1. Introduction

Two methods that have often been used to generate THz radiation with lasers are difference frequency generation (DFG) and optical rectification. DFG uses two relatively long (> 1 ns) pulses separated in frequency by the desired signal frequency. This approach has produced peak powers as high as 200 W (at 1.5 THz) using gallium selenide (GaSe) as the nonlinear medium [1]. Optical rectification uses a relatively broadband laser pulse whose pulse length is approximately half the inverse of the desired signal frequency. This technique produces a broadband signal, and has generally been carried out in the non-phase matched Cherenkov regime [2, 3]. More recently, phase matched optical rectification has been demonstrated experimentally [4, 5] and analyzed theoretically and numerically [6, 7]. The synthesis of more general THz waveforms via optical rectification has also been demonstrated [8], although high THz powers were not reported.

 

Fig. 1. Schematic of optical rectification of a short modulated laser pulse.

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We report here on a hybrid technique where a titanium:sapphire laser system is used to generate a pulse which is modulated at the signal frequency as in DFG, but which is only a few modulation periods in length. The concept is illustrated schematically in Fig. 1. Because the laser pulse is only a few picoseconds long, intensities of 1 GW/cm² can be incident on a GaSe crystal without damaging it, and therefore higher conversion efficiencies per square length can be achieved than in the case of DFG. However, the pulse is still long enough so that phase matching can be sustained over long distances (> 1 cm) without being spoiled by group velocity slippage between the ordinary and extraordinary waves. Furthermore, both the signal frequency and bandwidth can be easily adjusted.

2. Experimental setup

The modulated optical pulses used in our experiments are generated using the TFL laser at the Naval Research Laboratory. The modulation is produced using a spatial filtering technique [9]. A schematic of the setup is shown in Fig. 2. The TFL laser is a titanium:sapphire chirped pulse amplification (CPA) system capable of producing up to 500 mJ, 50 fs pulses with a 10 Hz repetition rate. In its normal configuration, the system consists of an oscillator, stretcher, regenerative amplifier, 5-pass amplifier, and compressor. In these experiments a pulse shaper is interposed between the regenerative amplifier and the 5-pass amplifier (this configuration was chosen to protect the regenerative amplifier). The pulse shaper can be considered a standard CPA stretcher with effective length zero, and with a pair of slits in the focal plane of the lens. Because the focal plane of the lens is an image of the frequency content of the laser pulse, the pair of slits selects two frequency bands for transmission through the rest of the system. After amplification and compression, these frequencies are combined in time to form a modulated pulse. The length of the pulse is determined by the width of the two slits, and the modulation frequency is determined by the separation between the two slits. The system bandwidth supports modulation frequencies up to 5 THz, and pulse lengths as short as 100 fs. The upper bound on the pulse length depends on how narrow the slits are, and therefore on how much energy can be sacrificed in the pulse shaper. Note that a dedicated system could be designed to run more efficiently by incorporating the frequency selection into the usual stretcher and providing a safe level of pumping to the regenerative amplifier.

 

Fig. 2. Schematic of experimental setup. G1 is an 1800 g/mm grating. L1 is an f/10 lens placed 1 focal length (f = 50 cm) from G1. M2 is f away from L1. L1 is displaced vertically so that the return beam passes over M1. G2 is a 1.4 g/mm grating used to disperse THz pulses. F1 is a 3 mm thick black polyethylene filter used to extinguish the laser radiation while transmitting the THz radiation.

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The modulated pulse is used to generate THz radiation by means of optical rectification in a 5 mm thick, 10 mm diameter GaSe crystal (Eksma). The laser spot size is matched to the crystal diameter to approximate a plane wave interaction as nearly as possible. The GaSe crystal is favorable for THz generation because of its large second order susceptibility, and its phase matching characteristics. Phase matching is achieved using a collinear type II configuration in which the higher frequency band of the laser (the pump) is an ordinary wave and the lower frequency band (the idler) is an extraordinary wave. This geometry is realized by rotating the laser polarization by 45 degrees with respect to the axis of the rotation stage on which the crystal is mounted. A side effect of this approach is that only half the laser energy participates in the phase matched interaction.

The diagnostics used to analyze the THz radiation include a liquid helium cooled silicon bolometer (IRLabs) and a 50×25 mm² aluminum grating with triangular grooves separated by 700 microns. The bolometer entrance is covered by 3 mm thick black polyethylene (PE) in order to extinguish the laser but transmit a detectable portion of the THz radiation. The collection optic inside the bolometer is an f/3.8 Winston cone with a 12.7 mm entrance diameter. The extinction of the laser by the black PE is sufficient so that even with the laser aligned directly into the bolometer, no signal is detected. In some experiments, the bolometer input is placed within a few cm of the GaSe. In others, the THz radiation is first reflected off the grating so that the various diffracted orders can be analyzed. In these cases, the grating is placed 15 cm from the GaSe crystal, and the bolometer is placed 15 cm from the grating.

3. Simulation model

Because of the substantial bandwidth (≈10%) of the THz signals considered in this work, the usual analysis of difference frequency generation [10] cannot be applied. In particular, each Fourier component of the THz signal will suffer a different phase mismatch, as illustrated in Fig. 3. The situation is complicated further by group velocity dispersion. To overcome these difficulties we have developed a simulation model which calculates the THz waveform produced by an arbitrary laser pulse in an arbitrary birefringent crystal. A detailed description of the model can be found in Ref. [7]. A brief summary follows.

 

Fig. 3. Theoretical phase mismatch developed in a 5 mm thick GaSe crystal vs. signal wavelength for a 0.8 micron wavelength pump, based on the dispersion relation of Ref. [11]. The crystal axis makes an angle of 1.67 degrees with respect to the wavevector. The pump is an ordinary wave and the idler is an extraordinary wave. The curve is insensitive to the polarization of the signal.

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The laser fields are described by a complex vector envelope (ℰ_x ,ℰ_y ,ℰ_z ) defined by

where $\widetilde{ℰ}$_i is the real valued i-component of the electric field and i ∈ (x,y,z). The coordinate system is defined so that waves polarized in the y-direction are ordinary waves. In normalized units (see appendix), the propagation equations for the laser envelope are

where

and

Here, χ_ij (ω) is the susceptibility coupling the j-component of the field to the i-component of the polarization at the frequency ω. The terms I_i are the nonlinear contributions to the polarization which are computed using the full second order susceptibility tensor [7]. To put the propagation equations into a form suitable for numerical evaluation, the following procedure is used. First, the equations are entered into a symbolic math program. They are then rewritten in terms of the new coordinates τ = t - n_gz and η = z using the substitutions ∂_z = ∂_η - n_g ∂_τ and ∂_t = ∂_τ . Generally, n_g is chosen to be the group index associated with the ordinary wave so that the laser pulse is approximately stationary in the simulation box. The choice k ₀ = ω ₀ n ₂₂ is also generally made. Next, the equations are expanded and all derivatives higher than the first order in η and the second order in τ are dropped. Finally, the equations are expressed in finite difference form which yields implicit equations for the fields at the new time which can be solved via the Crank-Nicholson method.

The THz field (E_x ,E_y ,E_z ) is treated as real valued since an envelope approximation is not generally appropriate. Dispersion is introduced by coupling the field to a population of harmonic oscillators representing the lattice vibration. The propagation equations are

where H_i represents the lattice vibration, S_i represents the nonlinear polarization, and

Here ψ_ij , n_θ , and n_o can be regarded as constants chosen to yield the correct dispersion characteristics when combined with the lattice vibration [7]. The lattice vibration is most conveniently computed in the coordinate system where the susceptibility tensor is diagonal so that it satisfies three independent equations of the form

where i varies over the new spatial coordinates, v_i is used to introduce damping, Ω_i is the resonant frequency, and ρ_i is a coupling constant. The nonlinear polarization S_i is computed from the laser field using the full second order susceptibility tensor. The propagation equations for the THz pulse can be put in the form of the flux conservative initial value problem which we solve using the Lax-Wendroff method. The harmonic oscillator equation representing the lattice vibration is written as two coupled first order equations which are solved implicitly to ensure numerical stability.

4. Results

To produce a 1.0 THz signal a mask was prepared with slit separation 2.9 mm and slit width 0.7 mm. The modulated pulse was amplified to a compressed energy of 2.5 mJ. Based on the Fresnel reflection coefficients, 1.9 mJ of laser energy was transmitted into the GaSe crystal. The bolometer was placed a few cm from the GaSe. The signal produced on the bolometer as a function of the external phase matching angle is shown in Fig. 4, along with a simulated signal for comparison. The dispersion parameters for the simulation were derived from the GaSe dispersion relation of Ref. [11], which is valid in both the optical and THz regimes. The strongly peaked signals at 5 and 6 degrees correspond to the expected phase matched interaction. The difference between the experimental and simulated peaks could be due to the experimental error in defining normal incidence, the experimental error in setting the modulation frequency, or differences between the actual and assumed dispersion relations.

Both the experimental and simulated curves reveal a secondary peak near 0 degrees. This is related to the generation of signal frequencies near the inverse of the total laser pulse length. In other words, for small angles, optical rectification of the beat envelope becomes approximately phase matched. This is illustrated in Fig. 5 which shows the simulated THz waveforms for external phase matching angles θ _ext = 0° and θ _ext = 4.8° In the former case, the 1.0 THz waveform is superposed with a single cycle ≈ 0.2 THz waveform. In the latter case the 1.0 THz signal is dominant.

In all cases, the phase matched THz signal varied as sin²(4θ_w ), where θ_w is the rotation angle of the waveplate used to vary the laser polarization. This is as expected for a type II interaction. The THz signal was independent of the azimuthal rotation ϕ of the crystal about its linear symmetry axis. Although ϕ is expected to determine which polarization components of the THz field are excited, it turns out that the phase matching angle for producing THz radiation with either polarization is approximately the same. Thus, the signal is not strongly dependent on ϕ.

The energy in the THz pulse is expected to scale as the square of the pump intensity. By interposing various combinations of NG glasses between the laser and the GaSe crystal the experimental intensity scaling shown in Fig. 6 was produced. For comparison, a simulated intensity scaling is shown on a different scale. For intensities between 10⁸ W/cm² and 10⁹ W/cm² the expected quadratic scaling is observed. For intensities higher than 10⁹ W/cm², however, the experimental THz energy does not increase appreciably with intensity. This appears to be due to the effects of two-photon absorption. In particular, taking 0.558 cm/GW as the two photon absorption coefficient [12], and using 1 GW/cm² as the pump intensity, one finds that 25% of the pump is absorbed in a 5 mm long crystal. Furthermore, for several picosecond long pulses, we estimate that the electron-hole plasma becomes dense enough to be opaque at THz frequencies.

 

Fig. 4. Comparison of experimental and numerical signals as a function of phase matching angle for a 1.0 THz modulation. The simulated signal is corrected for the 28% internal reflection expected to occur at the output of the crystal, and assumes uniform fluence throughout the 1 cm diameter crystal.

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To demonstrate tunability, the grating was interposed between the GaSe and the bolometer. The bolometer was positioned at a 90° angle with respect to the rays incident on the grating. The angle of incidence was varied by rotating the grating, which caused different diffracted orders to be intercepted by the bolometer. Fig 7 shows the results of scanning the grating angle for three different laser modulation frequencies. In each case, the peaks appear at the expected locations, which are denoted by the dashed vertical lines.

The energy in the THz pulse can be estimated using the responsivity of the bolometer which is quoted by the manufacturer as 2.39 × 10⁵ V/W. For incident radiation pulses much shorter than the bolometer response time, the bolometer signal is proportional to the pulse energy. The proportionality constant is the responsivity divided by the response time. Taking into account the preamplifier gain of 200 ×, the expected bolometer signal is 1.8 × 10¹⁰ V/J. This calibration factor gives a lower bound on the THz energy since the responsivity is expected to be less than the manufacturer’s value at THz wavelengths. The largest signal observed at 1.0 THz was 3.0 V, corresponding to an energy of > 0.17 nJ. However, the transmission through 3 mm of black PE was measured to be about 2.7%. The THz energy before transmission was therefore > 6 nJ, which corresponds to a peak power of > 1.5 kW. This is far less than the THz power produced in the simulation, which was « 150 kW. The discrepancy may be partially due to the overestimate of the responsivity of the bolometer at long wavelengths. The discrepancy may also be due to imperfections in the crystal or laser pulse.

 

Fig. 5. Simulated THz Waveforms at θ _ext = 0° and 4.8°. The electric field is evaluated inside the crystal. The electric field is larger outside the crystal because of the fact that in air a smaller fraction of the wave energy is carried by the magnetic field and dielectric polarization. Taking into account the 28% internal reflection at the crystal output, the transmitted intensity would be 200 kW/cm² and the peak electric field would be 12 kV/cm.

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Fig. 6. Experimental and simulated THz signal as a function of pump intensity in the crystal for 1.0 THz modulation. It must be emphasized that only the scaling can be compared. That is, the two vertical axes can be shifted with respect to one another until a calibration factor is specified. Note also that the pump intensity is averaged over the optical frequency, but not the modulation frequency.

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Fig. 7. Bolometer signal vs. grating angle for (a) 0.7 THz modulation (b) 1.0 THz modulation (c) 2.0 THz modulation. The dashed lines indicate the expected location of the three lowest diffracted orders assuming the signal frequency and the modulation frequency are the same.

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5. Conclusion

A femtosecond laser system can be modified to produce a short modulated pulse which can then be used in a GaSe crystal to produce high peak power terahertz radiation. Experimentally, a lower bound of 1.5 kW was established for the peak power produced at 1.0 THz in a 5 mm thick crystal. Simulations suggest that peak powers as high as 150 kW can be obtained. The terahertz radiation can be tuned by changing the physical separation between two slits. It should also be possible to tune the bandwidth of the terahertz radiation by changing the slit width. Rapid tuning can be accomplished by replacing the slits with a spatial light modulator.