, B. Maheau, “A localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  12. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  13. G. Gouesbet, G. Grehan, B. Maheau, “Computation of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef] [PubMed]
  14. A. Doicu, S. Schabel, F. Ebert, “Generalized Lorenz–Mie theory for non-spherical particles with applications in the Phase-Doppler anemometrie,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick, eds. (Nürnberg Messe GmbH, Nürnberg, Germany, 1995), pp. 119–128.
  15. B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).
  16. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
  17. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  18. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres–Part I–Multipole expansion and ray-optical solution,” IEEE Trans. Antennas. Propag. 19, 378–390 (1971).
    [CrossRef]
  19. A. Messiah, Quantum Mechanics (Editura Stiintifica, Bucuresti, 1974).
  20. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957).
  21. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
    [CrossRef]
  22. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  23. A. Doicu, T. Wriedt, “Plane wave spectrum method of electromagnetic beams,” in Proceedings of the First Workshop on Electromagnetic and Light Scattering—Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 33–37.

1994 (2)

1993 (2)

J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
[CrossRef]

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered an internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

1990 (1)

1989 (1)

1988 (3)

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

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

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

1986 (2)

1985 (1)

G. Gouesbet, G. Grehan, B. Maheau, “Scattering of a Gaussian beam by a Mie scatterer centre using a Bromwich formalism,” J. Opt. (Paris) 16, 89–93 (1985).

1982 (1)

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

1979 (1)

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

1971 (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres–Part I–Multipole expansion and ray-optical solution,” IEEE Trans. Antennas. Propag. 19, 378–390 (1971).
[CrossRef]

1962 (1)

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

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

1954 (1)

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Alexander, D. R.

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

Barber, P. W.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered an internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

Barton, J. P.

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

Bruning, J. H.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres–Part I–Multipole expansion and ray-optical solution,” IEEE Trans. Antennas. Propag. 19, 378–390 (1971).
[CrossRef]

Cruzan, O. R.

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

Davis, L. W.

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

Doicu, A.

A. Doicu, S. Schabel, F. Ebert, “Generalized Lorenz–Mie theory for non-spherical particles with applications in the Phase-Doppler anemometrie,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick, eds. (Nürnberg Messe GmbH, Nürnberg, Germany, 1995), pp. 119–128.

A. Doicu, T. Wriedt, “Plane wave spectrum method of electromagnetic beams,” in Proceedings of the First Workshop on Electromagnetic and Light Scattering—Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 33–37.

Ebert, F.

A. Doicu, S. Schabel, F. Ebert, “Generalized Lorenz–Mie theory for non-spherical particles with applications in the Phase-Doppler anemometrie,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick, eds. (Nürnberg Messe GmbH, Nürnberg, Germany, 1995), pp. 119–128.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957).

Friedman, B.

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).

Gouesbet, G.

G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
[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]

G. Gouesbet, G. Grehan, B. Maheau, “A localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, B. Maheau, 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 (1989).
[CrossRef]

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

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

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

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

G. Gouesbet, G. Grehan, B. Maheau, “Scattering of a Gaussian beam by a Mie scatterer centre using a Bromwich formalism,” J. Opt. (Paris) 16, 89–93 (1985).

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

Grehan, G.

Hill, S. C.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered an internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

Khaled, E. E. M.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered an internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

Lo, Y. T.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres–Part I–Multipole expansion and ray-optical solution,” IEEE Trans. Antennas. Propag. 19, 378–390 (1971).
[CrossRef]

Lock, J. A.

Maheau, B.

Messiah, A.

A. Messiah, Quantum Mechanics (Editura Stiintifica, Bucuresti, 1974).

Russek, J.

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).

Schabel, S.

A. Doicu, S. Schabel, F. Ebert, “Generalized Lorenz–Mie theory for non-spherical particles with applications in the Phase-Doppler anemometrie,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick, eds. (Nürnberg Messe GmbH, Nürnberg, Germany, 1995), pp. 119–128.

Schaub, S. A.

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

Wriedt, T.

A. Doicu, T. Wriedt, “Plane wave spectrum method of electromagnetic beams,” in Proceedings of the First Workshop on Electromagnetic and Light Scattering—Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 33–37.

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered an internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

IEEE Trans. Antennas. Propag. (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres–Part I–Multipole expansion and ray-optical solution,” IEEE Trans. Antennas. Propag. 19, 378–390 (1971).
[CrossRef]

J. Appl. Phys. (1)

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

J. Opt. (Paris) (3)

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

G. Gouesbet, G. Grehan, B. Maheau, “Scattering of a Gaussian beam by a Mie scatterer centre using a Bromwich formalism,” J. Opt. (Paris) 16, 89–93 (1985).

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

J. Opt. Soc. Am. A (5)

Phys. Rev. A (1)

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

Q. Appl. Math. (3)

B. Friedman, J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

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

Other (5)

A. Messiah, Quantum Mechanics (Editura Stiintifica, Bucuresti, 1974).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957).

A. Doicu, S. Schabel, F. Ebert, “Generalized Lorenz–Mie theory for non-spherical particles with applications in the Phase-Doppler anemometrie,” in Proceedings of the Fourth International Congress on Optical Particle Sizing, F. Durst, J. Domnick, eds. (Nürnberg Messe GmbH, Nürnberg, Germany, 1995), pp. 119–128.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

A. Doicu, T. Wriedt, “Plane wave spectrum method of electromagnetic beams,” in Proceedings of the First Workshop on Electromagnetic and Light Scattering—Theory and Applications, T. Wriedt, M. Quinten, K. Bauckhage, eds. (Universität Bremen, Bremen, Germany, 1996), pp. 33–37.

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Figures (5)

Fig. 1
Fig. 1

Geometry of the problem: beam coordinate system Oxyz and particle-location system Oxyz′. The Gaussian beam travels in the z direction and is polarized in the x direction.

Fig. 2
Fig. 2

Cumulative relative errors for the real and imaginary part of g n,TM m coefficients computed in the localized approximation method and by using Eq. (20).

Fig. 3
Fig. 3

Cumulative relative errors for the real and imaginary part of g n,TM m coefficients computed in the localized approximation method and by using the integral representation given in Eq. (26).

Fig. 4
Fig. 4

Cumulative relative errors for the real and imaginary part of g n,TM m coefficients computed in the localized approximation method and by using the simplified integral representation given in Eq. (30).

Fig. 5
Fig. 5

Cumulative relative errors for the real and imaginary part of integral U nm and the right-hand side of Eq. (37).

Tables (1)

Tables Icon

Table 1 Comparison between gn,TM m Coefficients Computed in the Localized Approximation and Addition Theorem Methods

Equations (39)

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f¯, g¯=Sf¯·g¯*dS,
n¯×E¯r¯=n¯×E¯Rr¯=e¯r¯, r¯S, e¯L2S,
e¯Nr¯=n=1Nm=-nn Cnmign,TEmRn¯×M¯mn1k¯r¯+gn,TMmRn¯×N¯mn1k¯r¯,
e¯-e¯N<δ.
ign,TEmR=1E0e¯, n¯×M¯mn1,  gn,TMmR=1E0e¯, n¯×N¯mn1
E¯NRr¯=E0n=1Nm=-nn Cnmign,TEmRM¯mn1kr¯+gn,TMmRN¯mn1kr¯,
gn,XmR=gn,Xm+δgn,XmR,  X=TE, TM.
e¯N0r¯=n=1Nm=-nnCnmign,TEmn¯×M¯mn1k¯r¯+gn,TMmn¯×N¯mn1k¯r¯
E¯N0r¯=E0n=1Nm=-nnCnmign,TEmM¯mn1kr¯+gn,TMmN¯mn1kr¯
E¯0r¯=E0n1m=-nn Cnmign,TEmM¯mn1kr¯+gn,TMmNmn1kr¯.
Cnm=Cn,m  0-1mn+m!n-m!Cn,m < 0.
Cn=in-12n+1nn+1.
gn,TM1R=gn,TM-1R=12gnR,  gn,TE1R=-gn,TE-1R=-i2gnR.
gn=exp-s2n+0.52.
M¯mn1kr¯Nmn1kr¯=n1m=-nnAmnmnBmnmnM¯mn1kr¯+BmnmnAmnmnN¯mn1kr¯.
gn,TMmign,TEm=12Cnmn1±Cn,-1Amn-1,n-Bmn-1,n+Cn1Amn1,n+Bmn1,ngn.
gn,TMm,locign,TEm,loc=-1m-1KnmΨ¯00 expikz012×expim-1φ0Jm-12Q¯ρ0ρnw02±expim+1φ0Jm+12Q¯ρ0ρnw02,
Ψ¯00=iQ¯ exp-iQ¯ρ02/w02exp-iQ¯n+0.52/k2w02,  Knm=-imin+0.5m-1,m0nn+1n+0.5,m=0,  ρn=n+0.51k, Q¯=1/i-2z0/l.
Amnmn=in-n22n+1nn+1n-m!n+m!×0π Jm-mkρ0 sin αexp×im-mφ0expik cos αz0×dPnmdαdPnmdα+mmPnmsin αPnmsin αsin αdα,  Bmnmn=in-n22n+1nn+1n-m!n+m!×0π Jm-mkρ0 sin αexp×im-mφ0expik cos αz0×mdPnmdαPnmsin α+mPnmsin αdPnmdαsin αdα.
gn,TMmign,TEm=14n-m!n+m!×0π/2 expik cos αz0×ΨnmαTm-1sin α±Ψn,-mαTm+1sin αS1αsin αdα+-1n+m+10π/2 exp-ik cos αz0×Ψn,-mαTm-1sin α±Ψn,mαTm+1sin αS2αsin αdα,
Tmα=Jmkρ0 sin αexpimφ0  Ψnmα=dPnmcos αdα+mPnmcos αsin α,
S1α=n12n+1nn+1dPn1dα+Pn1sin α×exp-s2n+0.52,  S2α=n12n+1nn+1-1ndPn1dα-Pn1sin α×exp-s2n+0.52.
S1αŜ1α=1s2fcorsin αcos α exp-14s2 sin2 α,
fcorsin α=1.
fcorsin α=1-s21-sin α3s1-sin α3 2 s.
gn,TMmign,TEm=14n-m!n+m!×0π/2 expik cos αz0×ΨnmαTm-1sin α±Ψn,-mαTm+1sin α1s2fcor×exp-14s2 sin2αsin α cos αdα.
Pnmcos α  n+m!n-m!Jmn+0.5αn+0.5m
Ψnmαsin αn-m!n+m!Ψˆnmα,
Ψˆnmα=1n+0.5mn+1+m×n+0.5n+1.5mJmn+1.5α-n+11-α22-mJmn+0.5·α
gn,TMmign,TEm=expikz04Unm expim-1φ0±Vnm expim+1φ0,
Unm=01ΨˆnmxJm-1kρ0x1s2×fcorxexp-x24iQ¯s2dx, Vnm=01Ψˆn,-mxJm+1kρ0x1s2×fcorxexp-x24iQ¯s2dx.
gn,TM-mx0, y0, z0=gn,TMmx0,-y0, z0, gn,TEmx0, y0, z0=-imgn,TMmy0, -x0, z0, for m  0  gn,TE-mx0, y0, z0=-gn,TEmx0,-y0, z0, m.
Vnm=-1m+1Un,-m,
Unm=-1m-1Vn,-m, m0.
0 exp-a2t2tμ-1Jνbt=Γν+μ2Γν+112aμ12baνM×ν+μ2, ν+1, -b24a2,
n+2M1, 2,-s2n+1.52-nM×1, 2, -s2n+0.522 exp-s2n+0.52
gn,TM1gn,TM1,loc=0.5gn,  gn,TM-1gn,TM-1,loc=0.5gn,  gn,TE1gn,TE1,loc=-0.5ign,  gn,TE-1gn,TE-1,loc=0.5ign, gn=iQ¯ exp-iQ¯s2n+0.52expikz0,
Unm=2-1m-1KnmΨ¯00Jm-12Q¯ρ0ρnw02.
Vnm=2-1m-1KnmΨ¯00Jm+12Q¯ρ0ρnw02,

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