Abstract

Effects of spatial fine structure on a femtosecond-order optical pulse are numerically studied by using nondispersive and linear dielectric diffraction gratings in the resonance domain. Whereas a pulse longer than 100 fs is expected to behave the same as a continuous wave, the distribution of energy into each diffraction order deviates gradually for shorter pulses. This is due to the wide spectral profile of the pulsed wave. Overlap of adjacent diffraction orders will also occur for a pulse shorter than 5 fs. Therefore extra attention should be paid to designing optical elements based on diffractive structure for use with ultra-short pulsed waves.

© 1999 Optical Society of America

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References

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  1. The term resonance domain is explained briefly in 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). A more rigorous definition would be the range of a grating period normalized by wavelength where neither effective medium theory nor scalar diffraction theory is accurate. For example, refer to Fig. 1.6 in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie Verlag, Berlin, 1997), p. 12.
    [CrossRef]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 36–38.
  3. Well known and organized examples are the following: R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980); T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985); 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.
    [CrossRef]
  4. H. Ichikawa, “Electromagnetic analysis of diffraction gratings by the finite-difference time-domain method,” J. Opt. Soc. Am. A 15, 152–157 (1998).
    [CrossRef]
  5. R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
    [CrossRef]
  6. Although semiconductor-based pulse lasers have much higher pulse frequencies, e.g., some reach several tens of GHz, their pulse durations are longer than 1 ps. See, for example, Abstracts of Fifth International Workshop on Femtosecond Technology (The Femtosecond Technology Research Association, Tsukuba, Japan, 1998).
  7. K. L. Shlager, J. B. Schneider, “A selective survey of the finite-difference time-domain literature,” IEEE Antennas Propag. Mag. 37, 39–56 (1995).
    [CrossRef]
  8. J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Otics, H. P. Herzig, ed. (Taylor & Francis, London, 1997), Chap. 2, pp. 31–52.
  9. 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).
  10. A. P. Zhao, A. Räisänen, S. R. Cvetkovic, “A fast and efficient FDTD algorithm for the analysis of planar microstrip discontinuities by using a simple source excitation scheme,” IEEE Microwave Guided Wave Lett. 5, 341–343 (1995).
    [CrossRef]
  11. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  12. J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).
  13. B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991), p. 787.
  14. I. J. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965).
    [CrossRef]

1998 (1)

1997 (1)

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

1995 (2)

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

A. P. Zhao, A. Räisänen, S. R. Cvetkovic, “A fast and efficient FDTD algorithm for the analysis of planar microstrip discontinuities by using a simple source excitation scheme,” IEEE Microwave Guided Wave Lett. 5, 341–343 (1995).
[CrossRef]

1994 (1)

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

1992 (1)

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).

1965 (1)

Berenger, J.-P.

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

Cvetkovic, S. R.

A. P. Zhao, A. Räisänen, S. R. Cvetkovic, “A fast and efficient FDTD algorithm for the analysis of planar microstrip discontinuities by using a simple source excitation scheme,” IEEE Microwave Guided Wave Lett. 5, 341–343 (1995).
[CrossRef]

Diels, J.-C.

J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).

Goodman, J. W.

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

Ichikawa, H.

Malitson, I. J.

Miller, J. M.

Noponen, E.

Räisänen, A.

A. P. Zhao, A. Räisänen, S. R. Cvetkovic, “A fast and efficient FDTD algorithm for the analysis of planar microstrip discontinuities by using a simple source excitation scheme,” IEEE Microwave Guided Wave Lett. 5, 341–343 (1995).
[CrossRef]

Rudolph, W.

J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991), p. 787.

Schneider, J. B.

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

Shlager, K. L.

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

Taghizadeh, M. R.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991), p. 787.

Turunen, J.

Vasara, A.

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).

Zhao, A. P.

A. P. Zhao, A. Räisänen, S. R. Cvetkovic, “A fast and efficient FDTD algorithm for the analysis of planar microstrip discontinuities by using a simple source excitation scheme,” IEEE Microwave Guided Wave Lett. 5, 341–343 (1995).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

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

IEEE Microwave Guided Wave Lett. (1)

A. P. Zhao, A. Räisänen, S. R. Cvetkovic, “A fast and efficient FDTD algorithm for the analysis of planar microstrip discontinuities by using a simple source excitation scheme,” IEEE Microwave Guided Wave Lett. 5, 341–343 (1995).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

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).

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. (1)

J. Opt. Soc. Am. A (2)

Other (6)

J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).

B. E. A. Saleh, M. C. Teich, Fundamental of Photonics (Wiley, New York, 1991), p. 787.

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

Well known and organized examples are the following: R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980); T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985); 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.
[CrossRef]

Although semiconductor-based pulse lasers have much higher pulse frequencies, e.g., some reach several tens of GHz, their pulse durations are longer than 1 ps. See, for example, Abstracts of Fifth International Workshop on Femtosecond Technology (The Femtosecond Technology Research Association, Tsukuba, Japan, 1998).

J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Otics, H. P. Herzig, ed. (Taylor & Francis, London, 1997), Chap. 2, pp. 31–52.

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Figures (9)

Fig. 1
Fig. 1

Grating structure employed in this paper. λ0, central wavelength in vacuum; ϵjr, relative permittivity of the jth medium. A definition of the coordinate system used is also shown.

Fig. 2
Fig. 2

Modulation of incident pulse. (a) Amplitude, (b) intensity. Thick curves, pulse modulation; thin curves, envelope; τ, FWHM pulse duration; t0, offset of peak position.

Fig. 3
Fig. 3

Time variation of the field for a 10-fs pulse. Note that the scale of the field amplitude is different in each graph. (a) Incident pulse, (b) transmitted pulse, (c) reflected pulse.

Fig. 4
Fig. 4

Diffraction efficiency spectra of transmitted waves versus diffraction angle. For convenience of comparison, the intensity of the spectrum for each pulse duration is normalized.

Fig. 5
Fig. 5

Change in the zeroth- and first-order diffraction efficiencies of transmitted waves with the period of a grating for a continuous wave. The grating depth is kept constant at a value of 1λ. Rj, reflected ±jth order; Tj, transmitted ±jth order.

Fig. 6
Fig. 6

Diffraction efficiencies of each order for several different pulse durations. Rj, reflected ±jth order; Tj, transmitted ±jth order; cw denotes continuous wave.

Fig. 7
Fig. 7

Total reflected and transmitted efficiencies for various pulse durations. R, reflection; T, transmission; cw denotes continuous wave.

Fig. 8
Fig. 8

Total reflected and transmitted efficiencies of continuous waves at various wavelengths. T, transmission; R, reflection.

Fig. 9
Fig. 9

Spectra of incident and diffracted pulses of 2 fs. In, incident pulse; T, transmitted pulse; R, reflected pulse. Total is the sum of the transmitted and reflected pulses.

Tables (2)

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Table 1 Details of Sample Points and Computationa

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Table 2 Dispersion of Fused Silica

Equations (20)

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f(t)=exp{-2 ln 2[(t-t0)2/τ2]}cos[2πν0(t-t0)],
Am(t)=Fx{f(x, t)},
ηm=|Am(t)|2cos θm
T sin θm=mλ0/ϵ2r.
Am(ν)=Ft{Fx{f(x, t)}},
ηm(ν)=|Am(ν)|2 cos θm(ν),
T sin θm(ν)=mc/ϵ2rν.
Eyn+1(i, k)
=Eyn(i, k)+Δtϵ(i, k)Hxn+1(i, k)-Hxn+1(i, k-1)Δz-Hzn+1(i, k)-Hzn+1(i-1, k)Δx,
Hxn+1(i, k)
=Hxn(i, k)+Δtμ0Eyn(i, k+1)-Eyn(i, k)Δz,
Hzn+1(i, k)
=Hzn(i, k)-Δtμ0Eyn(i+1, k)-Eyn(i, k)Δx,
Ex=Ez=Hy=0.
Δt[(Δx)-2+(Δz)-2]-1/2ϵqr/c,
Gn(iE+1, k)=Gn(1, k),
Eyn(i, 1)=exp{-2 ln 2[(nΔt-t0)2/τ2]}×cos[2πν0(nΔt-t0)],
Hxn(i, 1)=-exp{-2 ln 2[(t˜-t0)2/τ2]}×cos[2πν0(t˜-t0)]×ϵ1r/Z0,
(k1-1)Δzϵ1rct0,
2(k1-1)ϵ1r=(k2-k1)ϵqr+2(kE-k2)ϵ2r,

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