Relativistic Doppler reflection of terahertz light from a moving plasma front in an optically pumped Si wafer

The relativistic Doppler reflection of terahertz (THz) light from a photoinduced moving plasma front in a silicon wafer irradiated by an optical pump light was investigated. The intensity and the phase shift spectra of the reflected THz light were investigated by THz time-domain spectroscopy with broadband electro-optic sampling in a thin GaP crystal. The frequency up-shift factor, which is a ratio of averaged frequencies from after to before Doppler reflection, was measured to be $\mathrm{\Gamma }=2.02$ with a pump energy density of $3.3\mu \mathrm{J}/\mathrm{m}{\mathrm{m}}^2$ at a wavelength of 800 nm. The frequency up-shift increased as optical pump energy density increased. The frequency up-shift was reproduced by evaluating the moving plasma front whose spatial carrier density distribution depended on the pump energy density.

Figure 1

(a) Schematic diagram of the relativistic Doppler reflection. The time sequence of the THz probe and optical pump pulses and their propagation in the Si wafer are shown from (I) to (IV). (b) Schematic diagram of the experimental apparatus. λ/2 and λ/4 are the half and quarter wave plates, respectively, LN is the $\mathrm{LiNb}{\mathrm{O}}_3$ crystal, ITO is the indium-tin-oxide coated glass plate, WP is the Wollaston prism, BPD is the balanced photodiode, BS1 is the wedge plate, BS2 is the 1:1 beam splitter, L1 and L2 are BK-7 lenses with focal lengths of 250 and 150 mm, respectively. PL1, PL2, and PL3 are plastic lenses for the THz light (PAX, Tsurupica) with focal lengths 50, 100, and 100 mm, respectively, OPM is the off-axis parabolic mirror with a focal length of 50 mm, ND is the neutral density filter, and EO X-tal is the 0.4-mm-thick GaP EO sampling crystal. (c) THz waveform transmitted through the Si wafer without a pump irradiation.

Figure 2

(a) Temporal waveforms and (b) Fourier transformed spectra of the reflected THz light as a function of the pump-probe delay time $\mathrm{\Delta }t$. The thin black and thick red lines show the results measured by the 1-mm-thick ZnTe and 0.4-mm-thick GaP EO crystals, respectively. The red dashed line in (b) for $\mathrm{\Delta }t=0\mathrm{ps}$ shows the spectra measured at $\mathrm{\Delta }t=-2\mathrm{ps}$ with the GaP crystal as a reference. The pump energy density of the optical pump light was 2.7 and $3.3\mu \mathrm{J}/\mathrm{m}{\mathrm{m}}^2$ for the ZnTe and GaP crystals, respectively.

Figure 3

Time-dependent power spectra $I\left({\nu }_{\mathrm{THz}},\mathrm{\Delta }t\right)$ of the reflected THz light measured with a fine scan step of 67 fs by (a) 1-mm-thick ZnTe and (b) 0.4-mm-thick GaP EO sampling crystals, respectively. The pump energy density of the optical pump light was $2.7\mu \mathrm{J}/\mathrm{m}{\mathrm{m}}^{-2}$ for the ZnTe crystal and $3.3\mu \mathrm{J}/\mathrm{m}{\mathrm{m}}^2$ for the GaP crystal. The intensity color map is displayed with a logarithmic scale. (c) Time-dependent average frequency of spectra $\langle {\nu }_{\mathrm{THz}}\left(\mathrm{\Delta }t\right)\rangle$ and frequency up-shift factor $\mathrm{\Gamma }\left(\mathrm{\Delta }t\right)$. The thin black and thick red lines show the results measured by the ZnTe and GaP crystals, respectively.

Figure 4

Phase shift spectra $\mathrm{\Delta }\varphi \left({\nu }_{\mathrm{THz}},\mathrm{\Delta }t\right)$ of the reflected THz light as a function of the pump-probe delay time $\mathrm{\Delta }t$. The red lines show the observed spectra. The blue dashed lines show the spectra as calculated by the 1D FDTD. The black dot-dashed line in the graph for $\mathrm{\Delta }t=2\mathrm{ps}$ shows the analytical solution of the Maxwell equations assuming an exponential decrease of carrier density in the depth direction in the Si wafer. The pump energy density of the optical pump light was $3.3\mu \mathrm{J}/\mathrm{m}{\mathrm{m}}^2$.

Figure 5

Time dependent power spectra $I\left({\nu }_{\mathrm{THz}},\mathrm{\Delta }t\right)$ measured by the 0.4-mm-thick GaP EO sampling crystal at pump energy densities of (a) 0.38, (b) 1.1, and (c) $3.3\mu \mathrm{J}/\mathrm{m}{\mathrm{m}}^2$.

Figure 6

Pump energy density dependence of the maximum up-shift factor. Black filled squares and red filled circular dots are the maximum up-shift factor ${\mathrm{\Gamma }}_{\max }$ obtained by frequency averaging and phase shift analysis, respectively. Blue open triangular dots are the maximum up-shift factor ${\tilde{\mathrm{\Gamma }}}_{\max }$ modified by the effect of the interaction time between the moving plasma front and the THz light.

Figure 7

THz frequency dependencies of the (a) phase shift for the reflected THz light $\mathrm{\Delta }\varphi$, and (b) velocity of the plasma front $v_{\mathrm{P}}$. The pump energy density of the optical pump light is $3.3\mu \mathrm{J}/\mathrm{m}{\mathrm{m}}^2$. (c) Optical pump energy density dependence of the maximum velocity of the plasma front $v_{\mathrm{P}}$.

Figure 8

Waveforms of the reflected THz light before ($\mathrm{\Delta }t=-2\mathrm{ps}$, black line) and after ($\mathrm{\Delta }t=3\mathrm{ps}$, red lines) pump light irradiation with energy densities of 0.38, 0.96, 1.6, and $3.3\mu \mathrm{J}/\mathrm{m}{\mathrm{m}}^2$. The blue dashed line shows the round-trip time of the THz light in the Si without optical pump light irradiation. The blue thick arrows indicate the temporal advance induced by the static plasma with a finite thickness.