Abstract

We apply boundary integrals to the analysis of diffraction from both conductive and dielectric diffractive optical elements. Boundary integral analysis uses the integral form of the wave equation to describe the induced surface distributions over the boundary of a diffractive element. The surface distributions are used to determine the diffracted fields anywhere in space. In contrast to other vector analysis techniques, boundary integral methods are not restricted to the analysis of infinitely periodic structures but extend to finite aperiodic structures as well. We apply the boundary element method to solve the boundary integral equations and validate its implementation by comparing with analytical solutions our results for the diffractive analysis of a circular conducting cylinder and a dielectric cylinder. We also present the diffractive analysis of a conducting plate, a conducting linear grating, an eight-level off-axis conducting lens, an eight-level on-axis dielectric lens, and a binary dielectric lens that has subwavelength features.

© 1997 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (Wiley, New York, 1968).
  2. W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1978), Vol. XVI.
  3. O. Bryngdahl and F. Wyrowski, “Digital holography— computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1990), Vol. XXVIII.
  4. F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
  5. S.-H. Lee, ed., Selected Papers on Computer-Generated Holograms and Diffractive Optics, Vol. MS33 of Milestone Series (SPIE, Bellingham, Wash. 1992).
  6. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
  7. M. W. Farn, “Binary gratings with increased efficiency,” Appl. Opt. 31, 4453–4458 (1992).
    [CrossRef] [PubMed]
  8. E. N. Glytsis and T. K. Gaylord, “High-spatial-frequency binary and multilevel stairstep gratings: polarization-selective mirrors and broadband antireflection surfaces,” Appl. Opt. 31, 4459–4470 (1992).
    [CrossRef] [PubMed]
  9. H. Haidner, J. T. Sheridan, J. Schwider, and N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
    [CrossRef]
  10. E. Sidick, A. Knoesen, and J. N. Mait, “Design and rigorous analysis of high-efficiency array generators,” Appl. Opt. 32, 2599–2605 (1993).
    [CrossRef] [PubMed]
  11. E. Noponen, J. Turunen, and A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
  12. P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, and J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
    [CrossRef]
  13. M. Schmitz, R. Brauer, and O. Bryngdahl, “Phase gratings with subwavelength structures,” J. Opt. Soc. Am. A 12, 2458–2462 (1995).
    [CrossRef]
  14. See feature section, “ Diffractive Optics Modeling,”J. Opt. Soc. Am. A 12, 1025–1169 (1995).
  15. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich and S.-H. Lee, eds., Proc. SPIE 2404, 28–39 (1995).
    [CrossRef]
  16. R. F. Harrington, Field Computation by Moment Methods (Krieger, Malabar, Fla., 1968).
  17. F. Princemin, A. Sentenac, and J. J. Greffet, “Near field scattered by a dielectric rod below a metallic surface,” J. Opt. Soc. Am. A 11, 1117–1127 (1994).
    [CrossRef]
  18. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
  19. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1984), Vol. XXI.
  20. B. Kleeman, A. Mitreiter, and F. Wyrowski, “Integral equation method in diffractive optics,” presented at the Workshop on Diffractive Optics, Prague, Czech Republic, August 1995.
  21. H. El-Mikati, and J. B. Davies, “Improved boundary element techniques for two-dimensional scattering problems with circular boundaries,” IEEE Trans. Antennas Propag. AP-35, 539–544 (1987).
    [CrossRef]
  22. N. Morita, N. Kumagai, and J. R. Mautz, Integral Equation Methods for Electromagnetics (Artech, Norwood, Mass., 1991).
  23. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), pp. 195–199.
  24. S. Kagami and I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-32, 455–461 (1984).
  25. K. Yashiro and S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
    [CrossRef]
  26. P. D. Maker, D. W. Wilson, and R. E. Muller, “Fabrication and performance of optical interconnect analog phase holograms made by electron beam lithography,” in Optoelectronic Interconnects and Packaging, Vol. CR62 of Critical Reviews Series, R. T. Chen and P. S. Guilfoyle, eds. (SPIE, Bellingham, Wash., 1996), pp. 415–430.
  27. B. Lichtenberg and N. C. Gallagher, “Numerical modelling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
    [CrossRef]
  28. M. S. Mirotznik, D. W. Prather, and J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1308–1321( 1996).
    [CrossRef]
  29. J. L. Yao-Bi, L. Nicolas, and A. Nicolas, “2D electromagnetic scattering by simple shapes: a quantification of the error due to open boundary,” IEEE Trans. Magn. 29, 1830–1834 (1993).
    [CrossRef]
  30. B. Stupfel and R. Mittra, “A theoretical study of numerical absorbing boundary conditions,” IEEE Trans. Antennas Propag. 43, 478–486 (1995).
    [CrossRef]
  31. J. J. Wang, Generalized Moment Methods in Electromagnetics (Wiley, New York, 1991).
  32. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  33. C. B. Burchkardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  34. F. G. Kasper, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  35. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  36. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).
  37. J. J. Stammes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
    [CrossRef]
  38. J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).
  39. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 360.

1996

M. S. Mirotznik, D. W. Prather, and J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1308–1321( 1996).
[CrossRef]

1995

B. Stupfel and R. Mittra, “A theoretical study of numerical absorbing boundary conditions,” IEEE Trans. Antennas Propag. 43, 478–486 (1995).
[CrossRef]

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

M. Schmitz, R. Brauer, and O. Bryngdahl, “Phase gratings with subwavelength structures,” J. Opt. Soc. Am. A 12, 2458–2462 (1995).
[CrossRef]

See feature section, “ Diffractive Optics Modeling,”J. Opt. Soc. Am. A 12, 1025–1169 (1995).

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich and S.-H. Lee, eds., Proc. SPIE 2404, 28–39 (1995).
[CrossRef]

1994

F. Princemin, A. Sentenac, and J. J. Greffet, “Near field scattered by a dielectric rod below a metallic surface,” J. Opt. Soc. Am. A 11, 1117–1127 (1994).
[CrossRef]

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, and J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

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

1993

J. L. Yao-Bi, L. Nicolas, and A. Nicolas, “2D electromagnetic scattering by simple shapes: a quantification of the error due to open boundary,” IEEE Trans. Magn. 29, 1830–1834 (1993).
[CrossRef]

H. Haidner, J. T. Sheridan, J. Schwider, and N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

E. Sidick, A. Knoesen, and J. N. Mait, “Design and rigorous analysis of high-efficiency array generators,” Appl. Opt. 32, 2599–2605 (1993).
[CrossRef] [PubMed]

E. Noponen, J. Turunen, and A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
[CrossRef]

1992

1991

F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

1987

H. El-Mikati, and J. B. Davies, “Improved boundary element techniques for two-dimensional scattering problems with circular boundaries,” IEEE Trans. Antennas Propag. AP-35, 539–544 (1987).
[CrossRef]

1985

K. Yashiro and S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1984

S. Kagami and I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-32, 455–461 (1984).

1981

1978

1973

1966

Brauer, R.

Bryngdahl, O.

M. Schmitz, R. Brauer, and O. Bryngdahl, “Phase gratings with subwavelength structures,” J. Opt. Soc. Am. A 12, 2458–2462 (1995).
[CrossRef]

F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Burchkardt, C. B.

Collischon, M.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, and J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

Davies, J. B.

H. El-Mikati, and J. B. Davies, “Improved boundary element techniques for two-dimensional scattering problems with circular boundaries,” IEEE Trans. Antennas Propag. AP-35, 539–544 (1987).
[CrossRef]

El-Mikati, H.

H. El-Mikati, and J. B. Davies, “Improved boundary element techniques for two-dimensional scattering problems with circular boundaries,” IEEE Trans. Antennas Propag. AP-35, 539–544 (1987).
[CrossRef]

Farn, M. W.

Fukai, I.

S. Kagami and I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-32, 455–461 (1984).

Gallagher, N. C.

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

Gaylord, T. K.

Glytsis, E. N.

Greffet, J. J.

Haidner, H.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, and J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

H. Haidner, J. T. Sheridan, J. Schwider, and N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

Kagami, S.

S. Kagami and I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-32, 455–461 (1984).

Kasper, F. G.

Kipfer, P.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, and J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

Knoesen, A.

Knop, K.

Lichtenberg, B.

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

Lindolf, J.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, and J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

Mait, J. N.

M. S. Mirotznik, D. W. Prather, and J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1308–1321( 1996).
[CrossRef]

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich and S.-H. Lee, eds., Proc. SPIE 2404, 28–39 (1995).
[CrossRef]

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

E. Sidick, A. Knoesen, and J. N. Mait, “Design and rigorous analysis of high-efficiency array generators,” Appl. Opt. 32, 2599–2605 (1993).
[CrossRef] [PubMed]

Mirotznik, M. S.

M. S. Mirotznik, D. W. Prather, and J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1308–1321( 1996).
[CrossRef]

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich and S.-H. Lee, eds., Proc. SPIE 2404, 28–39 (1995).
[CrossRef]

Mittra, R.

B. Stupfel and R. Mittra, “A theoretical study of numerical absorbing boundary conditions,” IEEE Trans. Antennas Propag. 43, 478–486 (1995).
[CrossRef]

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Nicolas, A.

J. L. Yao-Bi, L. Nicolas, and A. Nicolas, “2D electromagnetic scattering by simple shapes: a quantification of the error due to open boundary,” IEEE Trans. Magn. 29, 1830–1834 (1993).
[CrossRef]

Nicolas, L.

J. L. Yao-Bi, L. Nicolas, and A. Nicolas, “2D electromagnetic scattering by simple shapes: a quantification of the error due to open boundary,” IEEE Trans. Magn. 29, 1830–1834 (1993).
[CrossRef]

Noponen, E.

Ohkawa, S.

K. Yashiro and S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

Prather, D. W.

M. S. Mirotznik, D. W. Prather, and J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1308–1321( 1996).
[CrossRef]

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich and S.-H. Lee, eds., Proc. SPIE 2404, 28–39 (1995).
[CrossRef]

Princemin, F.

Schmitz, M.

Schwider, J.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, and J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

H. Haidner, J. T. Sheridan, J. Schwider, and N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

Sentenac, A.

Sheridan, J. T.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, and J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

H. Haidner, J. T. Sheridan, J. Schwider, and N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

Sidick, E.

Stammes, J. J.

Streibl, N.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, and J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

H. Haidner, J. T. Sheridan, J. Schwider, and N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

Stupfel, B.

B. Stupfel and R. Mittra, “A theoretical study of numerical absorbing boundary conditions,” IEEE Trans. Antennas Propag. 43, 478–486 (1995).
[CrossRef]

Turunen, J.

Vasara, A.

Wyrowski, F.

F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Yao-Bi, J. L.

J. L. Yao-Bi, L. Nicolas, and A. Nicolas, “2D electromagnetic scattering by simple shapes: a quantification of the error due to open boundary,” IEEE Trans. Magn. 29, 1830–1834 (1993).
[CrossRef]

Yashiro, K.

K. Yashiro and S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

Appl. Opt.

IEEE Trans. Antennas Propag.

H. El-Mikati, and J. B. Davies, “Improved boundary element techniques for two-dimensional scattering problems with circular boundaries,” IEEE Trans. Antennas Propag. AP-35, 539–544 (1987).
[CrossRef]

S. Kagami and I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-32, 455–461 (1984).

K. Yashiro and S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

B. Stupfel and R. Mittra, “A theoretical study of numerical absorbing boundary conditions,” IEEE Trans. Antennas Propag. 43, 478–486 (1995).
[CrossRef]

IEEE Trans. Magn.

J. L. Yao-Bi, L. Nicolas, and A. Nicolas, “2D electromagnetic scattering by simple shapes: a quantification of the error due to open boundary,” IEEE Trans. Magn. 29, 1830–1834 (1993).
[CrossRef]

J. Mod. Opt.

M. S. Mirotznik, D. W. Prather, and J. N. Mait, “A hybrid finite element–boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1308–1321( 1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. Haidner, J. T. Sheridan, J. Schwider, and N. Streibl, “Design of a blazed grating consisting of metallic subwavelength binary grooves,” Opt. Commun. 98, 5–10 (1993).
[CrossRef]

Opt. Eng.

P. Kipfer, M. Collischon, H. Haidner, J. T. Sheridan, J. Schwider, N. Streibl, and J. Lindolf, “Infrared optical components based on a microrelief structure,” Opt. Eng. 33, 79–84 (1994).
[CrossRef]

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

Proc. IEEE

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Proc. SPIE

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich and S.-H. Lee, eds., Proc. SPIE 2404, 28–39 (1995).
[CrossRef]

Rep. Prog. Phys.

F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Other

S.-H. Lee, ed., Selected Papers on Computer-Generated Holograms and Diffractive Optics, Vol. MS33 of Milestone Series (SPIE, Bellingham, Wash. 1992).

J. W. Goodman, Introduction to Fourier Optics (Wiley, New York, 1968).

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1978), Vol. XVI.

O. Bryngdahl and F. Wyrowski, “Digital holography— computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1990), Vol. XXVIII.

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

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1984), Vol. XXI.

B. Kleeman, A. Mitreiter, and F. Wyrowski, “Integral equation method in diffractive optics,” presented at the Workshop on Diffractive Optics, Prague, Czech Republic, August 1995.

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

N. Morita, N. Kumagai, and J. R. Mautz, Integral Equation Methods for Electromagnetics (Artech, Norwood, Mass., 1991).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), pp. 195–199.

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

J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 360.

J. J. Wang, Generalized Moment Methods in Electromagnetics (Wiley, New York, 1991).

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Figures (6)

Fig. 1
Fig. 1

Geometry used for boundary integral equations: (a) diffracting structure, (b) interior and exterior limiting contours.

Fig. 2
Fig. 2

Validation of BEM solution to boundary integral equations: (a) geometry of dielectric circular cylinder, (b) comparison of results generated by the BEM with analytical solution.

Fig. 3
Fig. 3

Comparison of diffracted field determined by the BEM at 100λ from a perfectly conducting plate with that determined by scalar diffraction theory.

Fig. 4
Fig. 4

Diffraction from a perfectly conducting grating: (a) regional plot of blazed grating and magnitude of electric field, (b) FFT of electric-field magnitude at a single plane in the far field.

Fig. 5
Fig. 5

Diffraction from diffractive lenses determined by the BEM. For a perfectly conducting off-axis lens (a) is a regional plot of electric-field magnitude, and (b) is a line scan of electric-field intensity in its focal plane. Plots (c) and (d) are the same as (a) and (b) but are for an on-axis dielectric lens. Plots (e) and (f) are the same as (a) and (b) but are for an on-axis dielectric lens that has subwavelength features.

Fig. 6
Fig. 6

Comparison of electric field magnitude in the focal plane of a subwavelength lens for TE and TM polarization.

Tables (1)

Tables Icon

Table 1 Design and Modeling Parameters for Conducting and Dielectric Diffractive Lenses

Equations (43)

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

0=Escrs1-θ2π+CEscrGIrs, rnˆ-GIrs, rEscrnˆdl+Eincrs1-θ2π+CEincrGIrs, rnˆ-GIrs, rEincrnˆdl,
0=Escrsθ2π+CGOrs, rEscrnˆ-EscrGOrs, rnˆdl,
Esc(rs)=-C Qsc(r)GO(rs, r)dl.
Einc(xs, ys)=ωμ4C J(r)GO(rs, r)dl.
0=2Etot(r)+βI2Etot(r)forrI,
-f(r)=2Etot(r)+βO2Etot(r)forrO,
Etotr=C GIr, rEtotrnˆ-EtotrGIr, rnˆdl,rI,
Etotr=Eincr+C+C×EtotrGOr, rnˆ-GOr, rEtotrnˆdl,rO,
Einc(r)=C f(r)GO(r, r)dl
GIr, r=14jH02βI|r-r|=14jH02βIx-x2+y-y2,
GOr, r=14jH02βO|r-r|=14jH02βOx-x2+y-y2,
Etot(r)=Einc(r)+Esc(r).
C GOr, rEscrnˆ-EscrGOr, rnˆdl=0,
rO,
C GOr, rEincrnˆ-EincrGOr, rnˆdl=0,
rO,
0=Escrs+CEscrGIrs, rnˆ-GIrs, rEscrnˆdl+Eincrs+C EincrGIrs, rnˆ-GIrs, rEincrnˆdl,
0=Escrs+C GOrs, rEscrnˆ-EscrGOrs, rnˆdl.
0=Escrs-CEscrGOrs, rnˆ-GOrs, rEscrnˆdl-limε0-α/2α/2 -Escr2πε+lnβOε2πEscrnˆε dθ,
ε=limrrs |rs-r|.
0=Escrs1-θ2π+CEscrGIrs, rnˆ-GIrs, rEscrnˆdl+Eincrs1-θ2π+CEincrGIrs, rnˆ-GIrs, rEincrnˆdl,
0=Escrsθ2π+CGOrs, rEscrnˆ-EscrGOrs, rnˆdl.
Esc(r)=C [Esc(r)GO(r, r)nˆ-GO(r, r)Esc(r)nˆ]dl,
Escrξ=n=1N Eˆnscξ=n=1N Enscϕ1ξ+En+1scϕ2ξ,
Escrξnˆ=Qscrξ=n=1N Qˆnscξ=n=1N Qnscϕ1ξ+Qn+1scϕ2ξ.
ϕ1(ξ)=(1-ξ)/2,ϕ2(ξ)=(1+ξ)/2,
xˆn(ξ)=xnϕ1(ξ)+xn+1ϕ2(ξ),
yˆn(ξ)=ynϕ1(ξ)+yn+1ϕ2(ξ).
ωm=δrξ-rm=1,rξ=rm0,elsewhere,
ZIn,m-YIn,mZOn,mYOn,mEmscQmsc=-ZIn,mYIn,m00EmincQminc,
ZIn,m=1-θn2πδnm+-11 Δln2ϕ1ξGIrxˆn, yˆn, rmnˆ+Δln-12ϕ2ξGIrxˆn-1, yˆn-1, rmnˆdξ,
YIn,m=-11 Δln2ϕ1ξGIrxˆn, yˆn, rm+Δln-12ϕ2ξGIrxˆn-1, yˆn-1, rmdξ,
ZOn,m=θn2πδnm--11 Δln2ϕ1ξGOrxˆn, yˆn, rmnˆ+Δln-12ϕ2ξGOrxˆn-1, yˆn-1, rmnˆdξ,
YOn,m=-11 Δln2ϕ1ξGOrxˆn, yˆn, rm+Δln-12ϕ2ξGOrxˆn-1, yˆn-1, rmdξ.
ZIn,m-r,1YIn,mZOn,mr,2YOn,mHmscQmsc=-ZIn,mr,1YIn,m00×HmincQminc,
Einc(rs)=ωμ4C J(r)H0(2)×βO(xs-x)2+(ys-y)2dl,
rsC,
Jˆn(ξ)=n=1N Jnϕ1(ξ)+Jn+1ϕ2(ξ),
Einc(xs, ys)=ωμ8n=1N Δln -11 JˆnH0(2)×β[xs-xˆn(ξ)]2+[ys-yˆn(ξ)]2dξ,
[Em]=[Lm,n][Jn]
Em=Einc(xm, ym),
Lm,n=ωμ8-11Δlnϕ1(ξ)H0(2)×β[xm-xˆn(ξ)]2+[ym-yˆn(ξ)]2+Δln-1ϕ2(ξ)H0(2)×β[xm-xˆn-1(ξ)]2+[ym-yˆn-1(ξ)]2dξ.
Etot(x, y)=Einc(x, y)-ωμ8n=1N Δln-11 [Jnϕ1(ξ)+Jn+1ϕ2(ξ)]H0(2)×β[x-xˆn(ξ)]2+[y-yˆn(ξ)]2dξ.

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