Abstract

We report on the time-domain differentiation of light waves by metallic transmission gratings. Time-resolved terahertz experiments show that the first time derivative of an arbitrary waveform can be achieved by use of gratings of subwavelength period. The results are in accord with classical diffraction theory and may permit novel applications for tailoring few-cycle light pulses and ultrahigh-frequency optoelectronics.

© 2001 Optical Society of America

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  1. H. Bethe, Phys. Rev. 66, 163 (1944).
    [CrossRef]
  2. T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
    [CrossRef]
  3. L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry, and T. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
    [CrossRef]
  4. L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, Phys. Rev. Lett. 86, 1110 (2001).
    [CrossRef] [PubMed]
  5. E. Treacy, IEEE J. Quantum Electron. 5, 454 (1969).
    [CrossRef]
  6. K. Wynne and D. Jaroszynski, Opt. Lett. 24, 25 (1999).
    [CrossRef]
  7. Q. Wu and X.-C. Zhang, Appl. Phys. Lett. 67, 3523 (1995).
    [CrossRef]
  8. Q. Wu, M. Litz, and X.-C. Zhang, Appl. Phys. Lett. 68, 2924 (1996).
    [CrossRef]
  9. For scaling of the calculated data we considered the factor ωel/c as deduced in Eq.  (6).
  10. L. Weinstein, The Theory of Diffraction and the Factorization Method (Golem, Boulder, Col., 1969).
  11. T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000), and referencies therein.
    [CrossRef]
  12. M. MacDonald, A. Alexanian, R. York, Z. Popovic, and E. Grossman, IEEE Trans. Microwave Theory Technol. 48, 712 (2000).
    [CrossRef]
  13. J. White, C. Ludwig, and J. Kuhl, J. Opt. Soc. Am. B 12, 1687 (1995).
    [CrossRef]
  14. R. Kersting, G. Strasser, and K. Unterrainer, Electron. Lett. 36, 1156 (2000).
    [CrossRef]

2001 (2)

L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry, and T. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef]

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, Phys. Rev. Lett. 86, 1110 (2001).
[CrossRef] [PubMed]

2000 (3)

T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000), and referencies therein.
[CrossRef]

M. MacDonald, A. Alexanian, R. York, Z. Popovic, and E. Grossman, IEEE Trans. Microwave Theory Technol. 48, 712 (2000).
[CrossRef]

R. Kersting, G. Strasser, and K. Unterrainer, Electron. Lett. 36, 1156 (2000).
[CrossRef]

1999 (1)

1998 (1)

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

1996 (1)

Q. Wu, M. Litz, and X.-C. Zhang, Appl. Phys. Lett. 68, 2924 (1996).
[CrossRef]

1995 (2)

1969 (1)

E. Treacy, IEEE J. Quantum Electron. 5, 454 (1969).
[CrossRef]

1944 (1)

H. Bethe, Phys. Rev. 66, 163 (1944).
[CrossRef]

Alexanian, A.

M. MacDonald, A. Alexanian, R. York, Z. Popovic, and E. Grossman, IEEE Trans. Microwave Theory Technol. 48, 712 (2000).
[CrossRef]

Bethe, H.

H. Bethe, Phys. Rev. 66, 163 (1944).
[CrossRef]

Brabec, T.

T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000), and referencies therein.
[CrossRef]

de Fornel, F.

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, Phys. Rev. Lett. 86, 1110 (2001).
[CrossRef] [PubMed]

Ebbesen, T.

L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry, and T. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef]

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

Garcia-Vidal, F.

L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry, and T. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef]

Ghaemi, H.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

Grillot, F.

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, Phys. Rev. Lett. 86, 1110 (2001).
[CrossRef] [PubMed]

Grossman, E.

M. MacDonald, A. Alexanian, R. York, Z. Popovic, and E. Grossman, IEEE Trans. Microwave Theory Technol. 48, 712 (2000).
[CrossRef]

Jaroszynski, D.

Kersting, R.

R. Kersting, G. Strasser, and K. Unterrainer, Electron. Lett. 36, 1156 (2000).
[CrossRef]

Krausz, F.

T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000), and referencies therein.
[CrossRef]

Kuhl, J.

Lezec, H.

L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry, and T. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef]

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

Litz, M.

Q. Wu, M. Litz, and X.-C. Zhang, Appl. Phys. Lett. 68, 2924 (1996).
[CrossRef]

Ludwig, C.

MacDonald, M.

M. MacDonald, A. Alexanian, R. York, Z. Popovic, and E. Grossman, IEEE Trans. Microwave Theory Technol. 48, 712 (2000).
[CrossRef]

Martin-Moreno, L.

L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry, and T. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef]

Pellerin, K.

L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry, and T. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef]

Pendry, J.

L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry, and T. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef]

Popovic, Z.

M. MacDonald, A. Alexanian, R. York, Z. Popovic, and E. Grossman, IEEE Trans. Microwave Theory Technol. 48, 712 (2000).
[CrossRef]

Salomon, L.

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, Phys. Rev. Lett. 86, 1110 (2001).
[CrossRef] [PubMed]

Strasser, G.

R. Kersting, G. Strasser, and K. Unterrainer, Electron. Lett. 36, 1156 (2000).
[CrossRef]

Thio, T.

L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry, and T. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef]

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

Treacy, E.

E. Treacy, IEEE J. Quantum Electron. 5, 454 (1969).
[CrossRef]

Unterrainer, K.

R. Kersting, G. Strasser, and K. Unterrainer, Electron. Lett. 36, 1156 (2000).
[CrossRef]

Weinstein, L.

L. Weinstein, The Theory of Diffraction and the Factorization Method (Golem, Boulder, Col., 1969).

White, J.

Wolff, P.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

Wu, Q.

Q. Wu, M. Litz, and X.-C. Zhang, Appl. Phys. Lett. 68, 2924 (1996).
[CrossRef]

Q. Wu and X.-C. Zhang, Appl. Phys. Lett. 67, 3523 (1995).
[CrossRef]

Wynne, K.

York, R.

M. MacDonald, A. Alexanian, R. York, Z. Popovic, and E. Grossman, IEEE Trans. Microwave Theory Technol. 48, 712 (2000).
[CrossRef]

Zayats, A.

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, Phys. Rev. Lett. 86, 1110 (2001).
[CrossRef] [PubMed]

Zhang, X.-C.

Q. Wu, M. Litz, and X.-C. Zhang, Appl. Phys. Lett. 68, 2924 (1996).
[CrossRef]

Q. Wu and X.-C. Zhang, Appl. Phys. Lett. 67, 3523 (1995).
[CrossRef]

Appl. Phys. Lett. (2)

Q. Wu and X.-C. Zhang, Appl. Phys. Lett. 67, 3523 (1995).
[CrossRef]

Q. Wu, M. Litz, and X.-C. Zhang, Appl. Phys. Lett. 68, 2924 (1996).
[CrossRef]

Electron. Lett. (1)

R. Kersting, G. Strasser, and K. Unterrainer, Electron. Lett. 36, 1156 (2000).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. Treacy, IEEE J. Quantum Electron. 5, 454 (1969).
[CrossRef]

IEEE Trans. Microwave Theory Technol. (1)

M. MacDonald, A. Alexanian, R. York, Z. Popovic, and E. Grossman, IEEE Trans. Microwave Theory Technol. 48, 712 (2000).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (1)

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

H. Bethe, Phys. Rev. 66, 163 (1944).
[CrossRef]

Phys. Rev. Lett. (2)

L. Martin-Moreno, F. Garcia-Vidal, H. Lezec, K. Pellerin, T. Thio, J. Pendry, and T. Ebbesen, Phys. Rev. Lett. 86, 1114 (2001).
[CrossRef]

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, Phys. Rev. Lett. 86, 1110 (2001).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000), and referencies therein.
[CrossRef]

Other (2)

For scaling of the calculated data we considered the factor ωel/c as deduced in Eq.  (6).

L. Weinstein, The Theory of Diffraction and the Factorization Method (Golem, Boulder, Col., 1969).

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Figures (3)

Fig. 1
Fig. 1

Time-resolved transmission of few-cycle THz pulses through a grating compared with the incident pulse. The incident pulse was recorded by transmission of the THz pulse through an uncovered reference area on the same wafer. (a) Grating oriented perpendicular to the electric field of the incident THz pulse. (b) Orientation parallel to the electric field. The transmission signal is magnified by a factor of 10.

Fig. 2
Fig. 2

Incident THz pulse, measured transmission signal, and calculated time derivative of the incident pulse. The transients of the THz signals are shown in (a). The transmission signal was recorded on a grating of 10μm period. The signal of the incident pulse is scaled by a factor of 0.1. Amplitude spectra and transmission spectra of the signals above are shown in (b).

Fig. 3
Fig. 3

Comparison of experimental data and calculated transmission transients for several grating periods.

Equations (6)

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

Exinct=0Ekexpikz-ωtdk+c.c.
Extrt=m=-0BmEk×expikysinϕm+zcosϕm-iωtdk+c.c.,
B0k=-ikl1-ikl,
Extrt=20kl-k3l3sinkz-ωtEkdk+20k2l2-k4l4coskz-ωtEkdk
=lctExinc-lc22t2Exinc+lc33t3Exinc-lc44t4Exinc.
Extrt=ωelcωetExinc-ωelc22ωet2Exinc+.

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