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Physical Review Letters

Linear Pulse Propagation in an Absorbing Medium

Steven Chu, S. Wong

Published Mar 15, 1982• 651 citations
Publisher

Abstract

The pulse velocity in the linear regime in samples of GaP: N with a laser tuned to the bound $A$-exciton line is measured with use of a picosecond time-of-flight technique. The pulse is seen to propagate through the material with little pulse-shape distortion, and with an envelope velocity given by the group velocity even when the group velocity exceeds 3\ifmmode\times\else\texttimes\fi{}${10}^{10}$ cm/sec, $\mathrm{equals}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$, or becomes negative. The results verify the predictions of Garrett and McCumber.

DOI
10.1103/physrevlett.48.738

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The pulse velocity in the linear regime in samples of GaP: N with a laser tuned to the bound AAA-exciton line is measured with use of a picosecond time-of-flight technique. The pulse is seen to propagate through the material with little pulse-shape distortion, and with an envelope velocity given by the group velocity even when the group velocity exceeds 3\ifmmode\times\else\texttimes\fi{}1010{10}^{10}1010 cm/sec, equals\ifmmode±\else\textpm\fi\ensuremath∞\mathrm{equals}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}equals\ifmmode±\else\textpm\fi\ensuremath∞, or becomes negative. The results verify the predictions of Garrett and McCumber.
We compute the peak velocity of a broad Gaussian pulse propagating in an absorbing thin film modeled with a one-oscillator dielectric function. The exact numerical calculation is done for a GaP:N film in the frequency region of the bound A exciton. For absorption lengths greater than the film thickness, the pulse velocity essentially coincides with the group velocity, and with the experimental results of S. Chu and S. Wong [Phys. Rev. Lett. 48, 738 (1982)]. However, outside the region of strong absorption the pulse velocity depends sensitively upon the thickness of the slab. Our results give a better account of the Chu and Wong experiment than the calculation based on the group velocity. For film thickness with smaller than one absorption length the pulse velocity disagrees with the group velocity even in the strong absorption region, as a consequence of Fabry-P\'erot reflections of the pulse inside the film.
Propagation of light pulses in wide-gap semiconductors, GaN and ZnO, has been studied by time-of-flight spectroscopy. It has been demonstrated that significant light delay exists near an exciton resonance and that this delay is accompanied by pulse distortion and attenuation. Simulation of the pulse shape in the time-energy plane has permitted us to refine excitonic parameters inherent for bulk materials. The time-of-flight studies done over a wide temperature range demonstrate the strong influence of resonant scattering by phonons on the light delay in polar semiconductors. Optical dispersion at exciton resonances in semiconductors results in a delay of light transfer, which is induced by the variation of the group velocity — a phenomenon studied long ago(1-3). Thisdelayhasbeenobservedbytime-of-flightspec- troscopy in different semiconductors (4-6). The current inter- estinsuchretardationisenhancedbecauseitisoneoftheways toobtaintheso-calledslowlight,whichisneededforoptical communication and data processing. Wide-gap semiconduc- tors, like ZnO and GaN, possess strong excitonic resonances thatmakesthemsuitableforsuchapplication. Lightvelocities as low as 6000 km/s in ZnO (4) and 2100 km/s in GaN (5) have been recently measured. In view of these findings, the distortion of a pulse shape has attracted special attention (7). It is worth mentioning that early experiments showed no sig- nificant changes in the shape of a light pulse at the excitonic resonances in GaAs (8) and GaP (9), while the optical dis- persion near the resonances implies a different group velocity and attenuation for the higher- and lower-energy edges of the pulses (10), that inevitably means their distortion in the time- energy plane. The optical pulse modification is also important for a set of optoelectronic devices — high-frequency ones or those, where the light should propagate a long distance. The
It is shown that the prediction of Garrett and McCumber that a light pulse's velocity may exceed the speed of light in a resonantly absorbing medium is due to asymmetric absorption of energy from the light pulse. More energy is absorbed from the trailing half of the pulse than from the front half, causing the center of gravity of the pulse to move at a velocity greater than the phase velocity of light. It is also shown that in certain cases a pulse's maximum will propagate at the group velocity even when the pulse as a whole is distorted by dispersion.
We have examined the propagation of femtosecond laser pulses in an absorbing dye solution through a short to a long range of propagation distance. The transmitted pulses show strong spectral shift and a superluminal to subluminal transition in the propagation velocity keeping its initial shape almost intact. It is verified that the peak velocity is well described by a modified group velocity v(S) defined within the framework of the saddle-point method as well as by a recent prediction of the net group delay of surviving frequency.