Abstract

The effects of fabrication errors on the performance of collimating finite-thickness cylindrical diffractive lenses with eight discrete levels are investigated with a rigorous boundary-element method and a scalar approach. The photolithographic fabrication errors considered are mask alignment errors, exposure errors (that result in linewidth errors), and etch-depth errors. A cylindrical Gaussian beam of TE or TM polarization is incident upon the resulting lenses. Lenses of F/4, F/2, and F/1.4 are examined. The diffraction efficiencies of the lenses with fabrication errors are generally lower than the error-free lenses with the most severe performance degradation occurring for mask misalignment and exposure errors.

© 1998 Optical Society of America

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    [CrossRef]

1998 (1)

1997 (4)

1996 (2)

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]

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 (7)

1994 (5)

1993 (2)

1991 (1)

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. Commun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

1989 (1)

Anderson, R. J.

M. E. Motamedi, R. J. Anderson, R. de la Rosa, L. G. Hale, W. J. Gunning, R. L. Hall, M. Khoshnevisan, “Binary optics thin film microlens array,” in Miniature and MicroOptics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 22–32 (1992).
[CrossRef]

Babin, S.

S. Babin, H. Haidner, P. Kipfer, A. Lang, J. T. Sheridan, W. Stork, N. Streibl, “Artificial index surface relief diffraction optical elements,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 202–213 (1992).
[CrossRef]

Bergstrom, J.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

Bojko, R. J.

K. M. Flood, J. M. Finlan, R. J. Bojko, “Multiple phase level computer-generated holograms etched in fused silica,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 91–96 (1989).
[CrossRef]

Bryngdahl, O.

Buralli, D. A.

Cox, J. A.

J. A. Cox, B. Fritz, T. Werner, “Process error limitations on binary optics performance,” in Computer and Optically Generated Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 80–88 (1991).
[CrossRef]

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

Crosignani, B.

S. Solimeno, B. Crosignani, A. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 4.

de la Rosa, R.

M. E. Motamedi, R. J. Anderson, R. de la Rosa, L. G. Hale, W. J. Gunning, R. L. Hall, M. Khoshnevisan, “Binary optics thin film microlens array,” in Miniature and MicroOptics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 22–32 (1992).
[CrossRef]

Di Porto, A.

S. Solimeno, B. Crosignani, A. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 4.

Drabik, T. J.

Fainman, Y.

Farn, M. W.

M. W. Farn, J. W. Goodman, “Effect of VLSI fabrication errors on kinoform efficiency,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 125–136 (1990).
[CrossRef]

Ferstl, M.

M. Ferstl, B. Kuhlow, E. Pawlowski, “Effect of fabrication errors on multilevel Fresnel zone lenses,” Opt. Eng. 33, 1229–1235 (1994).
[CrossRef]

Finlan, J. M.

K. M. Flood, J. M. Finlan, R. J. Bojko, “Multiple phase level computer-generated holograms etched in fused silica,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 91–96 (1989).
[CrossRef]

Flood, K. M.

K. M. Flood, J. M. Finlan, R. J. Bojko, “Multiple phase level computer-generated holograms etched in fused silica,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 91–96 (1989).
[CrossRef]

Fritz, B.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

J. A. Cox, B. Fritz, T. Werner, “Process error limitations on binary optics performance,” in Computer and Optically Generated Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 80–88 (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]

Gaylord, T. K.

Glytsis, E. N.

Goodman, J. W.

M. W. Farn, J. W. Goodman, “Effect of VLSI fabrication errors on kinoform efficiency,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 125–136 (1990).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Grann, E. B.

Gunning, W. J.

M. E. Motamedi, R. J. Anderson, R. de la Rosa, L. G. Hale, W. J. Gunning, R. L. Hall, M. Khoshnevisan, “Binary optics thin film microlens array,” in Miniature and MicroOptics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 22–32 (1992).
[CrossRef]

Haidner, H.

S. Babin, H. Haidner, P. Kipfer, A. Lang, J. T. Sheridan, W. Stork, N. Streibl, “Artificial index surface relief diffraction optical elements,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 202–213 (1992).
[CrossRef]

Hale, L. G.

M. E. Motamedi, R. J. Anderson, R. de la Rosa, L. G. Hale, W. J. Gunning, R. L. Hall, M. Khoshnevisan, “Binary optics thin film microlens array,” in Miniature and MicroOptics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 22–32 (1992).
[CrossRef]

Hall, R. L.

M. E. Motamedi, R. J. Anderson, R. de la Rosa, L. G. Hale, W. J. Gunning, R. L. Hall, M. Khoshnevisan, “Binary optics thin film microlens array,” in Miniature and MicroOptics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 22–32 (1992).
[CrossRef]

Harrigan, M. E.

Herzig, H. P.

Hirayama, K.

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. Commun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Chap. 6.

Khoshnevisan, M.

M. E. Motamedi, R. J. Anderson, R. de la Rosa, L. G. Hale, W. J. Gunning, R. L. Hall, M. Khoshnevisan, “Binary optics thin film microlens array,” in Miniature and MicroOptics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 22–32 (1992).
[CrossRef]

Kingslake, R.

R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), p. 124.

Kipfer, P.

S. Babin, H. Haidner, P. Kipfer, A. Lang, J. T. Sheridan, W. Stork, N. Streibl, “Artificial index surface relief diffraction optical elements,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 202–213 (1992).
[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. Commun. 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.

Kuhlow, B.

M. Ferstl, B. Kuhlow, E. Pawlowski, “Effect of fabrication errors on multilevel Fresnel zone lenses,” Opt. Eng. 33, 1229–1235 (1994).
[CrossRef]

Kuittinen, M.

M. Kuittinen, J. Turunen, “Mask misalignment in photolithographic fabrication of resonance-domain diffractive elements,” Opt. Commun. 142, 14–18 (1997).
[CrossRef]

Kunz, R. E.

Lang, A.

S. Babin, H. Haidner, P. Kipfer, A. Lang, J. T. Sheridan, W. Stork, N. Streibl, “Artificial index surface relief diffraction optical elements,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 202–213 (1992).
[CrossRef]

Lee, J.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

Lee, S. H.

Lichtenberg, B.

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

Lohmann, A. W.

Mait, J. N.

Marchand, P.

Miller, J. M.

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]

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]

Moharam, M. G.

Montiel, F.

Morris, G. M.

Motamedi, M. E.

M. E. Motamedi, R. J. Anderson, R. de la Rosa, L. G. Hale, W. J. Gunning, R. L. Hall, M. Khoshnevisan, “Binary optics thin film microlens array,” in Miniature and MicroOptics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 22–32 (1992).
[CrossRef]

Nelson, S.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

Nevière, M.

Nishihara, H.

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics XXIV, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 1–40.
[CrossRef]

Noponen, E.

Ozaktas, H. M.

Pawlowski, E.

M. Ferstl, B. Kuhlow, E. Pawlowski, “Effect of fabrication errors on multilevel Fresnel zone lenses,” Opt. Eng. 33, 1229–1235 (1994).
[CrossRef]

Pommet, D. A.

Popelek, J.

J. Popelek, F. Urban, “The vector analysis of the real diffractive optical elements,” in Nonconventional Optical Imaging Elements, J. Nowak, M. Zajac, eds., Proc. SPIE2169, 89–99 (1994).
[CrossRef]

Prata, A.

Prather, D. W.

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]

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]

Ricks, D. W.

D. W. Ricks, “Scattering from diffractive optics,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR-49 of SPIE Critical Review Series (SPIE, Bellingham, Wash., 1993), pp. 187–211.

D. W. Ricks, “Light scattering from binary optics,” in Computer and Optically Generated Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 89–100 (1991).
[CrossRef]

Rogers, J. R.

Ross, N.

Rossi, M.

Schmitz, M.

Sheridan, J. T.

S. Babin, H. Haidner, P. Kipfer, A. Lang, J. T. Sheridan, W. Stork, N. Streibl, “Artificial index surface relief diffraction optical elements,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 202–213 (1992).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20.

Solimeno, S.

S. Solimeno, B. Crosignani, A. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), Chap. 4.

Stork, W.

S. Babin, H. Haidner, P. Kipfer, A. Lang, J. T. Sheridan, W. Stork, N. Streibl, “Artificial index surface relief diffraction optical elements,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 202–213 (1992).
[CrossRef]

Streibl, N.

S. Babin, H. Haidner, P. Kipfer, A. Lang, J. T. Sheridan, W. Stork, N. Streibl, “Artificial index surface relief diffraction optical elements,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 202–213 (1992).
[CrossRef]

Suhara, T.

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics XXIV, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 1–40.
[CrossRef]

Taghizadeh, M. R.

Turunen, J.

Urban, F.

J. Popelek, F. Urban, “The vector analysis of the real diffractive optical elements,” in Nonconventional Optical Imaging Elements, J. Nowak, M. Zajac, eds., Proc. SPIE2169, 89–99 (1994).
[CrossRef]

Urey, H.

Urquhart, K. S.

Vasara, A.

Wang, A.

Werner, T.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

J. A. Cox, B. Fritz, T. Werner, “Process error limitations on binary optics performance,” in Computer and Optically Generated Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 80–88 (1991).
[CrossRef]

Wilson, D. W.

Zhou, Z.

Appl. Opt. (7)

Electron. Commun. Jpn. Pt. 2 (1)

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. Commun. Jpn. Pt. 2 74, 11–20 (1991).
[CrossRef]

J. Mod. Opt. (1)

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 (11)

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
[CrossRef]

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]

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]

M. Schmitz, O. Bryngdahl, “Rigorous concept for the design of diffractive microlenses with high numerical apertures,” J. Opt. Soc. Am. A 14, 901–906 (1997).
[CrossRef]

Z. Zhou, T. J. Drabik, “Optimized binary, phase-only, diffractive optical element with subwavelength features for 1.55 μm,” J. Opt. Soc. Am. A 12, 1104–1112 (1995).
[CrossRef]

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]

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Figures (9)

Fig. 1
Fig. 1

Geometry of a cylindrical multilevel diffractive lens with fabrication errors that are due to mask misalignments. The incident Gaussian beam is shown with its minimum waist w 0 at y = f, where f is the focal length of the lens. The three regions of interest are also shown with their respective refractive indices n i (i = 1, 2, 3). Boundary Γ1 represents the particular lens profile. Distance t 1 represents the thickness of the substrate, and h max is the maximum etch depth of the continuous profile Fresnel diffractive lens. The distance t 2 denotes the distance of the observation plane from the flat boundary Γ2, h(x) represents the etched depth of the diffractive lens, and D is its diameter (width). The radii (widths) R 1 and R 2 represent the calculational grid half-widths for the BEM implementation at boundaries Γ1 and Γ2, respectively. The ideal planar wave fronts are shown in region 3, and L represents the slit width in the observation plane.

Fig. 2
Fig. 2

(a) Profile of a fabrication-error-free eight-level F/48 lens. (b) Profile of an eight-level F/4 lens resulting from misalignment errors of D 12 = 0.5 μm (between masks 1 and 2) and D 13 = -0.5 μm (between masks 1 and 3). (c) Profile of an eight-level F/4 lens resulting from underexposure at each mask step that was equivalent to a reduction in linewidth of 0.5 μm. (d) Profile of an eight-level F/4 lens resulting from overexposure at each mask step that was equivalent to an increment in linewidth of 0.5 μm. (e) Profile of an eight-level F/4 lens resulting from larger etch depths by 0.2 μm per mask step. (f) Profile of an eight-level F/4 lens resulting from smaller etch depths by 0.2 μm per mask step.

Fig. 3
Fig. 3

Diffraction efficiency of an eight-level F/2 lens as a function of the mask misalignments D 12 (between masks 1 and 2) and D 13 (between masks 1 and 3). The curves bracketed by mask 2 correspond to the diffraction efficiency of the lens with a misalignment between the 1 and 2 masks only. The curves bracketed by mask 3 correspond to the diffraction efficiency of the lens with a misalignment between the 1 and 3 masks only. The curves bracketed by masks 2 and 3 correspond to the diffraction efficiency of the lens with a misalignment between masks 1 and 2 as well as between masks 2 and 3 where D 13 = -D 12. (a) TE incident polarization and (b) TM incident polarization.

Fig. 4
Fig. 4

Diffraction efficiency of eight-level F/4, F/2, and F/1.4 lenses as a function of the mask misalignments D 12 (between masks 1 and 2) and D 13 (between masks 1 and 3). For all lenses the misalignments were between masks 1 and 2 as well as between masks 2 and 3 with D 13 = -D 12. The incident polarization is TE.

Fig. 5
Fig. 5

Diffraction efficiency of an eight-level F/2 lens as a function of the exposure errors Δw (in mask steps 1, 2, or 3). Positive values of Δw correspond to decreased linewidths (underexposure cases), whereas negative values of Δw correspond to increased linewidths (overexposure cases). The curves denoted by mask step i (i = 1, 2, 3) correspond to the diffraction efficiency of the lens with exposure errors during the mask i step only. (a) TE incident polarization and (b) TM incident polarization.

Fig. 6
Fig. 6

Diffraction efficiency of eight-level F/4, F/2, and F/1.4 lenses as a function of the exposure errors Δw in mask 1 step only. Positive values of Δw correspond to decreased linewidths (underexposure cases) whereas negative values of Δw correspond to increased linewidths (overexposure cases). The incident polarization is TE.

Fig. 7
Fig. 7

Diffraction efficiency of an eight-level F/2 lens as a function of the etch-depth errors Δd (in mask steps 1, 2, or 3). Positive values of Δd correspond to smaller than the design etch depths whereas negative values of Δd correspond to larger than the design etch depths. The curves denoted by mask step i (i = 1, 2, 3) correspond to the diffraction efficiency of the lens with etch-depth errors during the mask step i only. (a) TE incident polarization and (b) TM incident polarization.

Fig. 8
Fig. 8

Diffraction efficiency of eight-level F/4, F/2, and F/1.4 lenses as a function of the etch-depth errors Δd in mask step 3 only. Positive values of Δd correspond to shallower etch depths whereas negative values of Δd correspond to deeper etch depths. The incident polarization is TE.

Fig. 9
Fig. 9

Electric-field intensity at the observation plane (at a distance of t 2′ = 49.986 mm away from the Γ2 boundary), calculated by use of BEM, for a TE-polarized incident Gaussian beam and for eight-level F/2 lenses with and without fabrication errors. The misaligned lens has D 12 = -D 13 = 0.5 μm. The underexposed lens has a linewidth change of Δw = 0.5 μm at the first mask step. The etch-depth-error lens has a deeper etch depth by 0.1 μm (Δd = -0.1 μm).

Equations (7)

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- ϕ 1 t r 1 + Γ 1 ϕ Γ 1 r Γ 1 n ˆ 12 · G 1 r 1 ,   r Γ 1 - p 1 G 1 r 1 ,   r Γ 1 ψ Γ 1 r Γ 1 d l = - ϕ inc r 1 ,
ϕ 2 t r 2 + Γ 1 ϕ Γ 1 r Γ 1 n ˆ 12 · G 2 r 2 ,   r Γ 1 - p 2 G 2 r 2 ,   r Γ 1 ψ Γ 1 r Γ 1 d l = 0 ,
  ϕ inc r 1 = ϕ inc x ,   Y = ϕ 0 w f w Y 1 / 2 u x exp - x 2 w 2 Y - j k 1 Y - f - 1 2 tan - 1 Y Y 0 + φ + k 1 x 2 2 R Y ,
DE BEM = P d BEM y P inc ,
ϕ 3 F x = n 1 λ 0 | y 0 | 1 / 2 exp j π 4 - k 0 n 1 | y 0 | × i exp - j θ i x i x i + 1   ϕ inc x ,   0 × exp j 2 π n 1 xx λ 0 | y 0 | d x ,
DE F = P d F y P inc = a   x   | ϕ 3 F x | 2 d x P inc ,
w y = 0 = 24.3   μ m = w 0 1 + f 2 Y 0 2 1 / 2 = w 0 1 + f 2 λ 0 2 π 2 n 1 2 w 0 4 1 / 2 ,

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