Abstract

A novel approach based on the boundary-element method (BEM) with exact and approximate Green’s functions is presented for the analysis of the performance of finite-substrate-thickness cylindrical diffractive lenses. This approach can be applied to the analysis of lenses with thicknesses of thousands of wavelengths for both TE and TM polarization. Multiple interference resonance effects between the upper and the lower boundaries of the lens are included. Also, for the cases when multiple interference resonance effects can be neglected, another approach based on the BEM with a modified approximate Green’s function is developed. It is shown that the latter approach corresponds to the cascaded application of the BEM as previously published [Appl. Opt. 37, 34, 6591 (1998)]. To illustrate the advantages of the approaches presented, a cylindrical diffractive lens with a substrate thickness of 2 mm is analyzed. It is not possible to analyze this large-substrate-thickness lens with the previously published cascaded application of the BEM.

© 1999 Optical Society of America

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References

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  1. H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1987), Vol. XXIV, pp. 1–40.
  2. J. R. Leger, M. G. Moharam, T. K. Gaylord, eds., feature diffractive optics applications, Appl. Opt. 34, 2399–2559 (1995).
    [CrossRef] [PubMed]
  3. H. M. Ozaktas, H. Urey, A. W. Lohmann, “Scaling of diffractive and refractive lenses for optical computing and interconnections,” Appl. Opt. 33, 3782–3789 (1994).
    [CrossRef] [PubMed]
  4. K. S. Urquhart, P. Marchand, Y. Fainman, S. H. Lee, “Diffractive optics applied to free-space optical interconnects,” Appl. Opt. 33, 3670–3682 (1994).
    [CrossRef] [PubMed]
  5. E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
  6. F. Montiel, M. Nevière, “Electromagnetic theory of Bragg–Fresnel linear zone plates,” J. Opt. Soc. Am. A 12, 2672–2678 (1995).
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  7. A. Wang, A. Prata, “Lenslet analysis by rigorous vector diffraction theory,” J. Opt. Soc. Am. A 12, 1161–1169 (1995).
    [CrossRef]
  8. 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]
  9. B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. (Bellingham) 33, 3518–3526 (1994).
    [CrossRef]
  10. 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]
  11. 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]
  12. 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]
  13. E. N. Glytsis, M. E. Harrigan, K. Hirayama, T. K. Gaylord, “Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation,” Appl. Opt. 37, 34–43 (1998).
    [CrossRef]
  14. E. N. Glytsis, M. E. Harrigan, T. K. Gaylord, K. Hirayama, “Effects of fabrication errors on the performance of cylindrical diffractive lenses: rigorous boundary element method and scalar approximation,” Appl. Opt. 37, 6591–6602 (1998).
    [CrossRef]
  15. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 506–538.
  16. G. S. Smith, “Directive properties of antennas for transmission into a material half-space,” IEEE Trans. Antennas Propag. AP-32, 232–246 (1984).
    [CrossRef]
  17. C. M. Butler, “Current induced on a conducting strip which resides on the planar interface between two semi-infinite half-spaces,” IEEE Trans. Antennas Propag. AP-32, 226–231 (1984).
    [CrossRef]
  18. C. M. Butler, X. Xu, A. W. Glisson, “Current induced on a conducting cylinder located near the planar interface between two semi-infinite half-spaces,” IEEE Trans. Antennas Propag. AP-33, 616–624 (1985).
    [CrossRef]
  19. P. G. Cottis, J. D. Kanellopoulos, “Scattering from dielectric cylinders embedded in a two-layer lossy medium,” Int. J. Electron. 61, 477–486 (1986).
    [CrossRef]
  20. X. Xu, A. W. Glisson, “Scattering of TM excitation by coupled and partially buried cylinders at the interface between two media,” IEEE Trans. Antennas Propag. AP-35, 529–538 (1987).
  21. N. P. Zhuck, A. G. Yarovoy, “Two-dimensional scattering from an inhomogeneous dielectric cylinder embedded in a stratified medium: case of TM polarization,” IEEE Trans. Antennas Propag. 42, 16–21 (1994).
    [CrossRef]
  22. E. Nishimura, N. Morita, N. Kumagi, “Scattering of guided modes caused by an arbitrarily shaped broken end in a dielectric slab waveguide,” IEEE Trans. Microwave Theory Tech. MTT-31, 923–930 (1983).
    [CrossRef]
  23. 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., Part 2: Electron. 74, 11–20 (1991).
    [CrossRef]
  24. M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK, Publishers, Tokyo, 1992), pp. 43–47.
  25. 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]
  26. D. A. Buralli, G. M. Morris, J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. 28, 976–983 (1989).
    [CrossRef] [PubMed]
  27. M. Rossi, R. E. Kunz, H. P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995).
    [CrossRef] [PubMed]
  28. R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), p. 124.

1998 (2)

1997 (2)

1996 (2)

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]

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]

1995 (4)

1994 (4)

H. M. Ozaktas, H. Urey, A. W. Lohmann, “Scaling of diffractive and refractive lenses for optical computing and interconnections,” Appl. Opt. 33, 3782–3789 (1994).
[CrossRef] [PubMed]

K. S. Urquhart, P. Marchand, Y. Fainman, S. H. Lee, “Diffractive optics applied to free-space optical interconnects,” Appl. Opt. 33, 3670–3682 (1994).
[CrossRef] [PubMed]

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

N. P. Zhuck, A. G. Yarovoy, “Two-dimensional scattering from an inhomogeneous dielectric cylinder embedded in a stratified medium: case of TM polarization,” IEEE Trans. Antennas Propag. 42, 16–21 (1994).
[CrossRef]

1993 (1)

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., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

1989 (1)

1987 (1)

X. Xu, A. W. Glisson, “Scattering of TM excitation by coupled and partially buried cylinders at the interface between two media,” IEEE Trans. Antennas Propag. AP-35, 529–538 (1987).

1986 (1)

P. G. Cottis, J. D. Kanellopoulos, “Scattering from dielectric cylinders embedded in a two-layer lossy medium,” Int. J. Electron. 61, 477–486 (1986).
[CrossRef]

1985 (1)

C. M. Butler, X. Xu, A. W. Glisson, “Current induced on a conducting cylinder located near the planar interface between two semi-infinite half-spaces,” IEEE Trans. Antennas Propag. AP-33, 616–624 (1985).
[CrossRef]

1984 (2)

G. S. Smith, “Directive properties of antennas for transmission into a material half-space,” IEEE Trans. Antennas Propag. AP-32, 232–246 (1984).
[CrossRef]

C. M. Butler, “Current induced on a conducting strip which resides on the planar interface between two semi-infinite half-spaces,” IEEE Trans. Antennas Propag. AP-32, 226–231 (1984).
[CrossRef]

1983 (1)

E. Nishimura, N. Morita, N. Kumagi, “Scattering of guided modes caused by an arbitrarily shaped broken end in a dielectric slab waveguide,” IEEE Trans. Microwave Theory Tech. MTT-31, 923–930 (1983).
[CrossRef]

Buralli, D. A.

Butler, C. M.

C. M. Butler, X. Xu, A. W. Glisson, “Current induced on a conducting cylinder located near the planar interface between two semi-infinite half-spaces,” IEEE Trans. Antennas Propag. AP-33, 616–624 (1985).
[CrossRef]

C. M. Butler, “Current induced on a conducting strip which resides on the planar interface between two semi-infinite half-spaces,” IEEE Trans. Antennas Propag. AP-32, 226–231 (1984).
[CrossRef]

Cottis, P. G.

P. G. Cottis, J. D. Kanellopoulos, “Scattering from dielectric cylinders embedded in a two-layer lossy medium,” Int. J. Electron. 61, 477–486 (1986).
[CrossRef]

Fainman, Y.

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 506–538.

Gallagher, N. C.

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

Gaylord, T. K.

Glisson, A. W.

X. Xu, A. W. Glisson, “Scattering of TM excitation by coupled and partially buried cylinders at the interface between two media,” IEEE Trans. Antennas Propag. AP-35, 529–538 (1987).

C. M. Butler, X. Xu, A. W. Glisson, “Current induced on a conducting cylinder located near the planar interface between two semi-infinite half-spaces,” IEEE Trans. Antennas Propag. AP-33, 616–624 (1985).
[CrossRef]

Glytsis, E. N.

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., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

Kanellopoulos, J. D.

P. G. Cottis, J. D. Kanellopoulos, “Scattering from dielectric cylinders embedded in a two-layer lossy medium,” Int. J. Electron. 61, 477–486 (1986).
[CrossRef]

Kingslake, R.

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

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., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

Koshiba, M.

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

Kumagi, N.

E. Nishimura, N. Morita, N. Kumagi, “Scattering of guided modes caused by an arbitrarily shaped broken end in a dielectric slab waveguide,” IEEE Trans. Microwave Theory Tech. MTT-31, 923–930 (1983).
[CrossRef]

Kunz, R. E.

Lee, S. H.

Lichtenberg, B.

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

Lohmann, A. W.

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]

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]

Marchand, P.

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 506–538.

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]

Montiel, F.

Morita, N.

E. Nishimura, N. Morita, N. Kumagi, “Scattering of guided modes caused by an arbitrarily shaped broken end in a dielectric slab waveguide,” IEEE Trans. Microwave Theory Tech. MTT-31, 923–930 (1983).
[CrossRef]

Morris, G. M.

Nevière, M.

Nishihara, H.

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

Nishimura, E.

E. Nishimura, N. Morita, N. Kumagi, “Scattering of guided modes caused by an arbitrarily shaped broken end in a dielectric slab waveguide,” IEEE Trans. Microwave Theory Tech. MTT-31, 923–930 (1983).
[CrossRef]

Noponen, E.

Ozaktas, H. M.

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]

Rogers, J. R.

Rossi, M.

Smith, G. S.

G. S. Smith, “Directive properties of antennas for transmission into a material half-space,” IEEE Trans. Antennas Propag. AP-32, 232–246 (1984).
[CrossRef]

Suhara, T.

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

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.

Wilson, D. W.

Xu, X.

X. Xu, A. W. Glisson, “Scattering of TM excitation by coupled and partially buried cylinders at the interface between two media,” IEEE Trans. Antennas Propag. AP-35, 529–538 (1987).

C. M. Butler, X. Xu, A. W. Glisson, “Current induced on a conducting cylinder located near the planar interface between two semi-infinite half-spaces,” IEEE Trans. Antennas Propag. AP-33, 616–624 (1985).
[CrossRef]

Yarovoy, A. G.

N. P. Zhuck, A. G. Yarovoy, “Two-dimensional scattering from an inhomogeneous dielectric cylinder embedded in a stratified medium: case of TM polarization,” IEEE Trans. Antennas Propag. 42, 16–21 (1994).
[CrossRef]

Zhuck, N. P.

N. P. Zhuck, A. G. Yarovoy, “Two-dimensional scattering from an inhomogeneous dielectric cylinder embedded in a stratified medium: case of TM polarization,” IEEE Trans. Antennas Propag. 42, 16–21 (1994).
[CrossRef]

Appl. Opt. (7)

Electron. Commun. Jpn., Part 2: Electron. (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., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

IEEE Trans. Antennas Propag. (5)

X. Xu, A. W. Glisson, “Scattering of TM excitation by coupled and partially buried cylinders at the interface between two media,” IEEE Trans. Antennas Propag. AP-35, 529–538 (1987).

N. P. Zhuck, A. G. Yarovoy, “Two-dimensional scattering from an inhomogeneous dielectric cylinder embedded in a stratified medium: case of TM polarization,” IEEE Trans. Antennas Propag. 42, 16–21 (1994).
[CrossRef]

G. S. Smith, “Directive properties of antennas for transmission into a material half-space,” IEEE Trans. Antennas Propag. AP-32, 232–246 (1984).
[CrossRef]

C. M. Butler, “Current induced on a conducting strip which resides on the planar interface between two semi-infinite half-spaces,” IEEE Trans. Antennas Propag. AP-32, 226–231 (1984).
[CrossRef]

C. M. Butler, X. Xu, A. W. Glisson, “Current induced on a conducting cylinder located near the planar interface between two semi-infinite half-spaces,” IEEE Trans. Antennas Propag. AP-33, 616–624 (1985).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

E. Nishimura, N. Morita, N. Kumagi, “Scattering of guided modes caused by an arbitrarily shaped broken end in a dielectric slab waveguide,” IEEE Trans. Microwave Theory Tech. MTT-31, 923–930 (1983).
[CrossRef]

Int. J. Electron. (1)

P. G. Cottis, J. D. Kanellopoulos, “Scattering from dielectric cylinders embedded in a two-layer lossy medium,” Int. J. Electron. 61, 477–486 (1986).
[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 (6)

Opt. Eng. (Bellingham) (1)

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

Other (5)

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

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]

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 506–538.

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

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

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Figures (9)

Fig. 1
Fig. 1

Geometry of a cylindrical diffractive lens. The incident Gaussian beam is shown with its minimum waist w0 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 ni (i=1, 2, 3). Boundary Γ1 represents the particular lens profile. Distance t1 represents the lens substrate thickness, and t2 denotes the distance of the observation plane from the flat boundary Γ2. The quantity h(x) represents the etched depth of the diffractive lens, and D is its width. The radius (width) R1 represents the calculational grid half-width for the BEM implementation at boundary Γ1.12-14

Fig. 2
Fig. 2

Specification of points, lengths, and angles used in the approximate Green’s function (a) for observation point r in region S2 and (b) for r in region S3. When source point rΓ1 and observation point r are given, rΓ2 is determined from the fact that the reflected angle equals the incident angle in (a) and from Snell’s law in (b).

Fig. 3
Fig. 3

Field intensity profile at the observation plane a distance of t2=49.986 mm away from boundary Γ2 for a lens substrate thickness of 20 µm (a) for a TE-polarized incident Gaussian beam and (b) for a TM-polarized beam, where (A) represents the exact three-layer BEM, line (B) represents the approximate Green’s function BEM [including multiple interference effects and described by Eqs. (11)–(17)], line (C) represents the approximate Green’s function BEM [excluding multiple interference effects and described by Eqs. (11)–(17) with R=0], and (D) represents the cascaded application of BEM.13,14

Fig. 4
Fig. 4

Total field distribution for a TE-polarized incident Gaussian beam calculated by use of the BEM with the approximate Green’s function (a) with multiple interference resonance effects and (b) without multiple interference resonance effects. The instantaneous field shown corresponds to the real part of the complex phasor Ez. The amplitudes that correspond to the contours are -0.5 and +0.5 relative to the amplitude of the incident field.

Fig. 5
Fig. 5

Transmitted power in region S3 normalized by the incident power Pt/Pinc (solid curve), calculated by use of the BEM with the approximate Green’s function including multiple interference resonance effects. Dashed curve, the normalized transmitted power calculated analytically when a plane wave is incident normally upon a planar glass of refractive index n2 and thickness t1.

Fig. 6
Fig. 6

Field intensity profile at the observation plane a distance t2=50 mm away from boundary Γ2 for the 2-mm lens substrate thickness (a) for a TE-polarized incident Gaussian beam and (b) for a TM-polarized beam.

Fig. 7
Fig. 7

Same as Fig. 5 but for a lens substrate thickness of ∼2 mm.

Fig. 8
Fig. 8

Magnitude of the electric field for the lens with a substrate thickness of 2 mm in the plane-wave incidence from region 1. The absolute value of the complex amplitude of the electric field is shown. The magnitudes that correspond to the contours are 0.5–3, in steps of 0.5 relative to the magnitude of the incident field.

Fig. 9
Fig. 9

Relative percent error [defined as error = 100× (exact-approximate)/exact] of the reflected powers Pr/Pinc or the transmitted powers Pt/Pinc calculated by the exact three-layer BEM and the BEM with the approximate Green’s function that includes multiple interference resonance effects.

Tables (1)

Tables Icon

Table 1 Fraction of Reflected and Transmitted Power, Total Power, and Diffraction Efficiency for the F/4 Eight-Level Lens with Thickness 20 µma

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

ϕ1t(r1)-Γ1[ϕΓ1(rΓ1)nˆ1·G1(r1, rΓ1)
-p1G1(r1, rΓ1)ψΓ1(rΓ1)]dl=ϕinc(r1),
ϕ2t(r2)+m=12Γm[ϕΓm(rΓm)nˆm·G2(r2, rΓm)
-p2G2(r2, rΓm)ψΓm(rΓm)]dl=0,
ϕ3t(r3)-Γ2[ϕΓ2(rΓ2)nˆ2·G3(r3, rΓ2)
-p3G3(r3, rΓ2)ψΓ2(rΓ2)]dl=0,
ϕΓ1(r Γ1)=ϕ1t(r Γ1)=ϕ2t(r Γ1),
ψΓ1(r Γ1)=(1/p1)nˆ1·ϕ1t(r Γ1)=(1/p2)nˆ1·ϕ2t(r Γ1),
ϕΓ2(r Γ2)=ϕ2t(r Γ2)=ϕ3t(r Γ2),
ψΓ2(r Γ2)=(1/p2)nˆ2·ϕ2t(r Γ2)=(1/p3)nˆ2·ϕ3t(r Γ2).
ϕinc(r1)=ϕ0u(x)w(f)w(Y)1/2×exp-x2w2(Y)-jk0n1(Y-f)-12tan-1YY0+φ+k0n1x22R(Y),
DE=Pd(y)Pinc,
ϕt(r)+Γ1[ϕΓ1(r Γ1)nˆ1·G(r, r Γ1)
-p2G(r, r Γ1)ψΓ1(r Γ1)]dl=0,
G˜(r, r Γ1)=(-j/4)H0(2)(k0n2Q2)+(-j/4)RH0(2)(k0n2Q˜2),
R=(p3/p2)cos θ-[(n3/n2)2-sin2 θ]1/2(p3/p2)cos θ+[(n3/n2)2-sin2 θ]1/2,
Q2=|r-r Γ1|,Q˜2=|r-r Γ2|+|r Γ2-r Γ1|
G˜(r, r Γ1)=(-j/4)H0(2)(k0 n2 L2)TA exp(-jk0n3L3),
T=2(p3/p2)cos θ2(p3/p2)cos θ2+(n3/n2)cos θ3,
A=1+n2L3 cos2 θ2n3L2 cos2 θ3-1/2,
L2=|rΓ2-rΓ1|,L3=|r-rΓ2|

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