2008 OPTICS LETTERS / Vol. 26, No. 24 / December 15, 2001Time-domain differentiation of terahertz pulsesA. Filin, M. Stowe, and R. KerstingDepartment of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180Received August 15, 2001We report on the time-domain differentiation of light waves by metallic transmission gratings. Time-resolvedterahertz experiments show that the first time derivative of an arbitrary waveform can be achieved by use ofgratings of subwavelength period. The results are in accord with classical diffraction theory and may permitnovel applications for tailoring few-cycle light pulses and ultrahigh-frequency optoelectronics. © 2001 OpticalSociety of AmericaOCIS codes: 300.6340, 300.6530, 050.0050, 070.6020, 220.0220.The control of light by modification of either itsspectral distribution or its temporal shape is aprime objective of optics. Periodic structures suchas gratings have frequently been used as tools forthe control of light, and more-advanced structuressuch as photonic bandgaps have stimulated increasedefforts to understand the transmission and ref lectionof subwavelength structures. In the case of metallicstructures, light transmission is constrained whenthe dimension of the apertures becomes comparablewith the wavelength of the incident light or gets evensmaller.1Recent advances in submicrometer metallicstructures have shown that increased transmissioncan be achieved by coupling of light to surface plasmonresonances.2–4Most research on periodic metallicstructures has focused on spectral transmission orref lection. Although gratings have been used fordecades for time-domain pulse shaping,5less is knownabout their effects, which occur on subcycle timescales, because few experiments have had the requiredtime resolution. The first time-resolved experimentsto address such properties were performed at ter-ahertz (THz) frequencies and interpreted in termsof a superluminal transport through subwavelengthstructures.6In this Letter we present a time-resolved study oflight transmission through metallic gratings. Theexperiments were performed with THz time-domainspectroscopy. Femtosecond time resolution permitsthe observation of subcycle changes of the transmittedlight as the light propagates through gratings. Itwas found that the zero-order transmission signal isthe first time derivative of the incident light wave.Classical diffraction theory confirmed the experimen-tal findings and showed the scale invariance of thisanalog time differentiation of light pulses.Metallic transmission gratings were fabricatedupon semi-insulating silicon bye-beam evaporation.The 10 mm 3 10 mm gold gratings had periods of1040 mm and a filling factor of 50%. The 200-nmmetal films were signif icantly thicker than the skindepth of gold, which is approximately 30 nm at 1 THz.We measured the transmission through the gratingsby free-space THz time-domain spectroscopy. Coher-ent THz pulses were generated by the excitation ofan n-doped InAs crystal with 70-fs laser pulses of770-nm wavelength and 5-nJ pulse energy. The cen-ter frequency of the THz pulses was ⬃2.25 THz, whichcorresponded to a wavelength of ⬃130 mm. Thetransmitted pulses were detected in the time domainby electro-optic sampling method.7,8The detectionbandwidth of the setup was limited to ⬃3.5 THz bythe 300-mm-thick ZnTe crystal. Only the zero-orderdiffraction of the transmission was detected. Theaperture of the setup prevented higher orders fromcontributing to the signal. To avoid spectroscopicartifacts caused by water absorption, we performedthe experiments in a vacuum chamber at 10 Pa.The transmission through gratings of a subwave-length period strongly depends on the orientationof the grating lines relative to the incident elec-tric field of the THz pulse. Figure 1(a) shows thecase of perpendicular orientation. As expected,the THz pulses are perfectly transmitted throughthe grating. Both amplitude and phase of theFig. 1. Time-resolved transmission of few-cycle THzpulses through a grating compared with the incidentpulse. The incident pulse was recorded by transmissionof the THz pulse through an uncovered reference area onthe same wafer. (a) Grating oriented perpendicular to theelectric field of the incident THz pulse. (b) Orientationparallel to the electric f ield. The transmission signal ismagnified by a factor of 10.0146-9592/01/242008-03$15.00/0 © 2001 Optical Society of AmericaDecember 15, 2001 / Vol. 26, No. 24 / OPTICS LETTERS 2009incident light are preserved. In the remainder ofthis Letter we discuss exclusively gratings that haveparallel orientation with respect to the electric f ieldof the incident light. Figure 1(b) shows that in thiscase the transmitted field amplitude is significantlyreduced. Moreover, it appears that the transmit-ted pulse arrives earlier than a pulse transmittedthrough a reference area on the same wafer that isnot covered by metallic structures. A similar nega-tive phase shift was reported for THz pulses propa-gating through metallic patterns or close to a metalwire.6In this research the negative phase shift wasinterpreted in terms of superluminal propagation,although it was clearly pointed out that the groupvelocity is not superluminal. We performed a detailedanalysis of both the leading edge and the centroid ofthe transmitted pulse, which showed that no superlu-minal propagation occurs. Neither the center of theTHz pulse nor the leading edge arrived earlier whenthe reduced intensity was taken into account. A closerlook at Fig. 1(b) suggests an alternative interpreta-tion: The extrema of the transmission signal corre-spond to the inf lection points of the incident pulse,indicating that the transmission is the first timederivative of the incident pulse. These findings wereconfirmed by calculation of the derivative of theincident pulse.9Figure 2 shows a comparison of thecalculation and the measured data. The amplitudespectra of the transmission exhibit good agreementover a broad frequency range extending from 0.75 to3.5 THz.To achieve a quantitative insight into the observedphenomena, we calculated the transmission in the timedomain. In general, the transmission properties arewell understood for two extreme relations of perioddof the grating and wavelength l of the incident light:(i) If d兾l ! 0, the grating becomes a homogeneousmetallic sheet. As a result, the transmission of thegrating is zero and the wave is completely ref lected, ifperfect