Abstract

Laser beams emitted from concentric-circle-grating, surface-emitting semiconductor lasers are azimuthally polarized and have an electric field amplitude profile that can be represented as a J1 Bessel function multiplied by a Gaussian function, where the arguments of both of the functions are complex. The detailed propagation formulas for these laser beams and similar azimuthally polarized Laguerre–Gaussian beams through optical systems representable by complex ABCD beam matrices are obtained. As a specific example, azimuthally polarized J1 Bessel–Gaussian laser beam propagation through a laser amplifier with a radial gain profile is considered.

© 1997 Optical Society of America

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References

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  1. T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
    [CrossRef]
  2. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  3. N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
    [CrossRef]
  4. D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
    [CrossRef]
  5. N. G. Alexopoulos, S. R. Kerner, “Coupled power theorem and orthogonality relations for optical disk waveguides,” J. Opt. Soc. Am. 67, 1634–1638 (1977).
    [CrossRef]
  6. S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
    [CrossRef]
  7. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
    [CrossRef]
  8. R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: The azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
    [CrossRef] [PubMed]
  9. R. H. Jordan, D. G. Hall, “The azimuthally polarized Bessel–Gauss beam,” Opt. Photon. News 5(12), 19 (1994).
  10. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
    [CrossRef] [PubMed]
  11. P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
    [CrossRef]
  12. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 773–802.
  13. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [CrossRef]
  14. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1996).
    [CrossRef]
  15. L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
    [CrossRef]

1996 (3)

1994 (2)

R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: The azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
[CrossRef] [PubMed]

R. H. Jordan, D. G. Hall, “The azimuthally polarized Bessel–Gauss beam,” Opt. Photon. News 5(12), 19 (1994).

1992 (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

1990 (2)

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

1989 (1)

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

1980 (1)

S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
[CrossRef]

1977 (1)

1972 (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

1968 (1)

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

1965 (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 773–802.

Alexopoulos, N. G.

S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
[CrossRef]

N. G. Alexopoulos, S. R. Kerner, “Coupled power theorem and orthogonality relations for optical disk waveguides,” J. Opt. Soc. Am. 67, 1634–1638 (1977).
[CrossRef]

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Bour, D. P.

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

Carlson, N. W.

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

Casperson, L. W.

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1996).
[CrossRef]

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Cordero-Iannarella, R. F.

S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
[CrossRef]

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Evans, G. A.

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

Greene, P. L.

Hall, D. G.

P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
[CrossRef] [PubMed]

R. H. Jordan, D. G. Hall, “The azimuthally polarized Bessel–Gauss beam,” Opt. Photon. News 5(12), 19 (1994).

R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: The azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
[CrossRef] [PubMed]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Hardy, A.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

Jordan, R. H.

R. H. Jordan, D. G. Hall, “The azimuthally polarized Bessel–Gauss beam,” Opt. Photon. News 5(12), 19 (1994).

R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: The azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
[CrossRef] [PubMed]

Kerner, S. R.

S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
[CrossRef]

N. G. Alexopoulos, S. R. Kerner, “Coupled power theorem and orthogonality relations for optical disk waveguides,” J. Opt. Soc. Am. 67, 1634–1638 (1977).
[CrossRef]

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

Liew, S. K.

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

Parke, R.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Scifres, D. R.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 773–802.

Streifer, W.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

Tovar, A. A.

Welch, D. F.

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

Wicks, G. W.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Yariv, A.

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Appl. Phys. Lett. (4)

N. W. Carlson, G. A. Evans, D. P. Bour, S. K. Liew, “Demonstration of a grating-surface-emitting diode laser with low-threshold current density,” Appl. Phys. Lett. 56, 16–18 (1990).
[CrossRef]

D. F. Welch, R. Parke, A. Hardy, W. Streifer, D. R. Scifres, “Low-threshold grating-coupled surface-emitting lasers,” Appl. Phys. Lett. 55, 813–815 (1989).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. R. Kerner, N. G. Alexopoulos, R. F. Cordero-Iannarella, “On the theory of corrugated optical disk waveguides,” IEEE Trans. Microwave Theory Tech. MTT-28, 18–24 (1980).
[CrossRef]

J. Appl. Phys. (2)

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Opt. Photon. News (1)

R. H. Jordan, D. G. Hall, “The azimuthally polarized Bessel–Gauss beam,” Opt. Photon. News 5(12), 19 (1994).

Other (1)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), pp. 773–802.

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Figures (1)

Fig. 1
Fig. 1

Beam profiles of an azimuthally polarized J1 Bessel–Gaussian beam propagating through a laser amplifier with a positive radial gain profile. After some distance (∼1 Rayleigh length, z0), the beam profile stops changing shape as it approaches the waveguide’s fundamental mode.

Tables (1)

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Table 1 Some Common Beam Matrices

Equations (84)

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2E(r, ϕ z)+k2(r, ϕ, z)E(r, ϕ, z)
=-2k(r, ϕ, z)k(r, ϕ, z)·E(r, ϕ z).
E(r, ϕ, z)=Er(r, ϕ, z)ir+Eϕ(r, ϕ, z)iϕ.
2Er+k2(r, ϕ, z)Er=1r2Er+2Eϕϕ,
2Eϕ+k2(r, ϕ, z)Eϕ=1r2Eϕ-2Erϕ.
2Err2+1rErr+2Erz2+k2(r, z)Er=Err2,
2Eϕr2+1rEϕr+2Eϕz2+k2(r, z)Eϕ=Eϕr2.
Eϕ(r, z)=ψ(r, z)exp-i0zk0(z)dz.
2ψr2+1rψr-ψr2-2ik0(z)ψz
+k2(r, z)-k02(z)-idk0dzψ=0,
k(r, z)=k0(z)-k2(z)r2/2,
k2(r, z)k0(z)[k0(z)-k2(z)r2].
2ψr2+1rψr-ψr2-2ik0(z)ψz
-k0(z)k2(z)r2+idk0dzψ=0.
ψ(r, z)=J(r, z)exp{-i[Q(z)r2/2+P(z)]}.
2Jr2+1r-2iQ(z)rJr-2ik0(z)Jz-Jr2-Q2(z)+k0(z)Qz+k0(z)k2(z)r2J
-2k0(z)Pz+2iQ(z)+idk0dzJ=0.
Q2(z)+k0(z)dQdz+k0(z)k2(z)=0,
2k0(z)dPdz=-2iQ(z)-idk0dz-1W2(z),
2Jr2+1r-2iQ(z)rJr-2ik0(z)Jz
-Jr2+JW2(z)=0.
ρ(r, z)=rW(z),
ζ(r, z)=z.
2Jρ2+1ρ-2iρW2(z)Q(z)-k0(z)W(z)dWdzJρ
-2ik0(z)W2(z)Jζ+1-1ρ2J=0.
Q(z)k0(z)=1W(z)dWdz
ρ2d2Jdρ2+ρdJdρ+(ρ2-1)J=0.
E(r, z)=(E0, rir+E0, ϕiϕ)J1rW(z)×exp-ik0(z)q(z)r22+P(z)×exp-i0zk0(z)dz.
ψ(r, z)=L(r, z)exp{-i[Q(z)r2/2+P(z)]}.
ρ(r, z)=r(4p+2)1/2W(z),
ζ(r, z)=z,
2Lρ2+1ρ-2i(4p+2)ρW2(z)Q(z)-k0(z)W(z)dWdzLρ
-2ik0(z)W2(z)Lζ+(4p+2)-1ρ2L=0.
2i(4p+2)W2(z)Q(z)-k0(z)W(z)dWdz=2,
d2Ldρ2+1ρ-2ρdLdρ+(4p+2)-1ρ2L=0.
τ=ρ2,
4τd2Ldτ2+(4-4τ)dLdτ+(4p+2)-1τL=0.
L(τ)=τ1/2Lp1(τ)
τ d2Lp1dτ2+(1+1-τ)dLp1dτ+pLp1(τ)=0.
E(r, z)=(E0, rir+E0, ϕiϕ)r(4p+2)1/2W(z)×Lp1r2(4p+2)W2(z)×exp-ik0q(z)r22+P(z)×exp-i0zk0(z)dz.
Q2(z)+k0(z)dQdz+k0(z)k2(z)=0.
1q(z)Q(z)k0(z)=1u(z)dudz,
ddzk0(z)dudz+k2(z)u(z)=0.
u(z)=A(z)u(0)+B(z)u(0)/q(0).
u(z)/q(z)=C(z)u(0)+D(z)u(0)/q(0),
u(z)u(z)/q(z)=A(z)B(z)C(z)D(z) u(0)u(0)/q(0).
1q(z)=C(z)+D(z)/q(0)A(z)+B(z)/q(0).
Q(z)k0(z)=1W(z)dWdz.
W(z)=W(0)u(0)u(z),
W(z)=W(0)[A(z)+B(z)/q(0)].
2k0(z)dPdz=-2iQ(z)-idk0dz-1W2(z).
P(z)-P(0)=-i ln[A(z)+B(z)/q(0)]+i2ln[A(z)D(z)-B(z)C(z)]-12k0(0)W2(0)B(z)A(z)+B(z)/q(0),
A(z)D(z)-B(z)C(z)=k0(0)/k0(z)
2i(4p+2)W2(z)Q(z)-k0(z)W(z)dWdz=2.
i(4p+2)1u(z)dudz-1W(z)dWdz=1k0(z)W2(z).
W2(z)=W(0)2[A(z)+B(z)/q(0)]2+2ik0(0)(4p+2)B(z)[A(z)+B(z)/q(0)].
2dPdz=-2iu(z)dudz-ik0(z)dk0dz-(4p+2)1u(z)dudz-1W(z)dWdz.
P(z)-P(0)=-i ln[A(z)+B(z)/q(0)]+i2ln[A(z)D(z)-B(z)C(z)]+i(4p+2)4×ln1+2i(4p+2)k0(0)W2(0)×B(z)A(z)+B(z)/q(0).
u2u2/q2=A1B1C1D1 u1u1/q1
u3u3/q3=A2B2C2D2 u2u2/q2.
u3u3/q3=A2B2C2D2 A1B1C1D1 u1u1/q1.
M=A2B2C2D2 A1B1C1D1=M2M1.
M=MnMn-1M3M2M1.
1qout=C+D/q1A+B/q1.
1q=1R-iλn0πw2.
E2(r)=(E2, rir+E2, ϕiϕ)J1rW2×exp-ik02q2r22+P2exp-i0zk0(z)dz.
W2=W1(A+B/q1),
P2-P2=-i ln(A+B/q1)+i2ln(AD-BC)-12k01W12BA+B/q1.
E2(r)=(E2, rir+E2, ϕiϕ)21/2rW2×exp-ik02q2r22+P2×Lp12r2W22exp-i0zk0(z)dz,
W22=W12(A+B/q1)2+4ik01×B(A+B/q1),
P2-P1=-i ln(A+B/q1)+i2ln(AD-BC)+i(4p+2)4×ln1+4ik01W12BA+B/q1.
ABCD=1z01.
E(r, z)=E0J1r/W11+z/q1exp-ik01/q11+z/q1r22×exp[-i(P1+1/2k0W12 z/1+z/q1)]1+z/q1×exp(-ik0z),
ABCD=1z01 10-1/f1=1-z/fz-1/f1.
E(r, z)=E0J1r/W11-z/f+z/q1×exp-ik0-1/f+1/q11-z/f+z/q1r22×exp[-i(P1+1/2k0W12 z/1-z/f+z/q1)1-z/f+z/q1×exp(-ik0z),
k(r)=k0-k2r2/2,
k=2πnλ+iα.
ABCD=cos(γz)γ-1 sin(γz)-γ sin(γz)cos(γz),
γ=(1+i)α2λ4πn01/2.
E(r, z)=E0J1r/W1cos(γz)+γ-1 sin(γz)/q1×exp-ik0-γ sin(γz)+cos(γz)/q1cos(yz)+γ-1 sin(γz/q1r22×exp{-i(P1+1/2k0W12 γ-1 sin(γz)/cos(γz)+γ-1 sin(γz)/q1)}cos(γz)+γ-1 sin(γz)/q1exp(-ik0z).
1q1=-iγ
E(r, z)=E0J1[exp(iγz)r/W1]exp[-k0γr2/2]×exp-iγ-1 exp(iγz)sin(γz)2k0W12×exp[-i(k0-γ)z].
J1rW2W2w2r2W2
=2-3/221/2rW2L01rW2,

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