Abstract

A novel two-region formulation of the rigorous electromagnetic boundary element method (BEM) is developed and applied to practical diffractive cylindrical lenses of continuous profile and with discrete levels (32, 8, and 2). The performance of these diffractive lenses is presented for incident waves of TE and TM polarization, for a range of beam profiles, and for normal and nonnormal incidence. An optimum width of the Gaussian beam is determined. The BEM is shown to be accurate and versatile, providing the numerical and graphical results that are needed for analysis and design of diffractive elements.

© 1996 Optical Society of America

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References

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  1. Feature issue on “Diffractive optics applications,” Appl. Opt. 34, 2399–2559 (1995).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  3. D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
    [CrossRef]
  4. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
  5. B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
    [CrossRef]
  6. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modelling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
    [CrossRef]
  7. M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
    [CrossRef]
  8. M. S. Mirotznik, D. W. Prather, J. N. Mait, “Hybrid finite element–boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 2–13 (1996).
    [CrossRef]
  9. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Design of subwavelength diffractive optical elements using a hybrid finite element–boundary element method,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 14–23 (1996).
    [CrossRef]
  10. E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
  11. P. K. Banerjee, R. Butterfield, eds., Developments in Boundary Element Methods (Applied Science Publishers, London, 1979).
  12. E. Yamashita, ed., Analysis Methods for Electromagnetic Wave Problems (Artech House, Boston, 1990), pp. 33–77.
  13. P.-B. Zhou, Numerical Analysis of Electromagnetics Fields (Springer-Verlag, New York, 1993), pp. 287–323.
    [CrossRef]
  14. T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Com-mun. Jpn. Pt. 2 74, 11–20 (1991).
    [CrossRef]
  15. E. Noponen, J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1097–1109 (1994).
    [CrossRef]
  16. M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.
  17. P. D. Maker, R. E. Muller, “Phase holograms in polymethyl methacrylate,” J. Vac. Sci. Technol. B 10, 2516–2519 (1992).
    [CrossRef]
  18. 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 ’96, R. T. Chen, P. S. Guilfoyle, eds., Vol. CR62 of Critical Review Series (SPIE, Bellingham, Wash., 1996), pp. 415–430.
  19. D. W. Wilson, P. D. Maker, R. E. Muller, “Binary optic reflection grating for an imaging spectrometer,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 255–266 (1996).
    [CrossRef]

1996

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

1995

1994

1993

1992

P. D. Maker, R. E. Muller, “Phase holograms in polymethyl methacrylate,” J. Vac. Sci. Technol. B 10, 2516–2519 (1992).
[CrossRef]

1991

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Com-mun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

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]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Grann, E. B.

Ido, J.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Com-mun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

Kojima, T.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Com-mun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

Koshiba, M.

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

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.

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
[CrossRef]

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

M. S. Mirotznik, D. W. Prather, J. N. Mait, “Hybrid finite element–boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 2–13 (1996).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Design of subwavelength diffractive optical elements using a hybrid finite element–boundary element method,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 14–23 (1996).
[CrossRef]

Maker, P. D.

P. D. Maker, R. E. Muller, “Phase holograms in polymethyl methacrylate,” J. Vac. Sci. Technol. B 10, 2516–2519 (1992).
[CrossRef]

D. W. Wilson, P. D. Maker, R. E. Muller, “Binary optic reflection grating for an imaging spectrometer,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 255–266 (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 ’96, R. T. Chen, P. S. Guilfoyle, eds., Vol. CR62 of Critical Review Series (SPIE, Bellingham, Wash., 1996), pp. 415–430.

Mirotznik, M. S.

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

M. S. Mirotznik, D. W. Prather, J. N. Mait, “Hybrid finite element–boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 2–13 (1996).
[CrossRef]

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

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Design of subwavelength diffractive optical elements using a hybrid finite element–boundary element method,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 14–23 (1996).
[CrossRef]

Moharam, M. G.

Muller, R. E.

P. D. Maker, R. E. Muller, “Phase holograms in polymethyl methacrylate,” J. Vac. Sci. Technol. B 10, 2516–2519 (1992).
[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 ’96, R. T. Chen, P. S. Guilfoyle, eds., Vol. CR62 of Critical Review Series (SPIE, Bellingham, Wash., 1996), pp. 415–430.

D. W. Wilson, P. D. Maker, R. E. Muller, “Binary optic reflection grating for an imaging spectrometer,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 255–266 (1996).
[CrossRef]

Noponen, E.

Pommet, D. A.

Prather, D. W.

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

M. S. Mirotznik, D. W. Prather, J. N. Mait, “Hybrid finite element–boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 2–13 (1996).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Design of subwavelength diffractive optical elements using a hybrid finite element–boundary element method,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 14–23 (1996).
[CrossRef]

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

Turunen, J.

Vasara, A.

Wilson, D. W.

D. W. Wilson, P. D. Maker, R. E. Muller, “Binary optic reflection grating for an imaging spectrometer,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 255–266 (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 ’96, R. T. Chen, P. S. Guilfoyle, eds., Vol. CR62 of Critical Review Series (SPIE, Bellingham, Wash., 1996), pp. 415–430.

Zhou, P.-B.

P.-B. Zhou, Numerical Analysis of Electromagnetics Fields (Springer-Verlag, New York, 1993), pp. 287–323.
[CrossRef]

Appl. Opt.

Electron. Com-mun. Jpn. Pt. 2

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Com-mun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

J. Mod. Opt.

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

P. D. Maker, R. E. Muller, “Phase holograms in polymethyl methacrylate,” J. Vac. Sci. Technol. B 10, 2516–2519 (1992).
[CrossRef]

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 modelling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

M. S. Mirotznik, D. W. Prather, J. N. Mait, “Hybrid finite element–boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 2–13 (1996).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Design of subwavelength diffractive optical elements using a hybrid finite element–boundary element method,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 14–23 (1996).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

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 ’96, R. T. Chen, P. S. Guilfoyle, eds., Vol. CR62 of Critical Review Series (SPIE, Bellingham, Wash., 1996), pp. 415–430.

D. W. Wilson, P. D. Maker, R. E. Muller, “Binary optic reflection grating for an imaging spectrometer,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 255–266 (1996).
[CrossRef]

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

P. K. Banerjee, R. Butterfield, eds., Developments in Boundary Element Methods (Applied Science Publishers, London, 1979).

E. Yamashita, ed., Analysis Methods for Electromagnetic Wave Problems (Artech House, Boston, 1990), pp. 33–77.

P.-B. Zhou, Numerical Analysis of Electromagnetics Fields (Springer-Verlag, New York, 1993), pp. 287–323.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Profiles of continuous-, 32-, 8-, and 2-level diffractive lenses analyzed. All are nonparaxial designs with a minimum lateral feature size of 0.5 μm.

Fig. 2
Fig. 2

Representation of the two-dimensional space filled with two dielectrics used for the present scattering problem.

Fig. 3
Fig. 3

Field distribution of Ez for the continuous lens with a TE-polarized, normally incident plane wave. (a) Instantaneous field around the lens. The real part of the complex amplitude of Ez is shown. The amplitudes corresponding to the contours are −1, 0, and +1 relative to the amplitude of the incident field. At this instant of time the normalized field amplitude at x = 0 and y = 0 is 1.19. (b) Magnitude of the total field in the focal region. The absolute value of the complex amplitude of Ez is shown. The amplitudes corresponding to the contours are 1 to 5 in steps of 1 relative to the magnitude of the incident field. At this instant of time the normalized field magnitude at x = 0 and y=−100 μm is 5.58. The vertical dashed lines represent the diffraction-limited slit width of d0 = 8/π μm.

Fig. 4
Fig. 4

Same as Fig. 3, but for the 32-level lens.

Fig. 5
Fig. 5

Same as Fig. 3, but for the 8-level lens.

Fig. 6
Fig. 6

Same as Fig. 3, but for the 2-level lens.

Fig. 7
Fig. 7

Diffraction efficiency of the continuous-profile, 32-level, 8-level, and 2-level difractive lenses as a function of the detection slit width at the focal point for a TE-polarized, normally incident plane wave. The vertical dashed line represents the diffraction-limited slit width of d0 = 8/π μm.

Fig. 8
Fig. 8

Change of the diffraction efficiency of the four diffractive lenses with the standard deviation, σ, of a Gaussian window function for (a) TE-polarized and (b) TM-polarized, normally incident plane waves.

Fig. 9
Fig. 9

Magnitude of the total field Ez for the continuous lens with TE polarization in the focal region for normal and oblique incidences. The absolute value of the complex amplitude of Ez is shown. The amplitudes corresponding to the contours are 1 to 5 in steps of 1, with the intensity of the incident field being unity. The incident angle θ is (a) 0°, (b) 2.5°, (c) 5°, and (d) 10°.

Fig. 10
Fig. 10

Reflected and transmitted powers as a function of the aperture width D for l = 0, 0.5, 1, and 2 μm for the case of a TE-polarized, normally incident plane wave on a planar interface between two regions of refractive indices n1 = 1.5 and n2 = 1.0.

Fig. 11
Fig. 11

Magnitude of the total electric field as a function of y (for x = 0) in the case of a TE-polarized, normally incident plane wave on a planar interface between two regions of refractive indices n1 = 1.5 and n2 = 1.0. The BEM results are for a plane wave with a cosine-squared-edged window function of D = 50 μm and l = 1 μm.

Tables (7)

Tables Icon

Table 1 Fraction of Reflected and Transmitted Incident Power, Total Power, and Diffraction Efficiency for Rectangular Window Function

Tables Icon

Table 2 Diffraction Efficiency Dependence on Detection Slit Width at Focal Point

Tables Icon

Table 3 Fraction of Reflected and Transmitted Incident Power, Total Power, and Diffraction Efficiency for Cosine-Squared-Edged Window Function

Tables Icon

Table 4 Fraction of Reflected and Transmitted Incident Power, Total Power, and Diffraction Efficiency for Gaussian Window Function

Tables Icon

Table 5 Focal Position and Diffraction Efficiency of Continuous Lens for TE and TM Polarizations

Tables Icon

Table 6 Change of Solution with Number of Quadratic Elements

Tables Icon

Table 7 Change of Solution with Calculational Radius R for Continuous Lens

Equations (39)

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ϕ 1 t ( r 1 ) + Γ [ ϕ Γ ( r Γ ) n ̂ · G 1 ( r 1 , r Γ ) p 1 G 1 ( r 1 , r Γ ) ψ Γ ( r Γ ) ] d l = ϕ inc ( r 1 ) , r 1 S 1 ,
ϕ 2 t ( r 2 ) + Γ [ ϕ Γ ( r Γ ) n ̂ · G 2 ( r 2 , r Γ ) p 2 G 2 ( r 2 , r Γ ) ψ Γ ( r Γ ) ] d l = 0 , r 2 S 2 ,
G i ( r i , r Γ ) = ( j / 4 ) H 0 ( 2 ) ( k i | r i r Γ | ) ( i = 1 , 2 ) ,
ϕ 1 t = ϕ 2 t ϕ Γ ,
( 1 / p 1 ) n ̂ · ϕ 1 t = ( 1 / p 2 ) n ̂ · ϕ 2 t ψ Γ .
( θ Γ / 2 π 1 ) ϕ Γ ( r Γ ) + Γ [ ϕ Γ ( r Γ ) n ̂ · G 1 ( r Γ , r Γ ) p 1 G 1 ( r Γ , r Γ ) ψ Γ ( r Γ ) ] d l = ϕ inc ( r Γ ) ,
( θ Γ / 2 π ) ϕ Γ ( r Γ ) + Γ [ ϕ Γ ( r Γ ) n ̂ · G 2 ( r Γ , r Γ ) p 2 G 2 ( r Γ , r Γ ) ψ Γ ( r Γ ) ] d l = 0 ,
ϕ Γ = { N } T { ϕ Γ } e , ψ Γ = { N } T { ψ Γ } e ,
E z i s ( x , y i ) = a i ( ρ ) exp [ j ( ρ x ± β i y i ) ] d ρ ,
a i ( ρ ) exp ( j β i y i ) = 1 2 π E z i s ( x , y i ) exp ( j ρ x ) d x ,
E z i s ( x , y i ) = n = M / 2 M / 2 1 A i ( ρ n ) exp ( j ρ n x ) ,
A i ( ρ n ) = 1 M m = M / 2 M / 2 1 E z i s ( m Δ x , y i ) exp ( j ρ n m Δ x ) , Δ x = L / M ,
A i ( ρ n ) = a i ( ρ n ) Δ ρ exp ( j β i n y i ) , Δ ρ = 2 π / L ,
H x i s ( x , y i ) = ± 1 η i m = M / 2 M / 2 1 β i m k i A i ( ρ m ) exp ( j ρ m x ) ,
P s = Re { ± 1 2 L / 2 L / 2 E z i s ( x , y i ) [ H x i s ( x , y i ) ] * d x } = Re [ L 2 η i m = M / 2 M / 2 1 β i m * k i | A i ( ρ m ) | 2 ] ,
P f = Re { 1 2 d / 2 d / 2 E z 2 s ( x , f ) [ H x 2 s ( x , f ) ] * d x } = Re { d 2 η 2 m = M / 2 M / 2 1 n = M / 2 M / 2 1 β 2 m * k 2 [ A 2 ( ρ m ) ] * × A 2 ( ρ n ) sinc [ ( ρ m ρ n ) d / 2 ] } ,
P s = Re [ η i L 2 m = M / 2 M / 2 1 β i m k i | A i ( ρ m ) | 2 ] ,
P f = Re { η 2 d 2 m = M / 2 M / 2 1 n = M / 2 M / 2 1 β 2 m k 2 A 2 ( ρ m ) × [ A 2 ( ρ n ) ] * sinc [ ( ρ m ρ n ) d / 2 ] } .
φ ( x ) = k 0 n 2 ( f f 2 + x 2 ) ,
h ( x ) = [ φ ( x ) mod 2 π ] / [ ( n 2 n 1 ) k 0 ] ,
h q ( x p ) = int [ h ( x p ) / Δ + 0.5 ] Δ ,
h 2 ( x p ) = int [ h ( x p ) / Δ 2 ] Δ 2 ,
w ( x ) = { 1 , 0 | x | D / 2 l cos 2 ( | x | D / 2 + l 4 l π , D / 2 l | x | D / 2 + l , 0 , D / 2 + l | x | <
P inc = 1 2 E z inc ( H x inc ) * d x = 1 2 η 1 w 2 ( x ) d x = D l / 2 2 η 1 ,
P inc = 1 2 E x inc ( H z inc ) * d x = η 1 2 w 2 ( x ) d x = η 1 ( D l / 2 ) 2 .
w ( x ) = { exp ( x 2 / 2 σ 2 ) , 0 | x | l 0 , l < | x | < .
P inc = 1 2 E z inc ( H x inc ) * d x = 1 2 η 1 w 2 ( x ) d x = π σ erf ( l / σ ) 2 η 1 ,
P inc = 1 2 E x inc ( H z inc ) * d x = η 1 2 w 2 ( x ) d x = η 1 π σ erf ( l / σ ) 2 ,
2 ϕ 1 t + k 1 2 ϕ 1 t = 0 ,
ϕ 1 t = ϕ 1 s + ϕ inc ,
2 ϕ 1 s + k 1 2 ϕ 1 s = ( 2 ϕ inc + k 1 2 ϕ inc ) = 0 .
Φ inc = w ϕ inc .
2 Φ inc + k 1 2 Φ inc = ( 2 w ) ϕ inc + 2 ( w ) · ( ϕ inc ) + w ( 2 ϕ inc + k 1 2 ϕ inc ) = ( 2 w ) ϕ inc + 2 ( w ) · ( ϕ inc ) ,
2 ϕ 1 s + k 1 2 ϕ 1 s = [ ( 2 w ) ϕ inc + 2 ( w ) · ( ϕ inc ) ] .
ϕ 1 t ( r 1 ) + Γ [ ϕ Γ ( r Γ ) n ̂ · G 1 ( r 1 , r Γ ) p 1 G 1 ( r 1 , r Γ ) ψ Γ ( r Γ ) ] d l = Φ inc ( r 1 ) G 1 ( r 1 , r ) × { [ 2 w ( r ) ] ϕ inc ( r ) + 2 [ w ( r ) ] · [ ϕ inc ( r ) ] } d x d y .
w ( x ) = u ( x + D / 2 ) u ( x D / 2 ) ,
u ( x ) = { 0 , x < 0 1 , x > 0 .
( 2 w ) ϕ inc + 2 ( w ) · ( ϕ inc ) = [ δ ( x + D / 2 ) δ ( x D / 2 ) ] exp ( j k 1 y ) ,
Φ inc ( r 1 ) + G 1 x x = D / 2 exp ( j k 1 y ) d y G 1 x x = D / 2 exp ( j k 1 y ) d y .

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