Abstract

Diffraction gratings with feature sizes comparable to the wavelength are analyzed with a finite-difference time-domain method, which is a unique approach to electromagnetic problems in the time domain. The diffraction efficiencies obtained are in good agreement with other commonly used numerical methods in the frequency domain. As a further application, diffraction problems with pulsed light are also investigated.

© 1998 Optical Society of America

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References

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  1. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
    [CrossRef]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 36–38.
  3. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  4. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
  5. J. Turunen, F. Wyrowski, “Diffractive optics: from promise to fruition,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), Chap. 6, pp. 111–123.
  6. A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
    [CrossRef]
  7. A. Taflove, K. R. Umashankar, “Review of FD–TD numerical modeling of electromagnetic wave scattering and radar cross section,” Proc. IEEE 77, 682–699 (1989).
    [CrossRef]
  8. K. L. Shlager, J. B. Schneider, “A selective survey of the finite-difference time-domain literature,” IEEE Trans. Antennas Propag. Mag. 37, 39–56 (1995).
  9. W.-J. Tsay, D. M. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microwave Guid. Wave Lett. 3, 250–252 (1993).
    [CrossRef]
  10. A. C. Cangellaris, M. Gribbons, G. Sohos, “A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures,” IEEE Microwave Guid. Wave Lett. 3, 375–377 (1993).
    [CrossRef]
  11. D. T. Prescott, N. V. Shuley, “Extensions to the FDTD method for the analysis of infinitely periodic arrays,” IEEE Microwave Guid. Wave Lett. 4, 352–354 (1994).
    [CrossRef]
  12. J. B. Judkins, R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A 12, 1974–1983 (1995).
    [CrossRef]
  13. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  14. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
    [CrossRef]
  15. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  16. E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
  17. A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
    [CrossRef]
  18. J. Turunen, “Form-birefringence limits of Fourier-expansion methods in grating theory,” J. Opt. Soc. Am. A 13, 1013–1018 (1996).
    [CrossRef]
  19. L. Li, “Use of Fourier series in analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  20. P. M. Goorjian, A. Taflove, R. M. Joseph, S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
    [CrossRef]
  21. Z. Wang, Z. Xu, Z. Zhnag, “Diffraction integral formulas of the pulsed wave field in the temporal domain,” Opt. Lett. 22, 354–356 (1997).
    [CrossRef] [PubMed]
  22. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 921–924.
  23. W. T. Silfvast, “Lasers,” in Handbook of Optics, Vol. 1, M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 11, pp. 11.7–11.8.
  24. H. Kikuta, H. Yoshida, K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
    [CrossRef]

1997 (1)

1996 (2)

1995 (3)

J. B. Judkins, R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A 12, 1974–1983 (1995).
[CrossRef]

H. Kikuta, H. Yoshida, K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

K. L. Shlager, J. B. Schneider, “A selective survey of the finite-difference time-domain literature,” IEEE Trans. Antennas Propag. Mag. 37, 39–56 (1995).

1994 (2)

D. T. Prescott, N. V. Shuley, “Extensions to the FDTD method for the analysis of infinitely periodic arrays,” IEEE Microwave Guid. Wave Lett. 4, 352–354 (1994).
[CrossRef]

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1993 (3)

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

W.-J. Tsay, D. M. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microwave Guid. Wave Lett. 3, 250–252 (1993).
[CrossRef]

A. C. Cangellaris, M. Gribbons, G. Sohos, “A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures,” IEEE Microwave Guid. Wave Lett. 3, 375–377 (1993).
[CrossRef]

1992 (2)

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

P. M. Goorjian, A. Taflove, R. M. Joseph, S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

1989 (1)

A. Taflove, K. R. Umashankar, “Review of FD–TD numerical modeling of electromagnetic wave scattering and radar cross section,” Proc. IEEE 77, 682–699 (1989).
[CrossRef]

1988 (1)

A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
[CrossRef]

1985 (1)

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

1981 (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

1975 (1)

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Brodwin, M. E.

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

Cangellaris, A. C.

A. C. Cangellaris, M. Gribbons, G. Sohos, “A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures,” IEEE Microwave Guid. Wave Lett. 3, 375–377 (1993).
[CrossRef]

Gaylord, T. K.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 36–38.

Goorjian, P. M.

P. M. Goorjian, A. Taflove, R. M. Joseph, S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

Gribbons, M.

A. C. Cangellaris, M. Gribbons, G. Sohos, “A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures,” IEEE Microwave Guid. Wave Lett. 3, 375–377 (1993).
[CrossRef]

Hagness, S. C.

P. M. Goorjian, A. Taflove, R. M. Joseph, S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

Iwata, K.

H. Kikuta, H. Yoshida, K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

Joseph, R. M.

P. M. Goorjian, A. Taflove, R. M. Joseph, S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

Judkins, J. B.

Kikuta, H.

H. Kikuta, H. Yoshida, K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

Li, L.

Miller, J. M.

Moharam, M. G.

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

Mur, G.

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

Noponen, E.

Pozar, D. M.

W.-J. Tsay, D. M. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microwave Guid. Wave Lett. 3, 250–252 (1993).
[CrossRef]

Prescott, D. T.

D. T. Prescott, N. V. Shuley, “Extensions to the FDTD method for the analysis of infinitely periodic arrays,” IEEE Microwave Guid. Wave Lett. 4, 352–354 (1994).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 921–924.

Schneider, J. B.

K. L. Shlager, J. B. Schneider, “A selective survey of the finite-difference time-domain literature,” IEEE Trans. Antennas Propag. Mag. 37, 39–56 (1995).

Shlager, K. L.

K. L. Shlager, J. B. Schneider, “A selective survey of the finite-difference time-domain literature,” IEEE Trans. Antennas Propag. Mag. 37, 39–56 (1995).

Shuley, N. V.

D. T. Prescott, N. V. Shuley, “Extensions to the FDTD method for the analysis of infinitely periodic arrays,” IEEE Microwave Guid. Wave Lett. 4, 352–354 (1994).
[CrossRef]

Silfvast, W. T.

W. T. Silfvast, “Lasers,” in Handbook of Optics, Vol. 1, M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 11, pp. 11.7–11.8.

Sohos, G.

A. C. Cangellaris, M. Gribbons, G. Sohos, “A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures,” IEEE Microwave Guid. Wave Lett. 3, 375–377 (1993).
[CrossRef]

Taflove, A.

P. M. Goorjian, A. Taflove, R. M. Joseph, S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

A. Taflove, K. R. Umashankar, “Review of FD–TD numerical modeling of electromagnetic wave scattering and radar cross section,” Proc. IEEE 77, 682–699 (1989).
[CrossRef]

A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
[CrossRef]

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

Taghizadeh, M. R.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 921–924.

Tsay, W.-J.

W.-J. Tsay, D. M. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microwave Guid. Wave Lett. 3, 250–252 (1993).
[CrossRef]

Turunen, J.

Umashankar, K. R.

A. Taflove, K. R. Umashankar, “Review of FD–TD numerical modeling of electromagnetic wave scattering and radar cross section,” Proc. IEEE 77, 682–699 (1989).
[CrossRef]

Vasara, A.

Wang, Z.

Wyrowski, F.

J. Turunen, F. Wyrowski, “Diffractive optics: from promise to fruition,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), Chap. 6, pp. 111–123.

Xu, Z.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Yoshida, H.

H. Kikuta, H. Yoshida, K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

Zhnag, Z.

Ziolkowski, R. W.

IEEE J. Quantum Electron. (1)

P. M. Goorjian, A. Taflove, R. M. Joseph, S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
[CrossRef]

IEEE Microwave Guid. Wave Lett. (3)

W.-J. Tsay, D. M. Pozar, “Application of the FDTD technique to periodic problems in scattering and radiation,” IEEE Microwave Guid. Wave Lett. 3, 250–252 (1993).
[CrossRef]

A. C. Cangellaris, M. Gribbons, G. Sohos, “A hybrid spectral/FDTD method for the electromagnetic analysis of guided waves in periodic structures,” IEEE Microwave Guid. Wave Lett. 3, 375–377 (1993).
[CrossRef]

D. T. Prescott, N. V. Shuley, “Extensions to the FDTD method for the analysis of infinitely periodic arrays,” IEEE Microwave Guid. Wave Lett. 4, 352–354 (1994).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

IEEE Trans. Antennas Propag. Mag. (1)

K. L. Shlager, J. B. Schneider, “A selective survey of the finite-difference time-domain literature,” IEEE Trans. Antennas Propag. Mag. 37, 39–56 (1995).

IEEE Trans. Electromagn. Compat. (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

Opt. Rev. (1)

H. Kikuta, H. Yoshida, K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

Proc. IEEE (2)

A. Taflove, K. R. Umashankar, “Review of FD–TD numerical modeling of electromagnetic wave scattering and radar cross section,” Proc. IEEE 77, 682–699 (1989).
[CrossRef]

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

Wave Motion (1)

A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
[CrossRef]

Other (5)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 36–38.

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

J. Turunen, F. Wyrowski, “Diffractive optics: from promise to fruition,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), Chap. 6, pp. 111–123.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 921–924.

W. T. Silfvast, “Lasers,” in Handbook of Optics, Vol. 1, M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 11, pp. 11.7–11.8.

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Figures (11)

Fig. 1
Fig. 1

Unit cell in FDTD method. ○, Ex; □, Ey; △, Ez; ●, Bx; ■, By; △, Bz.

Fig. 2
Fig. 2

Grating structure employed here. λ, wavelength; jr, relative permittivity of jth media.

Fig. 3
Fig. 3

Computed area and sampling points. □, Ey; ●, Bx; △, Bz.

Fig. 4
Fig. 4

Wave reflected back from the edges of computed area. (a) continuous wave, (b) pulsed wave (least necessary), (c) pulsed wave (with some margin).

Fig. 5
Fig. 5

Convergence of the FDTD method (iE=80 and k2 -k1=80). Diffraction efficiencies are compared with the values for kE=10,000. Rj, refracted jth order; Tj, transmitted jth order.

Fig. 6
Fig. 6

Comparison of diffraction efficiencies by the FDTD method and the Fourier expansion method (kE=4000). 80,40 denotes iE=80 and k2-k1=40. FE means Fourier expansion method. R1, R2, and R3 for FE were not available.

Fig. 7
Fig. 7

Time variation of the electric fields in the exit plane of the grating. The value is normalized by the incident field.

Fig. 8
Fig. 8

Time variation of the z component of the modulus of the Poynting vector of propagating orders. Tj, transmitted jth order.

Fig. 9
Fig. 9

Sum of diffraction efficiencies of all transmitted orders for pulsed waves with different widths. ■, grating depth of λ; ○, grating depth of 2λ.

Fig. 10
Fig. 10

Comparison of diffraction efficiencies of pulsed light with continuous wave. iE=80, k2-k1=40 (grating depth is λ).

Fig. 11
Fig. 11

Comparison of diffraction efficiencies of pulsed light with continuous wave. iE=80, k2-k1=80 (grating depth is 2λ).

Tables (2)

Tables Icon

Table 1 Parameters of Solitary Gaussian Pulses Analyzeda

Tables Icon

Table 2 Diffraction Efficiencies of Pulsed Light

Equations (11)

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

Eyn+1(i, k)=1-σ(i, k)Δt(i, k)Eyn(i, k)+Δt(i, k)×Hxn+1(i, k)-Hzn+1(i, k-1)Δz-Hzn+1(i, k)-Hzn+1(i-1, k)Δx,
Hxn+1(i, k)=Hxn(i, k)+Δtμ(i, k)Eyn(i, k+1)-Eyn(i, k)Δz,
Hzn+1(i, k)=Hzn(i, k)+Δtμ(i, k)Eyn(i+1, k)-Eyn(i, k)Δx,
Ex=Ez=Hy=0.
Eyn(i, 1)=sin(2πcnΔt/λ),
Hxn(i, 1)=-sin[2πc(n-0.5)×Δt/λ-πΔz1r/λ]×1r/Z0.
2(k1-1)1r=(k2-k1)qr+2(kE-k2)2r,
Δtc-1[(Δx)-2+(Δz)-2]-1/2.
f(t)=exp[-4 ln 2(t-t0)2/τ2],
2(k1/2-1)1=(k2-k1)/2q+2ct/Δz0,
2(k1-1)1=(k2-k1)q+2(kE-1)2.

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