Abstract

We present the electromagnetic analysis of axially symmetric diffractive lenses. Analysis is performed by numerically solving the electric and magnetic field integral equations using the method of moments, and it exploits axial symmetry to reduce computational cost. Formulations for the analysis of lossless dielectric and perfectly conducting lenses are presented. The analysis of binary and eight-level lenses are performed to illustrate the utility of the technique.

© 2000 Optical Society of America

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References

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  1. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).
  2. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).
  3. B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
    [CrossRef]
  4. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
    [CrossRef]
  5. P. D. Maker, D. W. Wilson, R. E. Muller, “Fabrication and performance of optical interconnect analog phase holograms made by electron beam lithography,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guilfoyle, eds., Vol. CR62 of Critical Reviews Series (SPIE, Bellingham, Wash., 1996), pp. 415–430.
  6. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
    [CrossRef]
  7. K. Hirayama, E. N. Glytsis, T. K. Gaylord, D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
    [CrossRef]
  8. D. Davidson, R. Ziolkowski, “Body-of-revolution finite-difference time-domain modeling of space–time focusing by a three-dimensional lens,” J. Opt. Soc. Am. A 11, 1471–1490 (1994).
    [CrossRef]
  9. A. Wang, A. Prata, “Lenslet analysis by rigorous vector diffraction theory,” J. Opt. Soc. Am. A 12, 1161–1169 (1995).
    [CrossRef]
  10. D. W. Prather, S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
    [CrossRef]
  11. R. F. Harrington, Field Computation by Moment Methods (Krieger, Malabar, Fla., 1968).
  12. J. Mautz, R. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Ubertragungstech. 33, 71–80 (1979).
  13. A. W. Glisson, D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag. AP-28, 593–603 (1980).
    [CrossRef]
  14. N. Geng, L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antennas Propag. 47, 610–619 (1999).
    [CrossRef]
  15. A. W. Glisson, D. R. Wilton, “Simple and efficient numerical techniques for treating bodies of revolution,” (The University of Mississippi, Oxford, Miss., 1978).
  16. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  17. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  18. J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
    [CrossRef]

1999

D. W. Prather, S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
[CrossRef]

N. Geng, L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antennas Propag. 47, 610–619 (1999).
[CrossRef]

1997

1996

1995

1994

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

D. Davidson, R. Ziolkowski, “Body-of-revolution finite-difference time-domain modeling of space–time focusing by a three-dimensional lens,” J. Opt. Soc. Am. A 11, 1471–1490 (1994).
[CrossRef]

1981

1980

A. W. Glisson, D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag. AP-28, 593–603 (1980).
[CrossRef]

1979

J. Mautz, R. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Ubertragungstech. 33, 71–80 (1979).

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

Carin, L.

N. Geng, L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antennas Propag. 47, 610–619 (1999).
[CrossRef]

Davidson, D.

Gallagher, N. C.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Gaylord, T. K.

Geng, N.

N. Geng, L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antennas Propag. 47, 610–619 (1999).
[CrossRef]

Glisson, A. W.

A. W. Glisson, D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag. AP-28, 593–603 (1980).
[CrossRef]

A. W. Glisson, D. R. Wilton, “Simple and efficient numerical techniques for treating bodies of revolution,” (The University of Mississippi, Oxford, Miss., 1978).

Glytsis, E. N.

Harrington, R.

J. Mautz, R. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Ubertragungstech. 33, 71–80 (1979).

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Krieger, Malabar, Fla., 1968).

Hirayama, K.

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).

Lichtenberg, B.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Mait, J. N.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

Maker, P. D.

P. D. Maker, D. W. Wilson, R. E. Muller, “Fabrication and performance of optical interconnect analog phase holograms made by electron beam lithography,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guilfoyle, eds., Vol. CR62 of Critical Reviews Series (SPIE, Bellingham, Wash., 1996), pp. 415–430.

Mautz, J.

J. Mautz, R. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Ubertragungstech. 33, 71–80 (1979).

Mirotznik, M. S.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

Muller, R. E.

P. D. Maker, D. W. Wilson, R. E. Muller, “Fabrication and performance of optical interconnect analog phase holograms made by electron beam lithography,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guilfoyle, eds., Vol. CR62 of Critical Reviews Series (SPIE, Bellingham, Wash., 1996), pp. 415–430.

Prata, A.

Prather, D. W.

Shi, S.

Stamnes, J. J.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

van de Hulst, H. C.

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

Wang, A.

Wilson, D. W.

K. Hirayama, E. N. Glytsis, T. K. Gaylord, D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

P. D. Maker, D. W. Wilson, R. E. Muller, “Fabrication and performance of optical interconnect analog phase holograms made by electron beam lithography,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guilfoyle, eds., Vol. CR62 of Critical Reviews Series (SPIE, Bellingham, Wash., 1996), pp. 415–430.

Wilton, D. R.

A. W. Glisson, D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag. AP-28, 593–603 (1980).
[CrossRef]

A. W. Glisson, D. R. Wilton, “Simple and efficient numerical techniques for treating bodies of revolution,” (The University of Mississippi, Oxford, Miss., 1978).

Ziolkowski, R.

Arch. Elektron. Ubertragungstech.

J. Mautz, R. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Ubertragungstech. 33, 71–80 (1979).

IEEE Trans. Antennas Propag.

A. W. Glisson, D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propag. AP-28, 593–603 (1980).
[CrossRef]

N. Geng, L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antennas Propag. 47, 610–619 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Other

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

P. D. Maker, D. W. Wilson, R. E. Muller, “Fabrication and performance of optical interconnect analog phase holograms made by electron beam lithography,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guilfoyle, eds., Vol. CR62 of Critical Reviews Series (SPIE, Bellingham, Wash., 1996), pp. 415–430.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).

R. F. Harrington, Field Computation by Moment Methods (Krieger, Malabar, Fla., 1968).

A. W. Glisson, D. R. Wilton, “Simple and efficient numerical techniques for treating bodies of revolution,” (The University of Mississippi, Oxford, Miss., 1978).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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

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Figures (11)

Fig. 1
Fig. 1

Geometry of a homogeneous dielectric scattering object.

Fig. 2
Fig. 2

Approximation of an axially symmetric generating arc by linear segments.

Fig. 3
Fig. 3

Overlay of the computed and the analytically determined induced surface currents for a lossless dielectric sphere of 2λ diameter: (a) Jt, (b) Jϕ, (c) Mt, (d) Mϕ.

Fig. 4
Fig. 4

Overlay of the computed and the analytically determined near fields from a dielectric sphere at a plane z=5λ: (a) scattered fields, (b) total fields.

Fig. 5
Fig. 5

Overlay of the computed and the analytically determined induced surface currents for a perfectly conducting sphere of 1λ diameter: (a) Jt, (b) Jϕ.

Fig. 6
Fig. 6

Overlay of the computed and the analytically propagated scattered fields from a perfectly conducting sphere at (a) z=-3λ and (b) z=3λ.

Fig. 7
Fig. 7

Geometry and cross section of (a) two-level and (b) eight-level diffractive lenses of 30-µm focal length and 22.37-µm diameter.

Fig. 8
Fig. 8

Illustration of the electric field magnitude in the focal plane of (a) two-level and (b) eight-level dielectric lenses.

Fig. 9
Fig. 9

Comparison of line scans of the electric field magnitudes for two-level and eight-level lenses in the focal plane along the ϕ=90° axis.

Fig. 10
Fig. 10

Geometry of perfectly conducting diffractive lenses: (a) two-level lens of 27.4955-µm diameter, (b) eight-level lenses of 37.7995-µm diameter. Both lenses have a focal length of 30 µm.

Fig. 11
Fig. 11

Illustration of the electric field magnitude in the focal plane of (a) two-level and (b) eight-level perfectly conducting lenses.

Tables (2)

Tables Icon

Table 1 Computation Requirements for Dielectric Lenses

Tables Icon

Table 2 Computation Requirements for Perfectly Conducting Lenses

Equations (58)

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Ei=-jωμiAi+1jωϵi(·Ai)-×Fi,
Hi=-jωϵiFi+1jωϵi(·Fi)-×Ai,
Ai=14πSJs(r)Gi(r, r)ds
Fi=14πSMs(r)Gi(r, r)ds,
Gi(r, r)=exp(-jkiR)R,
nˆ×Einc=nˆ4π×SjωJs(μ1G1+μ2G2)ds+jωS(s·Js)G1ϵ1+G2ϵ2ds+×SMs(G1+G2)ds,
nˆ×Hinc=nˆ4π×SjωMs(ϵ1G1+ϵ2G2)ds+jωS(s·Ms)G1μ1+G2μ2ds-×SJs(G1+G2)ds,
Einc=uˆ exp(-jk1·r),
Hinc=k×uˆη0exp(-jk1·r),
Js=Jttˆ+Jϕϕˆ,Ms=Mttˆ+Mϕϕˆ.
EtincEϕincHtincHϕinc=β11ttβ11tϕβ12ttβ12tϕβ11ϕtβ11ϕϕβ12ϕtβ12ϕϕβ21ttβ21tϕβ22ttβ22tϕβ21ϕtβ21ϕϕβ22ϕtβ22ϕϕJtJϕMtMϕ.
β11tt(Jt)=jω4πSJt[sin γ sin γ cos(ϕ-ϕ)+cos γ cos γ](μ1G1+μ2G2)ds+j4πωtS 1ρt(ρJt)G1ϵ1+G2ϵ2ds,
β11tϕ(Jϕ)=-jω4πSJϕ sin γ sin(ϕ-ϕ)×(μ1G1+μ2G2)ds+j4πωtS 1ρϕ(Jϕ)G1ϵ1+G2ϵ2ds,
β11ϕt(Jt)=jω4πSJt sin γ sin(ϕ-ϕ)×(μ1G1+μ2G2)ds+j4πωρϕS 1ρt(ρJt)G1ϵ1+G2ϵ2ds,
β11ϕϕ(Jϕ)=jω4πSJϕ cos(ϕ-ϕ)(μ1G1+μ2G2)ds+j4πωρϕS 1ρϕ(Jϕ)G1ϵ1+G2ϵ2ds,
β12tt(Mt)=-14πSMt[(ρ sin γ sin γ-ρ cos γ sin γ)×sin(ϕ-ϕ)+(z-z)sin γ sin γ×sin(ϕ-ϕ)](G1+G2)ds,
β12tϕ(Mt)=-14πSMt[ρ cos γ-ρ cos γ cos(ϕ-ϕ)+(z-z)sin γ cos(ϕ-ϕ)](G1+G2)ds,
β12ϕt(Mt)=-14πSMt[ρ cos γ-ρ cos γ cos(ϕ-ϕ)-(z-z)sin γ cos(ϕ-ϕ)]×(G1+G2)ds,
β12ϕϕ(Mϕ)=-14πSMϕ(z-z)sin(ϕ-ϕ)×(G1+G2)ds.
Gi(r, r)=-Gi(r, r)R=-(1+jkiR)R3exp(-jkiR),i=1, 2.
β22pq(U; μ1, μ2, ϵ1, ϵ2)  β11pq(U; ϵ1, ϵ2, μ1, μ2),
β21pq(U) -β12pq(U),
Etinc=β11tt(Jt; μ1, ϵ1, μ2=0, ϵ2=)+β11tϕ(Jϕ; μ1, ϵ1, μ2=0, ϵ2=),
Eϕinc=β11ϕt(Jt; μ1, ϵ1, μ2=0, ϵ2=)+β11ϕϕ(Jϕ; μ1, ϵ1, μ2=0, ϵ2=).
tn-0.5=tn+tn-12,1nN+1.
Js(t, ϕ)=tˆm=-n=1NItmnB1Emn(t, ϕ)+ϕˆm=-n=1N+1IϕmnB2Emn(t, ϕ),
Ms(t, ϕ)=tˆm=-n=1NKtmnB1Hmn(t, ϕ)+ϕˆm=-n=1N+1KϕmnB2Hmn(t, ϕ),
B1Emn(t, ϕ)=12πρP1n(t)exp(jmϕ),
B2Emn(t, ϕ)=P2n(t)exp(jmϕ),
B1Hmn(t, ϕ)=η02πρP1n(t)exp(jmϕ),
B2Hmn(t, ϕ)=η0P2n(t)exp(jmϕ).
P1n(t)=1tn-0.5ttn+0.50otherwise,
P2n(t)=1tn-1ttn0otherwise.
T1Epq(t, ϕ)=P1q(t)ρexp(-jpϕ),
T2Epq(t, ϕ)=Δtqδ(t-tq-0.5)ρexp(-jpϕ),
T1Hpq(t, ϕ)=η0P1q(t)ρexp(-jpϕ),
T2Hpq(t, ϕ)=η0Δtqδ(t-tq-0.5)ρexp(-jpϕ).
T1Emq, EtmθincT2Emq, EϕmθincT1Hmq, HtmθincT2Hmq, Hϕmθinc=T1Emq, β11ttB1EmnT1Emq, β11tϕB1EmnT1Emq, β12ttB1EmnT1Emq, β12tϕB1EmnT2Emq, β11ϕtB2EmnT2Emq, β11ϕϕB2EmnT2Emq, β12ϕtB2EmnT2Emq, β12ϕϕB2EmnT1Hmq, β21ttB1HmnT1Hmq, β21tϕB1HmnT1Hmq, β22ttB1HmnT1Hmq, β22tϕB1HmnT2Hmq, β21ϕtB2HmnT2Hmq, β21ϕϕB2HmnT2Hmq, β22ϕtB2HmnT2Hmq, β22ϕϕB2HmnItmnIϕmnKtmnKϕmn.
Gi(ϕ, ϕ)=12πm=-+Gim exp[jm(ϕ-ϕ)],
Gi(ϕ, ϕ)=12πm=-+Gim exp[jm(ϕ-ϕ)],
Gim=-ππ exp(-jkiR0)R0cos mξdξ,
Gim=-ππ (1+jkiR0)exp(-jkiR0)R03cos mξdξ,
R0=[ρ2+ρ2-2ρρ cos ξ+(z-z)2]1/2.
T1Emq, β11tϕB1Emn=ω8πΔtq sin γn·[μ1ψ1(tn-0.5, tn; tq-0.5, m+1)+μ2ψ2(tn-0.5, tn; tq-0.5, m+1)-μ1ψ1(tn-0.5, tn; tq-0.5, m-1)-μ2ψ2(tn-0.5, tn; tq-0.5, m-1)]+ω8πΔtq sin γn+1×[μ1ψ1(tn, tn+0.5; tq-0.5, m+1)+μ2ψ2(tn, tn+0.5; tq-0.5, m+1)-μ1ψ1(tn, tn+0.5; tq-0.5, m-1)-μ2ψ2(tn, tn+0.5; tq-0.5, m-1)]-mΔtq4πωρn-0.5Δtnψ1(tn-1, tn; tq-0.5, m)ϵ1+ψ2(tn-1, tn; tq-0.5, m)ϵ2+mΔtq4πωϵ0ρn-0.5Δtn+1ψ1(tn, tn+1; tq-0.5, m)ϵ1+ψ2(tn, tn+1; tq-0.5, m)ϵ2
q=1, 2,, N+1;n=1, 2,, N,
ψi(t1, t2; tq, m)=t1t2-ππ exp(mξ)Gi(tq, l, ξ)dξdl.
Js(t, ϕ)=2Jtθ(t)cos ϕtˆ+2jJϕθ(t)sin ϕϕˆ,
Ms(t, ϕ)=2jMtθ(t)sin ϕtˆ+2Mϕθ(t)cos ϕϕˆ,
Esca=-Sjωμ14πJsG1+14πjωϵ1(s·Js)G1-14πMs×G1ds.
Esca=ρˆEρsca+ϕˆEϕsca+zˆEzsca.
Eρsca(ρ, ϕ, z)=m=0nαmNρmn(Itmn, Iϕmn, Ktmn, Kϕmn, ρ, z)
×cos mϕforθpolarizedj sin mϕforϕpolarized,
Eϕsca(ρ, ϕ, z)=m=0nαmNϕmn(Itmn, Iϕmn, Ktmn, Kϕmn, ρ, z)
×j sin mϕforθpolarizedcos mϕforϕpolarized,
Ezsca(ρ, ϕ, z)=m=0nαmNzmn(Itmn, Iϕmn, Ktmn, Kϕmn, ρ, z)
×cos mϕforθpolarizedj sin mϕforϕpolarized.
Nρmn=-jωμ116π2{sin γn[ψ1(tn-0.5, tn; t, m+1)+ψ1(tn-0.5, tn; t, m-1)]+sin γn+1[ψ1(tn, tn+0.5; t, m+1)+ψ1(tn, tn+0.5; t, m-1)]}Itmn+j16π2ωϵ11Δtn+1[2ρψ1(tn, tn+1; t, m)-ψ1ρ(tn, tn+1; t, m+1)-ψ1ρ(tn, tn+1; t, m-1)]-1Δtn[2ρψ1(tn-1, tn; t, m)-ψ1ρ(tn-1, tn; t, m+1)-ψ1ρ(tn-1, tn; t, m-1)]Itmn+jη08π2[cos γnU13ρ(tn-0.5, tn; t, m)+cos γn+1U13ρ(tn, tn+0.5; t, m)+sin γnU11(tn-0.5, tn; t, m)+sin γn+1U11(tn, tn+0.5; t, m)]Ktmn+ωμ18π[ψ1ρ(tn-1, tn; t, m+1)-ψ1ρ(tn-1, tn; t, m-1)]Iϕmn+m8πωϵ1[2ρψ1(tn-1, tn; t, m)-ψ1ρ(tn-1, tn; t, m+1)-ψ1ρ(tn-1, tn; t, m-1)]Iϕmn+η04πU12ρ(tn-1, tn; t, m)Kϕmn,
ψ1(t1, t2; tq, m)ψ1ρ(t1, t2; tq, m)U13ρ(t1, t2; tq, m)U11(t1, t2; tq, m)U12ρ(t1, t2; tq, m)=2t1t20πcos(mξ)ρ cos(mξ)ρ sin(ξ)sin(mξ)(z-z)sin(ξ)sin(mξ)ρ(z-z)cos(ξ)cos(mξ)G1(tq, l, ξ)dξdl.

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