Optical rectification of short laser pulses in electro-optic crystals is a widely used method for the generation of pulsed radiation in the terahertz (THz) frequency range. To achieve high conversion efficiencies, the group velocity of the driving laser pulse and the phase velocity of the generated THz pulse have to be matched. Depending on the method used for achieving phase matching, different emission geometries can be distinguished, such as surface emission, emission in a Čerenkov-like cone, tilted-pulse front generation [1], or collinear phase matching. The latter is certainly the most simple method to implement experimentally; however, it poses tight constraints on the type of crystal and the pump wavelength used.

In conjunction with Ti:sapphire laser systems, ZnTe is the most often used crystal material as it allows phase-matched THz generation around 2 THz [2]. However, the bandwidth is restricted to 3 THz due to the low phonon frequency and, in addition, two-photon absorption limits the usable pump fluences. A possible alternative to ZnTe is GaP, as it features a larger bandwidth due to the higher phonon frequency of 11 THz [3] and a significantly reduced probability for two-photon absorption [4]. However, with Ti:sapphire lasers, no phase-matched THz generation is possible in the collinear geometry due to the high refractive index mismatch between THz and optical frequencies. This mismatch leads to a limited coherence length, which manifests itself in interference minima in the emitted THz spectrum. A common way to increase the coherence length is quasi-phase-matching using periodically poled crystals. However, this technique is limited to a narrow spectral region and the fabrication of the emitter structures is getting more involved.

In this Letter, we report an alternative way to overcome the phase-matching limitation that is especially suited for high-power systems. The method is based on a transient modification of the THz refractive index that has been observed for high-energy pump pulses. We attribute this dynamic phase matching to the generation of hot phonons and the associated reduction of the polariton lifetime. Similar dynamic phase matching by nonlinear effects has already been demonstrated for higher-harmonic generation [5].

The generation of THz radiation by optical rectification in large area nonlinear crystals is excellently described by the model of Faure et al. [6]. The three main parameters entering the calculations are the refractive indices of GaP in the optical and THz frequency ranges, $n_{\mathrm{NIR}}$ and $n(\mathrm{\Omega })$, respectively, and the crystal thickness $L$. The optical refractive index is well described by the Sellmeier equation [7]:

At the carrier frequency $\lambda =0.78\mathrm{\mu m}$, the group refractive index for the pump pulse is $n_g=n_{\mathrm{NIR}}-\lambda (\partial n_{\mathrm{NIR}}/\partial \lambda )=3.66$. This value is higher than the value of the THz refractive index in the relevant frequency range of our experiment. This discrepancy is the cause of a short coherence length and the occurrence of sharp minima in the generated THz spectra. In the following, we suggest including the effect of coherent generation of optical phonons by the pump pulse on the polariton linewidth $\mathrm{\Gamma }$. The change in linewidth leads to an increase of the THz refractive index for frequencies below the phonon resonance. Thus, we expect an increase of the coherence length and an improved phase matching to the optical pump pulse.

Optical phonons are very efficiently generated in transparent crystals by impulsive stimulated Raman scattering of intense femtosecond laser pulses [8]. The resulting phonon distribution is far from equilibrium and will quickly thermalize. The main decay channel for both TO and LO phonons in GaP is the decay into two acoustic phonons at the zone boundary [9,10]. This downconversion process is mediated via the third-order anharmonicity of the crystal potential. This leads to an increased decay rate and thus an increased linewidth $\mathrm{\Gamma }$ of the polariton resonance. Neglecting the wave vector dependence of the anharmonic coefficients, the polariton linewidth takes the form [9]

To validate the model, we first have determined the correct parameters for our GaP crystal. For the experiments, we have used a regenerative Ti:sapphire amplifier (Spectra Physics) operating at a center wavelength of 780 nm with a repetition rate of 1 kHz. At the GaP crystal, the pulses have a duration of 130 fs with a maximum pulse energy of up to 3 mJ. Before the collimated pump beam is normally incident on the GaP crystal, it is expanded to a $1/e^2$ beam diameter of 35 mm. This corresponds to an averaged pump fluence up to $300\mathrm{\mu J}/{\mathrm{cm}}^2$. The crystal is a $(110)$-cut, double-side-polished GaP crystal (MolTech, Berlin) with a clear aperture of $30\mathrm{mm}\times 30\mathrm{mm}$ and a nominal thickness of $L=400\mathrm{\mu m}$. For this large illumination area, saturation of THz generation due to competing effects is greatly reduced [11]. To increase the amount of pump light inside the crystal and to reduce the residual pump light in the THz setup, the front and back facets of the crystal have been coated with an antireflection and a high-reflection coating, respectively. The coating of the crystal had no detrimental effect on the emitted THz light, as has been confirmed by additional THz transmission measurements. The emitted THz pulses have been detected using electro-optic sampling based on a 300 µm thick $(110)$-cut GaAs crystal. Electric peak field strengths over $40\mathrm{kV}/\mathrm{cm}$ have been achieved with a pump fluence of $300\mathrm{\mu J}/{\mathrm{cm}}^2$.

Figure 1 shows a typical THz spectrum obtained for a low pump fluence of $20\mathrm{\mu J}/{\mathrm{cm}}^2$ (dots). The overall bandwidth extends up to 5 THz and is mainly limited by the response function of the detection crystal [12]. The prominent dip around 3.3 THz (indicated by the vertical arrow) is a consequence of the aforementioned mismatch between the group velocity of the pump pulse and the phase velocity of the copropagating THz pulse.

 

Fig. 1. Comparison of experimental (dots) and calculated (solid curve) THz spectra for low pump fluence. The minimum indicated by the vertical arrow is attributed to the velocity mismatch between pump and THz pulses; the minima at higher frequencies arise from the response function of the detector crystal (asterisks). The time domain data is shown in the inset.

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In general, the refractive index data obtained in the visible spectral range is more reliable compared to the THz region. This becomes obvious when one compares the various values for $\epsilon (\omega )$ given in literature (see, for example, [7,12–14]). The main reason for these variations can be attributed to variations in crystal quality and background impurities. Thus, we have performed additional THz transmission measurements to extract the refractive index of the GaP crystal. The real part of the refractive index is shown in Fig. 2 (dots) together with a fit of the Lorentz model (solid curve). As starting values, we have used the data given in [12], ${\epsilon }_{\infty }=9.075$, ${\omega }_{\mathrm{TO}}=2\pi \times 11.011\mathrm{THz}$, ${\omega }_{\mathrm{LO}}=2\pi \times 12.082\mathrm{THz}$, and $\mathrm{\Gamma }=2\pi \times 0.129\mathrm{THz}$, which already gave excellent agreement in terms of the frequency dependence. To obtain a best fit, we had to vertically offset the model curve by adjusting both the crystal thickness and the high-frequency dielectric constant. To get a unique solution, we have used the THz emission spectrum shown in Fig. 1 as additional input. The best fit for both datasets has been obtained for ${\epsilon }_{\infty }=9.65$ and $L=412.5\mathrm{\mu m}$. The model spectrum is shown as a solid curve in Fig. 1 and is in excellent agreement with the experimental data. In the model calculations, we have also taken into account the response function of the detector crystal. The deviation of the model from the experimental data at low frequencies is attributed to diffraction effects in the optical setup, which have been neglected in the calculations.

 

Fig. 2. Real part of the THz refractive index. The solid curve is a fit of Eq. (2) to the experimental data (dots).

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The situation changes when higher pump fluences are used. Figure 3(a) shows normalized THz spectra obtained for various pump fluences ranging from 8 to $324\mathrm{\mu J}/{\mathrm{cm}}^2$. The most prominent effect of the increased pump power is the disappearance of the spectral dip at 3.3 THz. Thereby, the position of this feature remains unchanged. Its disappearance is a clear indication for an improved phase matching between pump and THz pulses and, consequently, an increased coherence length. While the model with constant damping rate is in excellent agreement with the THz emission under low pump fluence, it fails to explain the disappearance of the spectral dip at higher pump fluences.

 

Fig. 3. (a) Measured THz spectra for different pump fluences. (b) Calculated spectra for different effective phonon temperatures.

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Figure 3(b) shows THz spectra calculated using the power dependent damping rate [Eq. (3)] for different values of the effective phonon temperature $T$ ranging from 300 to ${10}^4\mathrm{K}$. As the THz pulse copropagates with the pump pulse, this seemingly high value of the effective temperature has only to be achieved in close temporal vicinity to the driving pulse. For the calculations, we have used the acoustic phonon frequencies ${\omega }_{LA(X)}=2\pi \times 7.46\mathrm{THz}$ and ${\omega }_{TA(X)}=2\pi \times 3.21\mathrm{THz}$ [15], and the value ${\mathrm{\Gamma }}_0=2\pi \times 0.044\mathrm{THz}$ for the scaling constant, which has been calculated using the room temperature value of the decay rate $\mathrm{\Gamma }$ in Eq. (3). There is excellent agreement between the model calculations and the experimental spectra shown in Fig. 3(a), which proves that this simple model reflects the dominant physics underlying the observed dynamical phase matching.

In summary, we have presented generation of spectrally broad THz pulses by optical rectification of intense femtosecond pulses in large area GaP crystals despite the huge index mismatch between THz and optical frequencies. This improved phase matching has been shown to depend on the power of the optical pump pulse and could be explained by the generation of coherent optical phonons and the associated increase of the polariton linewidth. This dynamic phase-matching mechanism is thereby not limited to GaP but applies to a broad range of nonlinear crystals used for generation of THz radiation.